Question

solve:
talia made a pattern for a pot - holder that is a regular 12 - sided polygon. the perimeter of the pot - holder pattern is 36 in. the distance to the center is 6 in. to the nearest in. what is the area of the pot - holder to the nearest in²?
a house number is displayed on a plaque in the shape of a regular 7 - sided polygon. the area of the plaque is 70 in². the perpendicular distance from a side to the center is 5 in. to the nearest inch. what is the perimeter of the plaque?

Explanation:

Step1: Recall the formula for the area of a regular polygon

The formula for the area \(A\) of a regular polygon is \(A=\frac{1}{2}Pa\), where \(P\) is the perimeter and \(a\) is the apothem (the distance from the center to a side).

Step2: Solve the first problem

Given \(P = 36\) in and \(a=6\) in. Substitute into the formula:
\(A=\frac{1}{2}\times36\times6\)
\(A = 18\times6\)
\(A=108\) in²

Step3: Solve the second problem

Given \(A = 70\) in² and \(a = 5\) in. Use the formula \(A=\frac{1}{2}Pa\) to solve for \(P\).
First, rewrite the formula for \(P\): \(P=\frac{2A}{a}\)
Substitute \(A = 70\) and \(a = 5\):
\(P=\frac{2\times70}{5}\)
\(P=\frac{140}{5}\)
\(P = 28\) in

Answer:

1. The area of the pot - holder is \(108\) in².
2. The perimeter of the plaque is \(28\) in.