Question

a community pool that is shaped like a regular pentagon needs a new cover for the winter mont. the radius of the pool is 20.10 ft. the pool is 23.62 ft on each side. to the nearest square foot, what is the area of the pool that needs to be covered? 192 ft² 960 ft² 1,921 ft² 3,842 ft²

Explanation:

Step1: Find the central angle of each sector

The central angle of a regular pentagon is \(\theta=\frac{360^{\circ}}{n}\), where \(n = 5\) (number of sides). So \(\theta=\frac{360^{\circ}}{5}=72^{\circ}\)

Step2: Find the area of one isosceles triangle

The formula for the area of a triangle with two sides \(a\) and \(b\) and included angle \(\theta\) (in degrees) is \(A=\frac{1}{2}ab\sin\theta\). Here \(a = b=\text{radius}=r = 20.10\) ft and \(\theta = 72^{\circ}\). So the area of one triangle is \(A_{1}=\frac{1}{2}\times20.10\times20.10\times\sin(72^{\circ})\)

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Step3: Find the area of the pentagon

Since the pentagon is composed of \(n = 5\) such congruent isosceles triangles. The area of the pentagon \(A=n\times A_{1}\)
\(A = 5\times192.558=962.79\approx960\) (rounded to the nearest ten)

Answer:

\(960\mathrm{ft}^{2}\)