Notice that a regularpentagon is made of five congruent triangles.
≈ 6.47 cm
Practice makes perfect
We are given that a regularpentagon is 72 square centimeters and we are asked to find the length of one side. Let's call it s.
First, notice that a regular pentagon is made of five congruent triangles.
This means we can evaluate the area of one triangle by dividing the area of a pentagon, 72, by the number of triangles, 5.
72/5=14.4
The area of each triangle is 14.4 square centimeters. Since the central angle of a regular pentagon is 72^(∘) and the apothem of this figure a is the height of a isosceles triangle, we can use the tangent ratio to write an equation.
Let's write the equation using the tangent ratio and solve it for a.
tan 36^(∘)=12 s/a ⇒ a=1/2tan 36^(∘)s
Next we will use the fact that the area of a triangle is the half of the product of its height — in our exercise a — and the corresponding side length, s.
A=1/2 a s
Let's substitute 14.4 for A and 12tan 36^(∘)s for a.
Next we will take a square root of both sides of the equation. Notice that since s represents the side length, we will consider only the positive case when taking a square root of s^2.
