Question

a model of a soccer ball is made up of regular pentagons and hexagons. the side length of one of the pentagons measures 2 inches and the apothem measures about 1.38 inches. what is the area of one of the pentagons? state your answer to the nearest tenth. square inches the side length of one of the hexagons measures 2 inches and the apothem measures about 1.73 inches. what is the area of one of the hexagons? state your answer to the nearest tenth. square inches

Explanation:

Step1: Calculate the area of the pentagon

The formula for the area of a regular polygon is \(A=\frac{1}{2}aP\), where \(a\) is the apothem and \(P\) is the perimeter.
For a regular pentagon with side length \(s = 2\) inches, the perimeter \(P=5s\). So \(P = 5\times2=10\) inches, and \(a = 1.38\) inches.
Then \(A_{pentagon}=\frac{1}{2}\times1.38\times10\)
\(A_{pentagon}=6.9\) square inches.

Step2: Calculate the area of the hexagon

For a regular hexagon, it can be divided into six equilateral triangles. The formula for the area of a regular hexagon is also \(A=\frac{1}{2}aP\).
The side length \(s = 2\) inches, so the perimeter \(P = 6s=6\times2 = 12\) inches, and \(a = 1.73\) inches.
Then \(A_{hexagon}=\frac{1}{2}\times1.73\times12\)
\(A_{hexagon}=10.4\) square inches.

Answer:

The area of the pentagon is \(6.9\) square inches.
The area of the hexagon is \(10.4\) square inches.