Problem 1 Finding Area

What is the area of a regular nonagon with 10-cm sides?

Draw a regular nonagon with center C. Draw C P ¯ c p bar  and C R ¯ c r bar  to form isosceles Δ P C R . cap delta p c r .  The measure of central ∠ P C R angle p c r  is 360 9 , 360 over 9 , comma  or 40. The perimeter is 9 · 10   , 9 middle dot 10 comma  or 90 cm. Draw the apothem C S ¯ . c s bar , .

m ∠ P C S = 1 2 m ∠ P C R = 20 m angle p c s equals , 1 half , m angle p c r , equals 20  and P S = 1 2 P R = 5  cm. p s equals . 1 half p r . equals 5 , cm.

Let a represent CS. Find a and substitute into the area formula.

tan 20 ° = 5 a Use the tangent ratio. a = 5 tan 20 ° Solve for  a . A = 1 2 a p     = 1 2 ⋅ 5 tan 20 ° ⋅ 90 Substitute  5 tan 20 ° for a and  90   for  p .   ≈ 618.1824194 Use a calculator. table with 5 rows and 3 columns , row1 column 1 , tangent , 20 degrees , column 2 equals , 5 over eh , column 3 cap use the tangent ratio. , row2 column 1 , eh , column 2 equals . fraction 5 , over tangent , 20 degrees end fraction , column 3 cap solve for . eh . , row3 column 1 , eh , column 2 equals , 1 half , eh p , column 3 , row4 column 1 , , column 2 equals , 1 half , dot . fraction 5 , over tangent , 20 degrees end fraction . dot 90 , column 3 cap substitute . fraction 5 , over tangent , 20 degrees end fraction . for a and . 90 , for , p . , row5 column 1 , , column 2 almost equal to . 618.1824194 , column 3 cap use a calculator. , end table

The area of the regular nonagon is about 618   cm 2 . 618 , cm squared , .

✓ Got It?

1. What is the area of a regular pentagon with 4-in. sides? Round your answer to the nearest square inch.