Geometry U1 L1-L7 Area and Volume

Lesson 1 Area Concepts of Polygons

Area Concepts of Polygons

When you measure the length of a line segment, standard units such as meter, yard, inches are used. These are often called linear units because they measure length.

The standard units of area are square units, such as square inches and square miles. 

A square meter is the area of a square whose sides are each one meter in

length.  Area is the number of non-overlapping square units needed to cover the interior

region of a two dimensional figure or the surface area of a three dimensional figure.

Just as distance is thought of in positive terms, so is area. When doing area calculations, if the result is a negative value, you know that a mistake has been made.

You can estimate the area of a figure by observing the number of square units it contains.

A   triangular region  is the union of a triangle and its interior. Every polygon can be broken into triangular regions. In other words, every polygon can be divided into a number of different triangles. The region of the polygon is the union of those nonoverlapping triangles.

The regions of each polygon can be broken into different, nonoverlapping triangles.

Notice that the only difference between the area of a geometric figure and its region is that the region includes the sides of the figure, but the area includes only the space inside the figure.

Postulates 17, 18 and 19 In any discussion of area, it is important to note the type of unit being used. Area answers should be in terms of the square unit being used. Many math and science problems deal with the conversion of units in length and

area. There are three new postulates in this lesson, beginning with Postulate 17.

This postulate states that for every polygonal region, there exists one unique positive number that represents the area of that number. The key concept in this postulate is that the number is positive.

Postulate 18 states if two polygonal figures are congruent , then they have the same area . This would be reasonable, as they would have the same values for each part of any formula that would be used to calculate the area. For example, congruent rectangles would have the same base and height to calculate the area with.

Postulate 19 states that the area of a polygonal region is the sum of the areas of the triangular regions that form the polygonal region . In other words, every polygon can be divided up into two or more triangles. We could calculate the area of each individual triangle and then add up the sums to find the area of the original polygon.

Consider a square one unit on each side. We shall agree that the area of this square is one square unit. When we find the area of a region, we are looking for the number of these square units that will fit onto the surface of that region. The unit of measurement used could be inches, feet, yards, or any other unit of length. The corresponding unit for area would be square inch, square foot, and square yard.

(Area Addition Postulate) 

Lesson 2 Area of Rectangles

POSTULATE 20: The area of a rectangle is the product of the length of a base and the length of the altitude to that base: A = bh

Any side of a rectangle can be considered to be the base. Then each side adjacent to that base is an altitude to that base. Another way of saying Postulate 20 is that the area is the product of the length and the width.

A useful corollary to Postulate 20 tells how to find the area of a square.

Corollary: The area of a square is the square of the length of its side: A = s 2.

Since a square is a rectangle with base and altitude equal, we can replace s for b and h in the formula for area of a rectangle.

Lesson 3 Area of Parallelograms

Any side of a parallelogram can be called its base. The altitude to that base is the length of the perpendicular segment between the base and its opposite side.

So when doing calculations involving the base of a parallelogram, it does not matter which side you use. Any side can be the base. However, once you pick which side you will call the base, the altitude of height of the parallelogram is the perpendicular segment between the base on the opposite side.

THEOREM 7-1

The area of a parallelogram is the product of any base and the altitude to that base: A = bh.

Given: RSTU with RS = b = baseSX   UT and SX = h = altitude

Prove: Area of RSTU = bh

Lesson 4 Area of Triangles and Rhombuses

THEOREM 7-2 The area of a triangle is one-half the product of the length of a base and the altitude to that base: A = ½bh

Given:   RST with RS = b, TX* > RS*, and TX = h = altitude

Prove: Area of   RST = ½bh

Any side of the triangle can be considered its base. The altitude of the triangle to that base is the height in our formula.

Theorem 7-2 has two very useful corollaries.

Corollary 1: The area of an equilateral triangle with side of length s is one-fourth the

product of the square of s and  : 

Corollary 2: The area of a rhombus is one-half the product of the lengths of its diagonals: 

Lesson 5 Area of Trapezoids

trapezoid is a quadrilateral with exactly one pair of parallel sides. Any side of a rectangle, triangle, or parallelogram can be taken as the base, but only the parallel sides of a trapezoid are called bases.

Remember that the altitude of a trapezoid is a segment perpendicular from any point in one base to a point in the line containing the other base.

Notice that the altitude (also called the height) creates a right angle with both bases.

THEOREM 7-3

The area of a trapezoid is one-half the product of the length of

an altitude and the sum of the lengths of the bases: A =   h(b1 + b2).

Given: Trapezoid ABCD withAB = b1, DC = b2, and

     DX = h

Prove: Area ABCD =   h (b1 + b2)

STATEMENT REASON1. Trapezoid ABCD with AB = b 1, DC = b 2, DX = h 1. Given

2. Draw DB 2. Auxiliary line

3. Area   ABD =   b 1 h; Area   DBC =   b 2 h3. Theorem 8-2, area of triangle

4. Area ABCD = Area   ABD + Area   DBC 4. Area addition postulate

5. Area ABCD =   b 1 h + b 2 h5. Substitution

6. Area ABCD =  h (b 1 + b 2)6. Distributive Property

Because this trapezoid is technically a parallelogram, you can compare the area of the trapezoid calculated with the trapezoid area formula with the result calculated with the

formula for the area of a parallelogram. For the area of a parallelogram, the length and width is needed. So, the length of the parallelogram is 8 units and the height of the parallelogram is 6 units. The area is the product of the length and width or 48 square units.

Other types of special trapezoids include rectangles, rhombuses, and squares. You can use the area of a trapezoid formula to find the area of each of these figures as well.

Lesson 6 Area of Regular Polygons

Recall that a regular polygon has all sides equal and all angles equal. An equilateral triangle is a regular polygon. A square is a regular polygon.

apothem of a regular polygon

the perpendicular distance from the center of the polygon to a side

center of a regular polygon

the common center of its inscribed and circumscribed circles

central angle of a regular polygon

an angle whose vertex is the center of the polygon and whose sides contain consecutive vertices of the polygon

radius of a regular polygon the distance from the center of the polygon to a vertex