Question

the diagram shows a regular pentagon with side length s. explore the area of this pentagon by following these steps.

1. press continue to draw an apothem with length a.
2. based on the variables on the diagram, how can the area of the shaded triangle be determined?

area = 1/2sa

1. because the triangles are congruent, the area of the entire pentagon is equal to 5 times the area of the shaded triangle.
2. the area of the pentagon is 5(1/2 sa), or 5s(1/2 a).

the of the pentagon is also equal to 5s

Explanation:

Step1: Recall the formula for the area of a triangle

The formula for the area of a triangle is \(A=\frac{1}{2}\times base\times height\). For the shaded triangle, the base is \(s\) (side - length of the pentagon) and the height is \(a\) (the apothem). So, the area of the shaded triangle is \(A_{triangle}=\frac{1}{2}sa\).

Step2: Consider the number of congruent triangles in a regular pentagon

A regular pentagon can be divided into 5 congruent isosceles triangles (by drawing lines from the center to each vertex).

Step3: Find the area of the pentagon

If the area of one triangle is \(\frac{1}{2}sa\), then the area of the pentagon \(A = 5\times\frac{1}{2}sa\). We know that the perimeter \(P\) of a regular pentagon with side - length \(s\) is \(P = 5s\). So, \(A=\frac{1}{2}\times(5s)\times a=\frac{1}{2}Pa\)

Answer:

perimeter