Write the apothem of a regular decagon in terms of its side. Then, use it to write an area formula that depends on the side length.
The area of Decagon B is 100 times the area of Decagon A.
Practice makes perfect
We will write a rule for the area of a decagon with a side length s. We need to begin by finding the central angle of decagon.
360/n ⇔ 360/10 = 36
We found that the central angle of the decagon is 36^(∘). Now we can draw the diagram of Decagon B. Recall that the apothem of a regular polygon bisects the central angle and the length of the opposite side.
Using the tangent ratio, we can write the apothem a of the decagon in terms of s.
tan 18 ^(∘) =s/2/a ⇒ a=s tan18^(∘)/2
Now the area of a decagon with a side length s can be found.
Now, let m represent the side length of Decagon A. Decagon A therefore has a perimeter of 10m. Then, since the perimeter of Decagon A is equal to the side length of Decagon B, Decagon B has a side length of 10m. Next, we will use the formula above to compare the area A_A of Decagon A and the area A_B of Decagon B.
Side Length, s
A=5/2s^2 tan18^(∘)
Decagon A
m
A_A=5/2( m)^2 tan18^(∘)
Decagon B
10m
A_B=5/2( 10m)^2 tan18^(∘)
The ratio of A_B to A_A tells us how many times bigger Decagon B is than Decagon A.
We found that the ratio of the areas of Decagon B to Decagon A is 100. Therefore, the area A_B of Decagon B is 100 times the area A_A of Decagon A.
A_B/A_A=100
⇕
A_B=100 * A_A
