Use trigonometric ratios to rewrite the apothem of Pentagon B and the side lengths of the pentagons in terms of the apothem of Pentagon A.
The area of Pentagon A is about 1.53 times the area of Pentagon B.
Practice makes perfect
Let a represent the apothem of Pentagon A and the radius of Pentagon B. We expect Pentagon A to be bigger than the other one because the radius of Pentagon A will be longer than its apothem.
Next, we will write the length of each side of the pentagons and the apothem of Pentagon B in terms of a. We know that the central angle of a pentagon is 72 ^(∘) and the apothem bisects this angle and the opposite side. Let's show what we know in the diagram.
In the diagram, x and y represent half of the side lengths of Pentagon A and Pentagon B, respectively. Additionally, b represents the apothem of Pentagon B. Using trigonometric ratios, we can write x, y, and b in terms of a.
Trigonometric Ratio
Rewrite
Pentagon A
tan 36^(∘)= x/a
x=atan36^(∘)
Pentagon B
sin 36^(∘)= y/a
y= a sin 36^(∘)
cos 36^(∘)= b/a
b= a cos 36^(∘)
We will now find the perimeters of the pentagons by multiplying the side lengths by the number of sides. Remember that x and y are half-lengths of the sides, so we need to multiply them by 2* 5, or 10. Then we can use the formula for the area of a regular polygon.
