a To calculate the area of an equilateral triangle, use the formula A= s^2sqrt(3)4.
B
b The area of a square can be calculated by raising the length of its side to the power of 2.
C
c Use the fact that each hexagon consists of 6 equilateral triangles.
D
d Note that the side lengths are decreasing as the number of sides in the polygons increases.
A
a ≈ 9.1 inches
B
b 6 inches
C
c 3.7 inches
D
d 4 inches, see solution.
Practice makes perfect
a It is given that the area of the regular polygon is 36in^2. It has 3 equilateral sides, so the polygon is an equilateral triangle. We can calculate its area using the following formula.
A=s^2sqrt(3)/4
Here, s is the length of the triangle's side. Let's substitute 36 for A into the formula and solve the equation for s.
c Now, let's find the length of a side of a regular polygon that has 6 sides, which is called a hexagon.
Notice that every hexagon contains of 6 equilateral triangles.
Dividing the area of the hexagon, which we know is 36in^2, we can find the area of one such triangle.
\begin{aligned}
A_\text{triangle}=\dfrac{36}{6}=6\text{ in}^2
\end{aligned}
In Part A, we already reviewed the formula for the area of an equilateral triangle.
A=s^2sqrt(3)/4
Let's substitute A with 6 and calculate the length of the triangle's, and respectively the hexagon's, side.
Therefore, each side of the hexagon is about 3.7 inches long.
d To estimate the length of the pentagon's side, let's start with gathering all the information that we have found.
\begin{aligned}
s_\text{triangle}&=9.1 \\
s_\text{square}&=6 \\
s_\text{pentagon}&={\color{#FF0000}{?}} \\
s_\text{hexagon}&=3.7
\end{aligned}
As we can see, the side lengths are decreasing as the number of sides in the polygons increases. Hence, the length of the side of the pentagon is between 6 and 3.7 inches.
