MATH · 2015. 9. 17. · The distance across a circle through the center. height. The perpendicular width of a plane figure. perimeter. The distance around the outside of a plane

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MATHSTUDENT BOOK

6th Grade | Unit 8

MATH 608Geometry and Measurement

INTRODUCTION |3

1. PLANE FIGURES 5PERIMETER |5AREA OF PARALLELOGRAMS |11AREA OF TRIANGLES |17AREA OF COMPOSITE FIGURES |21AREA OF CIRCLES |27PROJECT: ESTIMATING AREA |32SELF TEST 1: PLANE FIGURES |36

2. SOLID FIGURES 39SOLID FIGURES |39SURFACE AREA OF RECTANGULAR PRISMS |47VOLUME OF RECTANGULAR PRISMS |51FINDING MISSING DIMENSIONS |56PROJECT: TRIANGULAR PRISMS |60SELF TEST 2: SOLID FIGURES |64

3. REVIEW 66GEOMETRY AND MEASUREMENT |66PLANE FIGURES |67GLOSSARY |73

LIFEPAC Test is located in the center of the booklet. Please remove before starting the unit.

Unit 8 | Geometry and Measurement

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Geometry and Measurement | Unit 8

2|

Geometry and Measurement

IntroductionIn this unit, you will be introduced to the topics of geometry and measurement. You will learn about plane figures and solid figures and how they are measured. You will find that length is measured in various units, while area is measured in square units and volume is measured in cubic units. For plane figures, you will measure the perimeter and area of rectangles, parallelograms, triangles, circles, and composite figures. For solid figures, you will measure surface area and volume. You will use nets, two-dimensional views, and three-dimensional views to help visualize the figures. For each of these measurements, you will arrive at a general formula that will work for any rectangular prism. These basic tools will be useful in your future explorations in geometry.

ObjectivesRead these objectives. The objectives tell you what you will be able to do when you have successfully completed this LIFEPAC. When you have finished this LIFEPAC, you should be able to:

z Find the perimeter of a polygon.

z Review finding the circumference of a circle.

z Find the area of a parallelogram, a triangle, a circle, and simple composite figures.

z Classify solid figures.

z Find the surface area and volume of a rectangular prism.

z Find a missing dimension of a rectangular prism, given the surface area or volume.

z Use correct units for measurement.


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Survey the LIFEPAC. Ask yourself some questions about this study and write your questions here.


4|

1. PLANE FIGURES

PERIMETERBob bought a house on a large lot. He’s decided to install a fence around the property. How many feet of fence does he need? What infor-mation does Bob need to decide how much fence he needs?

Bob needs to find the perimeter of his prop-erty to know how many feet of fence to buy. In this lesson, you will learn how to find perimeter for different figures and solve real-life prob-lems such as Bob’s. You will also review finding the circumference of a circle.

ObjectivesReview these objectives. When you have completed this section, you should be able to:

z Find the perimeter of a polygon.

z Review how to find the circumference of a circle.

z Find the area of a parallelogram.

zUnderstand the relationship between the area of parallelograms and triangles.

z Find the area of a triangle.

z Find the area of simple composite figures.

z Find the area of a circle.

z Estimate the area of irregular figures.

Vocabularyarea. The measurement of the space inside a plane figure.

base. The length of a plane figure.

circumference. The distance around the outside of a circle.

composite figure. A geometric figure that is made up of two or more basic shapes.

diameter. The distance across a circle through the center.

height . The perpendicular width of a plane figure.

perimeter. The distance around the outside of a plane figure.

pi. The ratio of the circumference of a circle to its diameter; approximately 3.14.

radius. The distance from the center of a circle to any point on the circle.

semicircle. One half of a circle, divided by the diameter.

square units. The unit of measure for area.

trapezoid. A quadrilateral with one pair of parallel sides.

Note: All vocabulary words in this LIFEPAC appear in boldface print the first time they are used. If you are not sure of the meaning when you are reading, study the definitions given.


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Example:Bob wants to build a fence around his property. How many feet of fencing does he need?

Solution:To find the perimeter, we will add the side lengths.

100 feet + 150 + feet + 105 feet + 60 feet + 110 feet = 525 feet

So, Bob will need to buy 525 feet of fencing.

Key point!

Always label the units for any measurement. Doing this will make it clear what type of measurement it is (length, area, volume), as well as what units were used.

100 feet

105 feet

110 feet150 feet

60 feet

Perimeter is a measurement of length because it’s the distance around a figure. You could think of it as the length you would walk if you could walk around a figure.

To find perimeter, we need to know the length of each side of the figure. Then, we can add the side lengths. In the rectangle above, we can see that the lengths of the sides are 2 feet, 4 feet, 2 feet, and 4 feet. So, the perimeter is 12 feet:

2 feet + 4 feet + 2 feet + 4 feet = 12 feet

To find the perimeter of Bob’s property, we just need to know the lengths of each side of the lot.

4 feet

4 feet

2 feet 2 feet2 feet 2 feet

4 feet

4 feet

Did you know?

Perimeter comes from the Greek perimetros: from peri-, meaning “around,” and metron, meaning “measure.”

2 feet 2 feet

4 feet

4 feet


6| Section 1

Did you notice that the shape of Bob’s property is a pentagon? If it were a regular pentagon, or

any other regular polygon, we could shorten the process of finding the perimeter.

We could add the side lengths to find the perimeter.

60 ft + 60 ft + 60 ft + 60 ft + 60 ft = 300 ft

However, since we know that a regular penta-gon has five congruent sides, we can multiply one side length by five.

5(60 ft) = 300 ft

Since any regular polygon has congruent sides, we can just multiply the number of sides by the side length (s).

60 feet 60 feet

60 feet60 feet

60 feet

Example:Find the perimeter of each regular polygon below.

Solution:To find the perimeter, we will multiply the number of sides by s, the side length.

Regular Octagon

A regular octagon has eight congruent sides, so we will use the formula 8s to find the perimeter.

8s = perimeter

8(2 m) = 16 m

Regular Hexagon

A regular hexagon has six congruent sides, so we will use the formula 6s to find the perimeter.

6s = perimeter

6(4 feet) = 24 feet

Square

A square has four congruent sides, so we will use the formula 4s to find the perimeter.

4s = perimeter

4(3 inches) = 12 inches = 1 foot

2 m

4 ft

3 in


Section 1 |7

A square is a type of rectangle. Earlier, we found the perimeter of a rectangle, but we can shorten that process also.

Remember that the opposite sides of a rect-angle are congruent. So, instead of adding the length and the width two times:

4 feet + 2 feet + 4 feet + 2 feet = 12 feet

We can multiply the length plus the width by two:

(4 feet + 2 feet) + (4 feet + 2 feet) =

2(4 feet + 2 feet) =

2(6 feet) = 12 feet

For any rectangle, if l is the length and w is the width, the perimeter (p) is found using the for-mula p = 2(l + w).

4 feet

4 feet

2 feet 2 feet

Example:Bob plans to add a rectangular swimming pool to his property. It will be 25 feet wide and 50 feet long. He wants to include a border of dark blue tile around the edge of the pool. How many feet of tile will he need?

Solution:We need to find the distance around the pool: the perimeter. Since the pool is rectangular, we will use the formula p = 2(l + w) to find the perimeter.

p = 2(l + w)

p = 2(50 feet + 25 feet) Substitute the length and width.

p = 2(75 feet) Add.

p = 150 feet Multiply.

So, Bob will need 150 feet of tile.

50 feet

25 feet


8| Section 1

CIRCUMFERENCEAside from polygons, we can also find the perimeter of a circle, called the circumference. We know that pi (π) is the ratio of the circumference to the diameter (π c__

d ). So, the circumference is pi times the diameter:

C = πd

Example:Bob has decided to include a circular whirlpool next to the pool, with the same dark blue tile around the edge. If the pool is 6 feet across, how many feet of tile will he need?

Solution:We will use the formula C = πd to find the circumference. We know that the diameter of the pool is 6 feet, and we will use 3.14 to approximate pi.

C = πd

C = 3.14(6 feet)

C = 18.84 feet

So, Bob will need about 19 feet of tile for the whirlpool.

S-T-R-E-T-C-H

Now that Bob knows the length of tile he needs, he can calculate the cost. If each tile was 4 inches long and cost $1.79 each, can you find out how much the tiles for the pool and whirlpool will cost?

6 feet

Let’s Review!Before going on to the practice problems, make sure you understand the main points of this lesson.

9 Perimeter is the distance around a figure. It is found by adding the side lengths.

9 If the figure is a regular polygon, we can multiply the number of sides by the side length.

9 The perimeter of a circle is called the circumference. It is found using the formula C = πd.


Section 1 |9

Match the following items.

1.1 ________ the distance around the outside of a circle

________ the distance around the outside of a plane figure

Circle the letter of each correct answer.

1.2 A square has a side length of 5 inches. What is the perimeter of the square? a. 5 inches b. 10 inches c. 15 inches d. 20 inches

1.3 A rectangle is 6 meters long and 4 meters wide. What is the perimeter of the rectangle? a. 10 meters b. 20 meters c. 10 square meters d. 20 square meters

1.4 What is the perimeter of this figure? 8 cm

8 cm

16 cm

5 cm

5 cm4 cm

a. 16 cm b. 46 cm c. 48 cm d. 110 cm

1.5 What is the perimeter of a regular hexagon if you know that one side is 4 cm long? a. 24 cm b. 20 cm c. 32 cm d. can’t be determined

1.6 A square has a perimeter of 36 inches. How long is each side? a. 4 inches b. 6 inches c. 9 inches d. 12 inches

1.7 The length and width of each rectangle is given. Which rectangle will not have the same perimeter as the others?

a. l = 12, w = 12 b. l = 14, w = 9 c. l = 8, w = 16 d. l = 13, w = 11

1.8 What is the perimeter of this figure? 10 feet10 feet

2 feet 2 feet16 feet

a. 40 feet b. 36 feet c. 32 feet d. 28 feet

1.9 What is the circumference of a circle with a diameter of 100 m? a. 100 m b. 157 m c. 300 m d. 314 m

Answer true or false.

1.10 ______________ The circumference of a circle with diameter of 6 inches will be greater than the perimeter of a square with side length 6 inches.

a. circumference

b. perimeter


10| Section 1

AREA OF PARALLELOGRAMSEach of these figures is a parallelogram. We know that they have several things in common (opposite sides are parallel and congruent). There are also some differences (right angles, number of congruent sides). So, is finding the area for each parallelogram the same, or different?

In this lesson, we will discuss what area is and how it is measured, and we will explore the area of a parallelogram and how to find it.

AREA OF RECTANGLESArea is the amount of space that a plane fig-ure takes up. How do we measure, or count, the amount of space? The number of square spaces or units that are inside the figure is our measure of its area. So, area is measured in square units.

Can you find the area of the rectangle? How many squares are inside?

If we count the squares, we can see that there are 8. So, the rectangle’s area is 8 square units.

We use the same measures for area as we do for length: feet, inches, meters, etc. However, we refer to square feet, square inches, square meters.

If the rectangle’s length and width are mea-sured in inches, then each square inside the rectangle is 1 inch long and 1 inch wide: 1 square inch. The area is 8 square inches.

1 in

1 in

1 in2


Section 1 |11

Can you find the area of this rectangle?

We could count each of the squares inside the rectangle one at a time to find the area, but there is an easier way. Notice that there are 8 square feet in each row of the rectangle because it is 8 feet long. There are 7 rows of 8 squares because the rectangle is 7 feet wide.

We can multiply 8 by 7 to find the number of squares.

8 × 7 = 56

So, the area of the rectangle is 56 square feet.

So, to find the area (A) of any rectangle, we can just multiply the length (l) by the width (w).

A = l × w

It is important to understand area because it comes up in everyday situations.

Key point!

We could also have looked at the rectangle as 8 columns of 7 squares each: 7 × 8 = 56.

Remember, multiplication is commutative; order does not matter:

7 × 8 = 8 × 7

8 feet

7 feet

8 squares in each row

7 rows


12| Section 1

Key point!

Always label the units for any measurement. Doing this will make it clear what type of measure-ment it is (length, area, volume), as well as what units were used.

You can tell right away if the measurement is area if the units are square units.

Example:Bob wants to paint a wall in his kitchen. He needs to know the area of the wall so he can decide how much paint to buy. The wall is 8 feet high and 6.5 feet long. What is the area of the wall?

Solution:To find the area of the wall, we will multiply the length by the width because the wall is a rectangle.

A = lw

The length of the wall is 6.5 feet. Its width is 8 feet.

6.5 feet × 8 feet = 52 ft2

So, Bob will need enough paint to cover 52 ft2.

Did you know?

Units of area are often written as exponents: ft2, m2, cm2...

When we find area we multiply length times width, so just as 4 × 4 = 42, or 4 squared, feet × feet = ft2, or square feet.

6.5 ft.

8 ft.

Even though the wall is not made up entirely of squares, the area will be 52 square feet. This is because the half-squares can be added up to form whole squares. In this case, there are 8 half-squares, which is the equivalent of 4 whole squares.


Section 1 |13

Example:Bob has another wall he’d like to paint, but the label on the can of paint says it can cover only 120 square feet. If the wall is 8 feet high, how long of a wall could he paint?

Solution:Using the formula A = lw, we can solve for the length of the wall.

A = lw

120 ft2 = l(8 ft) Substitute known measures.

We know that 8 feet multiplied by another length of feet is 120 ft2. So, if we divide 120 ft2 by 8 ft, we’ll find the length.

120 ft2 ÷ 8 ft = l

15 ft = l

Bob has enough paint as long as the wall is less than 15 feet long.

AREA OF PARALLELOGRAMSWe know that a rectangle is a parallelogram, so can we find the area of a parallelogram the same way? How many square units do you think there are in this parallelogram?

Not all of the area is whole squares, but we can combine some pieces to make whole squares. There are 6 whole squares, and 4 half squares.

6 + 1__2 + 1__

2 + 1__2 + 1__

2

Group halves together to make 1.

6 + ( 1__2 + 1__

2 ) + ( 1__2 + 1__

2 )

Add.

6 + 1 + 1 = 8

So, the parallelogram has an area of 8 square units. Notice that this is the same area as the first rectangle we looked at. Also, notice that each figure is 4 units long and 2 units tall.

Using this method, we can find the area of a parallelogram in a similar way to finding the area of a rectangle.

So, the area (A) of a parallelogram is the base (b) multiplied by the height (h).

A = bh

A = bh = (4 cm)(2 cm) = 8 cm2

4 cm

2 cm

4 cm

A = bh = (4 cm)(2 cm) = 8 cm2

2 cm


14| Section 1

Example:Sue is re-tiling her bathroom. She is consider-ing these two tiles whose area is measured in square inches. She would like to use the larger tile so that she will need to buy fewer tiles. Which tile should she buy?

Solution:We need to find the area of each parallelogram to see which tile is larger. We will multiply the base by the height to find the area.

A = bh

The base of the first parallelogram is 3 inches. Its height is also 3 inches.

3 inches × 3 inches = 9 in2

The base of the second parallelogram is 5 inches. Its height is 2 inches.

5 inches × 2 inches = 10 in2

So, Sue should buy the second tile because it has a larger area.


9 Area is the space inside a figure and is measured in square units.

9 Area for a parallelogram is found by multiplying the base by the height.

3 in

3 in 2 in

5 in

This might help!

For rectangles, we use length and width to name the dimensions. For parallelograms, and other figures, we use base and height.

length

side length

base

height

Base is the length of the bottom side, not the length of the entire figure. Height is the verti-cal width of the figure, not the side length.


Section 1 |15

Match these items.

1.11 ________ the measurement of the space inside a plane figure

________ the length of a plane figure

________ the perpendicular width of a plane figure

________ the unit of measure for area

Circle the letter of each correct answer.

1.12 A rectangle is 6 centimeters long and 4 centimeters wide. What is the area of the rectangle? a. 10 cm2 b. 20 cm2 c. 24 cm2 d. 48 cm2

1.13 What is the area of a sheet of binder paper? (Binder paper is 8 1__2 inches by 11 inches.)

a. 88 inches b. 88 1__2 in2 c. 93 1__

2 in2 d. 94 in2

1.14 Which of the following is a unit of area? a. cm2 b. feet c. m3 d. inches

1.15 What is the area of this parallelogram if each 4 feet

3 feet

square is 1 square foot?

a. 12 b. 12 feet c. 12 ft2 d. 12 ft3

1.16 What is the area of a parallelogram with a height of 4 inches and a base of 5 inches?

a. 20 in b. 20 in2 c. 25 in d. 25 in2

1.17 A parallelogram has an area of 48 m2. If the base is 12 m long, what is the height? a. 4 m b. 8 m c. 12 m d. 36 m1.18 Which figure has sides that are perpendicular to each other?

a. b.

c. d.

1.19 Parallelogram A has a base of 4 cm and a height of 9 cm. Which figure described below has the same area as parallelogram A?

a. A rectangle 6 cm long and 5 cm wide. b. A parallelogram with a base of 8 cm and a height of 4 cm. c. A rectangle 9 cm long and 4 cm wide. d. A parallelogram with a base of 5 cm and a height of 8 cm.

a. square units

b. base

c. area

d. height


16| Section 1

AREA OF TRIANGLESYou know that area is measured in square units. Can you find the area of this triangle? It would be a little tricky to count the whole square and parts of squares accurately.

Luckily, there is an easier way! In this lesson we will explore finding the area of a triangle.

To find the area of a triangle, we will use our knowledge of the area of a parallelogram. We can make a parallelogram with two congruent triangles, if the triangles share one correspond-ing side:

Since we know how to find the area of a paral-lelogram, we can use that information to find the area of a triangle.

Since the triangles that make the parallelo-gram are congruent, their areas are equal. So, the area of each triangle is 1__

2 the area of the parallelogram:

We know that the area of a parallelogram is found by multiplying the base by the height: A = bh

1__2 + =1__

21

Since the area of the triangle is 1__2 the area of

the parallelogram, the formula for the area of the triangle is:

A = 1__2 bh

We can find the area of the first triangle we looked at by thinking of it as half of a parallelogram.

base: Counting the units, we can see that the base of the parallelogram is 5 units long.

height: Counting the vertical units, we can see that parallelogram has a height of 2 units.

The area of the parallelogram is 10 square units, multiplying the base by the height (5 × 2 = 10).

The area of the triangle is half of that:1__2 (10) = 5

So, the area of the triangle is 5 square units.

4 cm4 cm 4 cm

3 cm 3 cm 3 cm

4 cm

3 cm

Remember that the height of a figure is the ver-tical width. Each of these triangles has a height of 4 cm:

In fact, each of these triangles has the same area because they all have the same width! A = 1__

2bh

Substitute the base and height. A = 1__

2 (3 cm)(4 cm)

Multiply base times height. A = 1__

2 (12 cm2)

Divide by 2. A = 6 cm2

So, each triangle has an area of 6 square centimeters.


Section 1 |17

FIND THE AREA OF TRIANGLES

7 in

3.5 in

Example:There are three slices of pizza for Chris and his two friends. If Chris wants the largest slice, which slice should he choose?

Solution:We need to find the area of each slice to find which is largest. We will use the formula for area of a triangle.

First slice:

A = 1__2 bh

A = 1__2 (3 in)(8 in) Substitute the base and height.

A = 1__2 (24 in2) Multiply base times height.

A = 12 in2 Divide by 2.

Second slice:

A = 1__2 bh

A = 1__2 (4 in)(6 in) Substitute the base and height.

A = 1__2 (24 in2) Multiply base times height.

A = 12 in2 Divide by 2.

Third slice:

A = 1__2 bh

A = 1__2 (3.5 in)(7 in) Substitute the base and height.

A = 1__2 (24.5 in2) Multiply base times height.

A = 12.25 in2 Divide by 2.

So, the third slice is the largest, but just barely!

8 in

3 in

6 in

4 in

Think about it!

Notice that the height of a right triangle (half of a rectangle) is the same as the side length. This is because that side of the triangle is vertical.


18| Section 1


9 Any parallelogram can be divided into two congruent triangles.

9 The area for a triangle is found by taking one half of the base multiplied by the height: 1__2 bh

Match each word to its definition.

1.20 ________ the measurement of the space inside a plane figure

________ the length of a plane figure

________ the perpendicular width of a plane figure

________ the unit of measure for area

Example:Sarah has a crowded backyard, but she thinks she has room for a triangular garden. There is room for the triangular garden to be 4 feet wide. She would like the garden to have an area of 17 square feet. How long should the garden be?

Solution:We can use the formula for the area of a triangle and solve for the length of the garden. The garden is 4 feet wide, so that is the base.

A = bh

17 ft2 = (4 ft)h Substitute the base and area.

17 ft2 = (2 ft)h Divide by 2.

2 ft times the height is 17 ft2. So, if we divide 17 ft2 by 2 ft, we’ll find the height.

17 ft2 ÷ 2 ft = 8.5 ft

So, Sarah’s garden needs to be 8.5 feet long.

a. height

b. base

c. area

d. square units

Key point!

Always label the units for any measurement. Doing this will make it clear what type of measure-ment it is (length, area, volume), as well as what units were used.

You can tell right away if area is being measured if the units are square units.

If we know the area of a triangle and either the base or the height, we can solve for the other.


Section 1 |19

Circle the letter for each correct answer.

1.21 If the area of parallelogram ABCD is 27 square feet, what is the area of triangle ABC?

a. 27 ft2 b. 54 ft2 c. 13.5 ft2 d. 13 ft

1.22 The base of a triangle and a parallelogram are the same length. Their heights are also the same. If the area of the parallelogram is 48 m2, what is the area of the triangle?

a. 12 m2 b. 24 m2 c. 48 m2 d. 96 m2

1.23 The base of a parallelogram and a triangle are the same length, and both figures have the same area. What is true about height of the triangle?

a. It is the same as the parallelogram’s height. b. It is half of the parallelogram’s height. c. It is twice the parallelogram’s height. d. Its height is twice its base.

1.24 A triangle is 6 centimeters long and 4 centimeters high. What is the area of the triangle? a. 12 cm2 b. 20 cm2 c. 24 cm2 d. 48 cm2

1.25 What is the area of this triangle? a. 24 in2 b. 16 in2

c. 15 in2 d. 12 in2

1.26 The area of a triangle is 3.6 cm2. If the triangle has a base of 6 cm, what is the height? a. 0.6 cm b. 1.2 cm c. 12 cm d. 3 cm

1.27 Which of these triangles does not have the same base length as the others?

a. A b. B c. C d. D

1.28 What is the area of the triangle if the base is 5 centimeters and the height is 6 centimeters?

a. 15 cm2 b. 12 feet c. 4 m3 d. 16 inches

A B

C D

5 in5 in

6 in

4 in

A. B.

C. D.


20| Section 1

AREA OF COMPOSITE FIGURESKelsey has a small bedroom that includes her closet, her bed, a desk, a dresser, and some shelves. Her parents are going to put carpet in the room, so they need to find the area of the floor to know how much carpet to buy.

The room is not a rectangle, a parallelogram, or even a triangle. So, how will they find the area of the floor? In this lesson, we will answer this question and learn how to solve problems like these using what we already know about finding the area.

Before we look at some new figures, let’s take a minute to review what we already know about area.

We know that the area of a parallelogram is found by multiplying the base by the height.

The area of a triangle is found by taking half of the base multiplied by the height. We found that the area of a triangle was half the area of a parallelogram.

If we need to find the area of more complicated shapes, called composite figures, we can usu-ally divide the figure into basic shapes. Then, we can find the area of each shape and add them together.

Let’s take a look at the shape of Kelsey’s bed-room without the furniture, closet, or door, and with measurements marked.

12 feet

6 feet6 feet

6 feet

9 feet

15 feet

Can you see a way to divide the figure into rect-angles? There are three ways we could find the area using rectangles.

In Example 1, the room is divided into two rectangles:

One is 15 feet long and 6 feet wide, and the other is 9 feet long and 6 feet wide. We can find the area of each rectangle and add their areas together. A = lw + lw

Substitute the lengths and widths of the rectan-gles. A = (15 feet)(6 feet) + (9 feet)(6 feet)

Multiply, then add.

A = 90 ft2 + 54 ft2 = 144 ft2

15 feet

9 feet

6 feet 6 feet

Example 1


Section 1 |21

In Example 2, the room is again divided into two rectangles:

One is 12 feet long and 9 feet wide, and the other is 6 feet long and 6 feet wide. We can find the area of each rectangle and add their areas together. A = lw + lw

Substitute the lengths and widths of the rectan-gles. A = (12 feet)(9 feet) + (6 feet)(6 feet)

Multiply, then add. A = 108 ft2 + 36 ft2 = 144 ft2

In Example 3, we can look at the room as one rectangle, 15 feet long and 12 feet wide. Then, we can subtract the part of the rectangle that is not included in the room: a rectangle 6 feet long and 6 feet wide. A = lw – lw

Substitute the lengths and widths of the rectan-gles. A = (15 feet)(12 feet) – (6 feet)(6 feet)

Multiply, then subtract. A = 180 ft2 – 36 ft2 = 144 ft2

In each example, the area comes out to be 144 square feet. So, when we work with composite figures, there is often more than one way to break up the shape. You can choose whichever way works best for you!

15 feet

12 feet

6 feet

6 feet

9 feet

12 feet

Example 2

Example 3


22| Section 1

TRAPEZOIDSA figure that is similar to a parallelogram is a trapezoid. It is a quadrilateral that has one pair of parallel sides. The parallel sides are also called bases. This trapezoid has a height of 4 cm and bases of 4 cm and 10 cm.

Example:What is the area of the figure shown?

Solution:To find the area, we will divide the figure into two rectangles. We can find the area of each rectangle and then add the areas together.

The first rectangle is 8 cm long and 4 cm wide, and the second rectangle is 4 cm long and 4 cm wide.

A = lw + lw

Substitute the lengths and widths of the rectangles.

(8 cm)(4 cm) + (4 cm)(4 cm)

Multiply, then add.

32 cm2 + 16 cm2 = 48 cm2

So, the area of the figure is 48 square centimeters.

4 cm

4 cm

4 cm

4 cm

4 cm8 cm

2 cm

2 cm

4 cm

4 cm

4 cm8 cm

4 cm

4 cm

10 cm


Section 1 |23

We can find the area of the trapezoid by divid-ing it into a parallelogram and a triangle.

By drawing a line parallel to the right side, a parallelogram is created because we already know that the bases are parallel, so each pair of opposite sides is parallel. We also know that the bottom side of the parallelogram is 4 cm because opposite sides of a parallelogram are

congruent. Since the bottom base is 4 cm, the length of the triangle base is 6 cm (10 cm – 4 cm = 6 cm).

Now we can find the area by adding the area of the parallelogram and the area of the triangle:

A = bh + 1__2 bh

Substitute the base and height of each figure.

(4 cm)(4 cm) + (6 cm)(4 cm)

Multiply, then add.

16 cm2 + 12 cm2 = 28 cm2

So, the trapezoid has an area of 28 square centimeters.

4 cm

4 cm

10 cm

Example:Tom is going to paint the wall of a bedroom in his attic and he needs to know how much paint to buy. The wall is 8 feet high and includes a window.

Solution:The wall is a trapezoid, so we can divide it into a par-allelogram and triangle to find the area. For now, we’ll ignore the window and subtract its area at the end.

The bottom base of the parallelogram is 6 feet, since it is the same length as its opposite side. The base of the triangle is 10 feet, subtracting the parallelogram base from the trape-zoid base (16 feet – 6 feet = 10 feet).

Now we can find the area by adding the area of the parallelogram and the area of the triangle:

A = bh + 1__2 bh

(6 feet)(8 feet) + 1__2 (10 feet)(8 feet) Substitute the base and height of each figure.

48 feet2 + 40 feet2 = 88 feet2 Multiply, then add.

The wall is 88 square feet. However, the window area needs to be subtracted. The window is 2 feet long and 3 feet wide.

A = lw = (2 feet)(3 feet) = 6 ft2

Now we can subtract the window area from the wall area.

88 ft2 – 6 ft2 = 82 ft2

So, the trapezoid has an area of 82 square centimeters.

8 ft

6 ft

2 ft

3 ft

16 ft


24| Section 1


9 The area of composite figures can be found by dividing the figure into rectangles, parallelograms, and triangles.

9 The area of a trapezoid can be found by dividing it into a parallelogram and a triangle.


1.29 ________ a geometric figure that is made up of two or more basic shapes

________ a quadrilateral with one pair ofparallel sides

Place a check mark next to each correct answer (you may select more than one answer).

1.30 How could this figure be divided to find its area, assuming side lengths are known?

�

�

�

�

a. trapezoid

b. composite figure


Section 1 |25

Circle each correct answer.

1.31 What is the area of this figure? a. 128 in2 b. 118 in2

c. 108 in2 d. 48 in2

1.32 A wall is 12 feet long and 8 feet tall. There is a square window 4 feet long in the wall. What is the area of the wall surface?

a. 96 ft2 b. 92 ft2 c. 80 ft2 d. 32 ft2

1.33 What is the area of this figure? a. 20 m2 b. 24 m2

c. 32 m2 d. can’t be determined

1.34 If this picture frame can display a picture 6 inches long and 4 inches wide, what is the area of the frame itself?

a. 24 in2 b. 30 in2 c. 54 in2 d. 78 in2

1.35 The corners of a square are cut off two centimeters from each corner to form an octagon. If the octagon is 10 centimeters wide, what is its area?

a. 84 cm2 b. 92 cm2

c. 100 cm2 d. can’t be determined

1.36 What is the area of this trapezoid? a. 60 ft2 b. 80 ft2

c. 64 ft2 d. 24 ft2

1.37 What is the area of the figure? a. 40 ft2 b. 84 ft2

c. 96 ft2 d. can’t be determined

4 m 4 m

2 m

8 m

9 in

6 in

6 in4 in

10 cm

2 cm

2 cm

6 ft

8 ft

10 ft

10 feet10 feet

2 feet 2 feet16 feet

8 in

10 in

6 in

16 in


26| Section 1

AREA OF CIRCLESSteve has installed a sprinkler in the middle of his lawn. He is thinking about adding sprinklers in the corners and possibly the sides of the lawn. However, he would like to know the area that the sprinkler covers.

How can Steve find the area of this circular space? It’s definitely not a parallelogram or a triangle, and it’s not a composite figure. In this lesson we will explore how to find the area of a circle.

We can think of a circle as a regular polygon with many, many sides. Notice that as the poly-gon has more sides, it looks more like a circle.

In fact, this is how the area of a circle was calculated long ago. A Greek named Archime-des discovered that he could find the area of

the congruent triangles in the polygon and get closer and closer to the area of a circle as the number of sides increased. In a way, a circle is a composite figure!

Remember that pi, usually shown as the Greek letter p (π), is the ratio of the circumference to the diameter in a circle.

π = c__d

This ratio, and Archimedes’ discovery, leads us to the formula for the area of a circle.

dodecagon 20-gon 36-gon


Section 1 |27

The area of any regular polygon can be found by changing it into a parallelogram as long as the distance to the center is known.

The area of the parallelogram is the base mul-tiplied by the height. The height is the distance to the center of the octagon and the base is 1__

2of the perimeter of the octagon.

A = 1__2 ph

If the octagon were a circle, it would look like this instead. Now the height of the parallelo-gram is the radius of the circle, and the base of the parallelogram is 1__

2 of the circumference of the circle.

A = 1__2 C × r

The circumference is pi multiplied by the diam-eter, which is twice the radius.

C = π × d

d = 2r

Substitute the values for C and d.

A = 1__2 π × 2r × r

Simplify.

A = π × r2

Archimedes knew that as the polygon, or circle, was divided into more and more narrower sec-

tions, the area was getting closer and closer to pi r squared.

So, the area of a circle is found by multiplying pi by the square of the radius.

A = πr2

Although pi is a ratio, the decimal form does not repeat. Pi has been calculated to millions of decimal places. We will use 3.14, or 22___

7 , as a close approximation for pi, or we can leave the answer in terms of pi.

1__2

p

1__2

h

A = ph

1__2

C

r

A = π r2 A = π r2 A = π r2

= π(10cm)2 = (3.14)(10cm)2 = 22___7 (10 cm)2 Substitute values for r and p.

= 100π cm2 = (3.14)100π cm2 = 22___7 100 cm2 Multiply 10 × 10.

≈ 314 cm2 = 2200 _____7 cm2 Multiply by the value for π.

≈ 314.3 cm2 Divide by 7.

Reminder

The radius is half the length of the diameter. If the diameter is 20 m, the radius is 10 m.

r = 10

d = 20

Let’s find the area of a circle using all three forms.

We’ll use the formula for the area of a circle.


28| Section 1

Sometimes we need to find the area of half of a circle, called a semicircle. To find the area of a semicircle, we just divide the area of the whole circle by two, since there are two halves. The diameter is the edge of the semicircle.

The second and third answers are approxima-tions, and the accuracy we need depends on the situation. In fact, if we want a quick esti-mate for the area of a circle, we can use 3 for pi. This is also an easy way to check that your answer makes sense.

FIND THE AREA OF A CIRCLE AND SEMICIRCLESAlthough 100π is an exact answer, using the answer in terms of π is not very practical in real life situations where we need a measurement. However, it is very useful in many mathematical situations.

For real life situations, we will use 3.14, or 22___7 .

Example:Steve’s sprinkler sprays water 7 feet. What area of the lawn does it water?

Solution:The sprinkler will cover a circular area, and the radius of the circle (the distance it sprays water) is 7 feet. We will use the formula for the area of a circle, and 22___

7 for pi.

A = πr2

A = 22___7 (7 ft)2 Substitute values for r and p.

A = 22___7 (49 ft2) Multiply 7 × 7.

A = 22(7 ft2) Divide 49 by 7.

A = 154 ft2 Multiply.

So, the sprinkler will water an area of about 154 square meters.

Let’s double check that our answer makes sense. We’ll use the formula again, but we’ll use 3 for pi. When we squared the radius, we got 49 square feet. Multiplying by 3 for pi, we get 147 square feet (3 × 49 = 147). So, our answer makes sense because pi is a little more than 3, and 154 is a little more than 147. For this situation, perhaps an estimate would have been enough accuracy.

semicircle

A = πr2___2


Section 1 |29

Example:Sally is painting the area above her door with expensive gold leaf paint. The door is 3 feet wide. She needs to know the area so she knows how much paint to buy.

Solution:An estimate will not be useful here. Sally does not want to spend any more money on the paint than she needs to, so she wants an accu-rate answer.

We will use the formula for the area of a circle, and then divide that result by two since the area is a semicircle. The diameter of the semi-circle is 3 feet, so the radius is 1.5 feet (3 ÷ 2 = 1.5). We’ll use 3.14 for pi.

A = πr2 A = (3.14)(1.5 ft)2 Substitute values for pi and r.A = (3.14)2.25 ft2 Multiply 1.5 by 1.5.A ≈ 7.065 ft2

This is the area of the circle with a diameter of 3 feet. However, Sally is painting a semicir-cle, so we need to divide the result by 2.

7.065 ft2 ÷ 2 = 3.5325 ft2

So, Sally needs to buy enough paint to cover about 3.5 square feet.

Be careful!

Sometimes the diameter is given instead of the radius. Be sure to divide the diameter by two to find the radius and use it in the formula.

S-T-R-E-T-C-H

If a circle has an area of 144π square units, what is the radius of the circle?


9 The area of a circle is found using the formula A = πr2.

9 The area of a semicircle is found by dividing the area of the circle by 2: A = πr2____2


30| Section 1


1.38 ________ the distance across a circle through the center

________ the ratio of the circumference of a circle to its diameter; approximately 3.14

________ the distance from the center of a circle to any point on the circle

________ One half of a circle, divided by the diameter

Circle each correct answer.

1.39 Which measure is not the area of a circle with radius 20 mm?

a. 400π mm2 b. 1256 mm2 c. 628.57 mm2 d. 8800_____7 m2

1.40 A circle has an area of π cm2. What is its radius? a. 1 cm b. 2 cm

c. 1__2 cm d. can’t be determined

1.41 What is the area of this circle? a. 64π m2 b. 50.24 m2

c. 100.48 m2 d. 176____7 m2

1.42 The logo for Chris’s Calculator Company is 3 semicircles. The logo will be placed on the company building and will be 4 feet tall. What is the area of the logo?

a. 18.84 ft2 b. 37.68 ft2

c. 50.24 ft2 d. 75.36 ft2

1.43 A dog is tied to a 14 foot long leash in the middle of the yard. How much area does the dog have to run around? (use 22___

7 for pi)

a. 88 ft2 b. 616 ft2 c. 44 ft2 d. 308 ft2

1.44 A circle has a diameter of 6 cm. What is its area? a. 18.84 cm2 b. 37.68 cm2 c. 28.26 cm2 d. 9.42 cm2

d = 8 m

4 feet

a. diameter

b. pi

c. semicircle

d. radius


Section 1 |31

1.45 A circle has a radius of 5 inches. A semicircle has a radius of 10 inches. How do the areas compare?

a. The areas are equal. b. The semicircle has twice the area of the circle. c. The circle has twice the area of the semicircle. d. The semicircle has four times the area of the circle.

1.46 The area behind the free throw line on a basketball court is a semicircle with a 6 foot radius. What is the area of the semicircle?

a. 56.52 ft2 b. 132____7 ft2

c. 113.04 ft2 d. 264____7 ft2

6 feet

PROJECT: ESTIMATING AREAIn this section, you have learned about area and found the area of different plane figures: triangles, rectangles, parallelograms, circles, and composite figures. However, how do we find the area of a figure that isn’t any of the shapes we’ve explored?

In this project, you will explore how to find the area of figures like this.

Materials Pencil Grid Paper


32| Section 1

ESTIMATING AREAThe area of an irregular figure can be found accurately using a type of mathematics called Calculus. The method involves filling the space with narrower and narrower rectangles. Then, the areas of the rectangles are added. As the rectangles get narrower, less space is unac-counted for. If you continue your math studies, you will learn this method later in high school.

For now, we will use estimation to find the area of irregular figures. By placing a grid over the figure and using squares the size of the unit we need to measure, we can get a reasonable estimate.

= 1 cm2

We can use a few different strategies to esti-mate the area. First, notice that the figure is within a 4 × 4 square, so we know the area is less than 16 cm2, (4 cm × 4 cm = 16 cm2). This gives us a rough estimate.

We could count the number of empty squares and subtract that area from 16. There are about 2 whole squares and 5 half-squares that are empty.

= 1 cm2

1

2

1

2

4 5

3

Start by multiplying.

2 + 5( 1__2 ) =

2 + 5__2 =

Change 5__2 to a mixed number and add.

2 + 2 1__2 = 4 1__

2The figure is within an area of 16 cm2, and about 4 of the squares are empty. So, the area

of the figure is about 11 1__2 cm2.

16 – 4 1__2 = 11 1__

2


Section 1 |33

FIND THE AREA OF IRREGULAR FIGURES1. On a piece of grid paper, draw three differ-

ent irregular figures and estimate the area of each one.

2. Draw two different irregular figures with about 12 square units.

We could also estimate the area by counting the squares inside the figure. It looks like there are about 8 whole squares and 6 half-squares that are filled.

Multiply, then add.

8 + 6(1__2 ) =

8 + 3 = 11

Using this method, our result was 11 cm2. So, we arrived at a slightly different answer (our earlier estimate was 11

1__2 cm2), but both results

are an estimate, so either answer is valid.

= 1 cm2

1

432

65 7

8

2

3

1

5 6

4

Answer the following questions, corresponding to the numbered steps above.

1.47 List three figures that you have found the area of in this unit.

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

1.48 What strategy did you use to draw each figure? _______________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________


34| Section 1

1.49 What method(s) did you use to estimate the area of the figures you drew? ___________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

1.50 How accurate do you think your estimates were? _____________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

1.51 Can you think of any real life situations where we would need to find the area of irregular

figures? _______________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

________________________________________________________________________________________________

TEACHER CHECKinitials date

Review the material in this section in preparation for the Self Test. The Self Test will check your mastery of this particular section. The items missed on this Self Test will indicate spe-cific areas where restudy is needed for mastery.


Section 1 |35

Circle each correct answer (each answer, 4 points).

Use parallelogram ABCD to answer questions 1.01 – 1.04.

1.01 What is the perimeter? a. 14 cm b. 13 cm

c. 12 cm d. 11 cm

1.02 What is the area? a. 7 cm2 b. 9 cm2

c. 12 cm2 d. 12.25 cm2

1.03 What is the area of triangle ABC? a. 7 cm2 b. 3.5 cm2

c. 6 cm2 d. 4.5 cm2

1.04 What is the perimeter of ACD? a. 9.5 cm b. 7.5 cm2

c. 8.5 cm2 d. 5.5 cm

1.05 What is the circumference of a circle with a diameter of 5 meters? (Use 3.14 for pi.) a. 78.5 m b. 31.4 m c. 15.7 m d. 8.14 m

1.06 What is the perimeter of the figure? a. 28 m b. 22 m c. 20 m d. 14 m

1.07 A regular pentagon has a perimeter of 60 feet. How long is each side? a. 5 feet b. 6 feet c. 10 feet d. 12 feet

1.08 If the area of the parallelogram is 15 cm2, what is the area of the green triangle?

a. 30 cm2 b. 15 cm2

c. 7.5 cm2 d. 8 cm2

1.09 What is the height of the triangle? a. 2 units b. 3 square units

c. 3 units d. can’t be determined

SELF TEST 1: PLANE FIGURES

3 cm

3 cm

3 cm

3.5 cm3.5 cm

A B

CD

7 m

4 m

3 m

5 m

1 m


36| Section 1

1.010 The area of a triangle is 18 square feet. If the base is 3 feet, what is the height of the triangle?

a. 6 feet b. 3 feet c. 12 feet d. 9 feet

1.011 The area of a rectangle is 51 square inches. If the width of the rectangle is 6 inches, what is the length?

a. 19.5 inches b. 13 inches c. 9 inches d. 8.5 inches

1.012 Steve is adding wallpaper to a living room wall and he needs to know how much wallpaper to buy. If the wall is 8.5 feet tall and 12.5 wide, how much wallpaper should he buy?

a. 106.25 ft2 b. 96.25 ft2 c. 42 ft d. 108 ft2

1.013 A square has a perimeter of 12 cm. What is its area? a. 9 cm2 b. 18 cm2 c. 36 cm2 d. 144 cm2

1.014 What is the area of the triangle? a. 15 cm2 b. 14 cm2

c. 12 cm2 d. 24 cm2

1.015 What is the area of a circle with a diameter of 12 m? a. 18.84 m2 b. 37.68 m2 c. 75.36 m2 d. 113.04 m2

1.016 What is the area of the trapezoid? a. 88 in2 b. 128 in2

c. 96 in2 d. 48 in2

1.017 A square picture frame has a round circle cut out to show the picture. What is the area of the picture frame?

a. 177.5 cm2 b. 193.2 cm2 c. 334.5 cm2 d. 256 cm2

1.018 An arched entrance to a stadium is made by combining a square and a semicircle. What is the area of the opening?

a. 178.5 ft2 b. 150 ft2

c. 139.25 ft2 d. 100 ft2

1.019 What is the area of the composite figure? a. 84 m2 b. 72 m2

c. 108 m2 d. 96 m2

8 in

6 in

16 in

10 ft

10 cm

16 cm

4 m

6 m

12 m

8 m

8 cm

6 cm


Section 1 |37

SCORE TEACHERinitials date

6885

1.020 What is the area of the hexagon? a. 60 m2 b. 80 m2

c. 100 m2 d. 120 m2

Answer true or false (each answer, 5 points).

1.021 ____________ To find the perimeter and area of a square, use the same formula.

5 m

6 m

6 m

10 m

5 m


38| Section 1

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MATH · 2015. 9. 17. · The distance across a circle through the center. height. The perpendicular width of a plane figure. perimeter. The distance around the outside of a plane