Polygons, Circles, and Solids. Common Polygons Triangles Slide 1- 3

Section 2.3

Polygons, Circles, and Solids

Common Polygons

Triangles

Slide 1- 3

Quadrilaterals

Slide 1- 4

Sum of Interior angles of Regular Polygon

• Interior sum of angles of regular polygon is (n-2)180, n is the number of sides• What is the measure of the interior angles of

a STOP sign?• 1080 degree.

Slide 1- 5

A rectangle has four sides that meet to form 90° angles. Each set of opposite sides is parallel and congruent (has the same length).

5 cm

9 cm

5 cm

9 cm

In a rectangle, if one right angle is shown, the other three are also right angles.

90°angles

Each longer side of a rectangle is called the length (l) and each shorter side is called the width (w).

Slide 8.3- 6

Slide 8.3- 7

ParallelExample 1

Finding the Perimeter of a Rectangle

Slide 8.3- 8

Find the perimeter of each rectangle.

a.6 m

16 m

6 m

16 m

P = 2 • l + 2 • wP = 2 • 16 m + 2 • 6 mP = 32 m + 12 mP = 44 m

The perimeter of the rectangle is 44 m.

ParallelExample 1continued

Finding the Perimeter of a Rectangle

Slide 8.3- 9

Find the perimeter of each rectangle.

b. A rectangle 7.8 ft by 12.3 ft

P = 2 • l + 2 • w

Either method will give you the same result.

P = 2 • 12.3 ft + 2 • 7.8 ftP = 24.6 ft + 15.6 ftP = 40.2 ft

Or, you can add up the lengths of the four sides.P = 12.3 ft + 12.3 ft + 7.8 ft + 7.8 ftP = 40.2 ft

Slide 8.3- 10

The perimeter of a rectangle is the distance around the outside edges.

The area of a rectangle is the amount of surface inside the rectangle.

8 m

5 m

1 m

1 m

We have five rows of eight square meters for a total of 40 square meters.

1 square meter or (m)2

Slide 8.3- 11

Squares of many sizes can be used to measure area. For smaller areas, you might use the ones shown below.

Slide 8.3- 12

1 square inch(1 in.2)

1 in.

1 in.

1 squarecentimeter

(1 cm2)

1 cm1 cm

1 squaremillimeter(1 mm2)

1 mm1 mm

(Approximate-size drawings)

Other sizes of squares that are often used to measure area:

1 square meter (1 m2) 1 square foot (1 ft2)

1 square kilometer (1 km2) 1 square yard (1 yd2)

1 square mile (1 mi2)

ParallelExample 2

Finding the Area of a Rectangle

Slide 8.3- 13

Find the area of each rectangle.

a.

7 yd

15 yd

A = l • wA = 15 yd • 7 ydA = 105 yd2

ParallelExample 2continued

Finding the Area of a Rectangle

Slide 8.3- 14

Find the area of each rectangle.

b.

A = l • wA = 18 cm • 3 cmA = 54 cm2

18 cm

3 cm

Slide 8.3- 15

ParallelExample 3

Finding the Perimeter and Area of a Square

Slide 8.3- 16

a. Find the perimeter of a square where each side measures 7 m.

Use the formula.

P = 4 • sP = 4 • 7 mP = 28 m

Or add up the four sides.

P = 7 m + 7 m + 7 m + 7 mP = 28 m

Same answer

ParallelExample 3continued

Finding the Perimeter and Area of a Square

Slide 8.3- 17

b. Find the area of a square where each side measures 7 m.

A = s • s

A = 7 m • 7 m

A = s2

A = 49 m2 Square units for area.

ParallelExample 4

Finding the Perimeter and Area of a Composite Figure

Slide 8.3- 18

a. The floor of a room has the shape shown.

6 ft

6 ft

30 ft

21 ft

24 ft

15 ft

Suppose you want to put new wallpaper border along the top of the walls. How much material do you need?

Find the perimeter of the room by adding up the length of the sides.

P = 30 ft + 21 ft + 24 ft + 15 ft + 6 ft + 6 ft = 102 ft

ParallelExample 4continued

Finding the Perimeter and Area of a Composite Figure

Slide 8.3- 19

b. The carpet you like cost $24.25 per square feet. How much will it cost to carpet the room?

6 ft

6 ft

30 ft

21 ft

24 ft

15 ft

ParallelExample 4continued

Finding the Perimeter and Area of a Composite Figure

Slide 8.3- 20

b. Finally, multiply to find the cost of the carpet.

c. 36+504= 540 sq feet.

6 ft

6 ft

21 ft

24 ft

A parallelogram is a four-sided figure with opposite sides parallel, such as the ones below. Notice that the opposite sides have the same length.

Slide 8.4- 21

ParallelExample 1

Finding the Perimeter of a Parallelogram

Slide 8.4- 22

Find the perimeter of a the parallelogram.

P = 15 cm + 9 cm + 15 cm + 9 cm

15 cm

15 cm

9 cm9 cm

= 48 cm

Slide 8.4- 23

ParallelExample 2

Finding the Area of a Parallelogram

Slide 8.4- 24

Find the area of the parallelogram.

The base is 10 m and the height is 3 m. Use the formula to solve.

10 m

10 m

4 m4 m 3 m

A = b ∙ hA = 10 m ∙ 3 mA = 30 m2

Slide 8.5- 25

A triangle is a figure with exactly three sides.

To find the perimeter of a triangle, add the lengths of the three sides.

Find the perimeter of the triangle.

P = 12 ft + 16 ft + 20 ft

= 48 ft

ParallelExample 1

Finding the Perimeter of a Triangle

Slide 8.5- 26

12 ft

16 ft

20 ft

Pythagorean Theorem

• it only works on Right Triangles.• Where a and b are legs and c is the hypotenuse.

Slide 1- 27

Slide 8.5- 28

The height of a triangle is the distance from one vertex of the triangle to the opposite side (base).

The height line must be perpendicular to the base; that is, it must form a right angle with the base.

Slide 8.5- 29

Find the area of each triangle.

a.

ParallelExample 2

Find the Area of a Triangle

Slide 8.5- 30

1

2A b h

52 ft 14 ft1

2A

1

2A

1

5226

14 ft ft

2364 ftA

Find the area of each triangle.

c.

ParallelExample 2continued

Find the Area of a Triangle

Slide 8.5- 31

0.5 bA h

0.5 12.75 8.5A

54.1875A

18

2

312

4

Slide 8.6- 32

r r

d

ParallelExample 1

Finding the Diameter and Radius of a Circle

Slide 8.6- 33

Find the unknown length of the diameter or radius in each circle.

a.

r = 12 in.d = ?

Because the radius is 12 in., the diameter is twice as long.

d = 2 • r

d = 2 • 12 in.

d = 24 in.

ParallelExample 1continued

Finding the Diameter and Radius of a Circle

Slide 8.6- 34

Find the unknown length of the diameter or radius in each circle.

b.

r = ?

d = 7 m

The radius is half the diameter.

r = d2

r = 7 m2

r = 3.5 m or 3 m12

Slide 8.6- 35

The perimeter of a circle is called its circumference. Circumference is the distance around the edge of a circle.

Slide 8.6- 36

Dividing the circumference of any circle by its diameteralways gives an answer close to 3.14.

This means that going around the edge of any circle is a little more than 3 times as far as going straight across the circle.

3.14159265359

This ratio of circumference to diameter is called .

Slide 8.6- 37

ParallelExample 2

Finding the Circumference of Circles

Slide 8.6- 38

Find the circumference of each circle. Use 3.14 as the approximate value for . Round answers to the nearest tenth.

a.

24 m

The diameter is 24 m, so use the formula with d in it.

C = • d

C = 3.14 • 24 m

C ≈ 75.4 m Rounded

ParallelExample 2

Finding the Circumference of Circles

Slide 8.6- 39

Find the circumference of each circle. Use 3.14 as the approximate value for . Round answers to the nearest tenth.

b.

6.5 cm

In this example, the radius is labeled,so it is easier to use the formula withr in it.

C = 2 • • r

C = 2 • 3.14 • 6.5 cm

C ≈ 40.8 cm Rounded

Slide 8.6- 40

ParallelExample 3

Finding the Area of Circles

Slide 8.6- 41

Find the area of each circle. Use 3.14 for . Round answers to the nearest tenth.

a.

A circle with a radius of 14.2 cm.

Rounded; square units for area

A = • r • r

A ≈ 3.14 • 14.2 cm • 14.2 cm

A ≈ 633.1 cm2

ParallelExample 3continued

Finding the Area of Circles

Slide 8.6- 42

Find the area of each circle. Use 3.14 for . Round answers to the nearest tenth.

b.

Now find the area.

24 ftFirst find the radius.

r = d2

r = = 12 ft24 ft2

A ≈ 3.14 • 12 ft • 12 ft

A ≈ 452.2 ft2

HW section 2.3

• 13-59

Slide 1- 43