1. 10.4 Areas of Regular Polygons
Geometry

2. 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.

3. Important!!!

4. 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.

5. 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 s s s
A = ¼ s2

6. Finding the area of an Equilateral Triangle
Find the area of an equilateral triangle with 8 inch sides.
A = ¼ s2
Area of an equilateral Triangle
A = ¼
Substitute values.
A = ¼ • 64
Simplify.
A = • 16
Multiply ¼ times 64.
A = 16
Simplify.

7. Area Theorems A = bh Area of a Rectangle
The area of a rectangle is the product of its base and height. h b
A = bh

8. Area Theorems A = bh Area of a Parallelogram
The area of a parallelogram is the product of a base and height. h b
A = bh

9. Area Theorems A = ½ bh Area of a Triangle
The area of a triangle is one half the product of a base and height. h b
A = ½ bh

10. 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.

11. Areas of Trapezoids Area of a Trapezoidb2 h b1
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)

12. 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

13. 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 d2 d1

14. 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

15. 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 ) Substitute
A = ½ (40) Simplify
A = 20 Simplify
The area of trapezoid WXYZ is 20 square units

16. 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 20 20 A C 15 D E

17. Finding the area of a rhombus
A = ½ d1 d2
A = ½ (30)(40)
A = ½ (120)
A = 60 square units 15 20 20 A C 15 D E

18. Practice

20. A = ¼ s2 Find the sum of bases Find the height of one isosceles
b1 + b2 = 24*2 = 48
2. A = ½ * 48 * 9 = 216 m²
Find the height
of one isosceles
triangle by using
Pythagorean Formula
H = 10² - 6² = 8²
2. A = ½ * 8 * 12 = 96 m²

25. 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.

26. Using the Area Theorems9
Find the area of ABCD.
Solution:
Use AB as the base. So, b=16 and h=9
Area=bh=16(9) = 144 square units.