-Use the drawing provided to explain the following theorem.
"The area A of a regular polygon whose apothem has length a and whose perimeter
is P is given by ."
Given: Regular polygon with center O and length s for each side;
apothem so that Prove:

Question

-Use the drawing provided to explain the following theorem.
"The area A of a regular polygon whose apothem has length a and whose perimeter
is P is given by  ."
Given: Regular polygon  with center O and length s for each side;
apothem  so that  Prove:

Quizplus · Accepted Answer

The Answer is From center O, we draw radii

From center O, we draw radii , , , , and . Because the radii are congruent to each other and the sides of the regular polygon are all congruent to each other as well, , , , , and are all congruent to each other by SSS. Each of the congruent triangles has an altitude length of . Further, the length of each base of a triangle is s, the length of side of the polygon. Therefore, the area of the regular polygon is Because the sum of the sides equals perimeter P, we have .

,

, and

. Because the radii are congruent to each other and the sides of the regular polygon are all congruent to each other as well,

,

, and

are all congruent to each other by SSS.
Each of the congruent triangles has an altitude length of

. Further, the length of each
base of a triangle is s, the length of side of the polygon. Therefore, the area of the regular polygon is

Because the sum of the sides equals perimeter P, we have

.

-Use the Drawing Provided to Explain the Following Theorem