Question

Which of the following is the approximate measure of one of the sides of the Pentagon?

Answer 1

5) The circumcircle is the circle we can draw around the pentagon, where the pentagon is inside the circle.

The diameter of that circumcircle is 483.28 meters.

We have to find the length of one of the sides.

The diameter can be converted to a radius dividing it by 2:

`r=(D)/(2)=(483.28)/(2)=241.64`

We will need to also know the measure of the interior angle of the pentagon.

For regular polygons we can calculate it as:

`\begin{gathered} \alpha=((n-2))/(n)\cdot180\degree \\ \alpha=((5-2))/(5)\cdot180\degree \\ \alpha=(3)/(5)\cdot180\degree \\ \alpha=108\degree \end{gathered}`

We can then draw a right triangle that will let us calculate the side length (a):

As the radius R bisects the interior angle, we have an angle in the triangle that is half the interior angle: 108/2 = 54°.

Then, we can use trigonometric ratios to find a:

`\begin{gathered} \cos(54\degree)=(Adjacent)/(Hypotenuse)=(a\/2)/(R) \\ R\cdot\cos(54\degree)=a\/2 \\ a=2R\cdot\cos(54\degree) \\ a\approx2\cdot241.64\cdot0.5877 \\ a\approx284.06 \end{gathered}`

Answer: 284.06 meters.