Question

How long is each side of the Pentagon on the map?

Answer 1

Answer:

The apothem of the Pentagon is about 12.9 inches long. The area of a regular pentagon is equal to its perimeter times its apothem divided by 2. Therefore, the perimeter of the Pentagon on the map is about 94.4 inches. So, each side is about 18.9 inches long on the map.

Step-by-step explanation:

We know that this pentagon is a regular polygon, meaning we can divide this pentagon into five equal triangles. In order to find the length of the apothem, we need to find half the measure of the central angle of those triangles. Dividing 360° by 5, we get 72°. Next, divide 72° by 2 to get half the measure of the central angle, which is 36°. Then, we do: cos(36°) = $\frac{a}{16}$ or taking cos(36°) and multiplying it with 16, which equals 12.94427.

The formula to find the area of this polygon is: A = $\frac{ap}{2}$. This means the area of the polygon equals the apothem times its perimeter divided by 2. Using this formula, we can replace the variable with what we already know to solve for the perimeter.

608.7 = $\frac{12.9p}{2}$ ← Multiply both sides of the equation by its reciprocal $\frac{2}{12.9}$

p = 608.7 × $\frac{2}{12.9}$ ← Calculate 608.7 × $\frac{2}{12.9}$

p = $\frac{4058}{43}$ ≈ 94.73209

Because we know the perimeter is about 94.4 inches, we can divide that by 5 to find the length of the sides of the pentagon.

94.4 ÷ 5 = 18.88

Rounding 18.88 to the nearest tenths, is 18.9 inches.

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