10.4 Areas of Regular Polygons

Geometry

Objectives/Assignment

• Find the area of an equilateral triangle.

• Find the area of a regular polygon, such as the area of a square, rectangle, rhombus etc.

Important!!!

Finding the area of an equilateral triangle• The area of any triangle with base

length b and height h is given by

A = ½bh. The following formula for equilateral

triangles; however, uses ONLY the side length.

Area of an equilateral triangle

• The area of an equilateral triangle is one fourth the square of the length of the side times

A = ¼ s2

3

3

s s

s

A = ¼ s23

Finding the area of an Equilateral Triangle• Find the area of an equilateral

triangle with 8 inch sides.

A = ¼ s23

A = ¼ 823

A = ¼ • 643

A = • 163

A = 16 3

Area of an equilateral TriangleSubstitute values.

Simplify.

Multiply ¼ times 64.

Simplify.

Area Theorems

• Area of a Rectangle

• The area of a rectangle is the product of its base and height.

h

b

A = bh

Area Theorems

• Area of a Parallelogram• The area of a

parallelogram is the product of a base and height.

A = bh

h

b

Area Theorems

• Area of a Triangle• The area of a

triangle is one half the product of a base and height.

A = ½ bh

h

b

Justification

• You can justify the area formulas for triangles follows.

• The area of a triangle is half the area of a parallelogram with the same base and height.

Areas of Trapezoids

Area of a Trapezoid The area of a

trapezoid is one half the product of the height and the sum of the bases.

A = ½ h(b1 + b2)

b1

b2

h

Area of a Kite The area of a kite

is one half the product of the lengths of its diagonals.

A = ½ d1d2

d1

d2

Areas of Rhombuses

Area of a Rhombus

The area of a rhombus is one half the product of the lengths of the diagonals.

A = ½ d1 d2

d1

d2

Finding the Area of a Trapezoid

• Find the area of trapezoid WXYZ.

• Solution: The height of WXYZ is h=5 – 1 = 4

• Find the lengths of the bases.b1 = YZ = 5 – 2 = 3

b2 = XW = 8 – 1 = 7

W(8, 1)X(1, 1)

Z(5, 5)Y(2, 5)

Finding the Area of a Trapezoid

Substitute 4 for h, 3 for b1, and 7 for b2 to find the area of the trapezoid.

A = ½ h(b1 + b2) Formula for area of a trapezoid.

A = ½ (4)(3 + 7 ) SubstituteA = ½ (40) SimplifyA = 20 Simplify

The area of trapezoid WXYZ is 20 square units

8

6

4

2

5 10 15

W(8, 1)X(1, 1)

Z(5, 5)Y(2, 5)

Finding the area of a rhombus

• Use the information given in the diagram to find the area of rhombus ABCD.

• Solution— Use the formula for

the area of a rhombus d1 = BD = 30 and d2 = AC =40

15

15

20 20A

B

C

DE

Finding the area of a rhombus

A = ½ d1 d2

A = ½ (30)(40)A = ½ (120)A = 60 square units

15

15

20 20A

B

C

DE

Practice

A = ¼ s2 31. Find the height of one isosceles triangle by using Pythagorean Formula

H = 10² - 6² = 8²

2. A = ½ * 8 * 12 = 96 m²

1. Find the sum of bases

b1 + b2 = 24*2 = 48

2. A = ½ * 48 * 9 = 216 m²

Justification

• You can justify the area formulas for parallelograms as follows.

• The area of a parallelogram is the area of a rectangle with the same base and height.

Using the Area Theorems

• Find the area of ABCD.

Solution:

• Use AB as the base. So, b=16 and h=9

Area=bh=16(9) = 144 square units.

12

16

E

A

C

D

B

9