Finding the Area of a Regular Polygon
Geometry
Name: ________________
One method for finding the area of a regular polygon is to:
1. Divide the shape into equal triangles, and find the area of one triangle.
2. Multiply the area of one triangle by the number of triangles that make up the figure.
Find the area of the regular nonagon.
First Step: Divide the shape into equal triangles, and find the area of one triangle.
Redraw one of the triangles that make up the nonagon in the box below.
To find the area of the triangle, you will need the base and the height of the
triangle. Do you have either of these lengths given?
To solve for the base and height, you will have to use trigonometry. Begin by
drawing in the height of the triangle if you have not already done so. Label half
of the base x and the height y.
Next find the angles of the triangle using our formulas for central or interior angles. Please show your work in this space
below, and then label the angles on your triangle above.
𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑎𝑛𝑔𝑙𝑒 =
𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒 =
360
𝑛
180(𝑛 − 2)
𝑛
Now you have enough information to use trigonometry to find the base and height. Show your work for solving for x and y in
𝑜𝑝𝑝
this space below.
sin 𝜃 =
1
ℎ𝑦𝑝
cos 𝜃 =
𝑎𝑑𝑗
ℎ𝑦𝑝
tan 𝜃 =
𝑜𝑝𝑝
𝑎𝑑𝑗
Once you have solved for x and y, you can find the area of the triangle (𝐴 = 2 𝑏ℎ). Show your work below.
Second Step: Multiply the area of one triangle by the number of triangles that make up the figure.
A = _________
Finding the Area of a Regular Polygon
Geometry
Name: ________________
Follow the same steps to find the area of the figure in part b. Show all work!