Chapter 11.4, Problem 1CYP

To determine

To calculate: The center, radius, an apothem and a central angle of the polygon. Also, to find the measure of the central angle

Answer to Problem 1CYP

Center is $R$.

Radius is $RK\text{ or }RL$.

Apothem is $RS$.

Central angle is $\angle KRL$.

The measure of JK is $\angle KRL={60}^{\circ }$.

Explanation of Solution

Given information: A regular hexagon is inscribed in a circle R.

Calculation:

A regular hexagon is inscribed in a circle R. Therefore, the radius and the center of the circle are the radius and center of the hexagon.

Thus, the radius and center of the hexagon are same as that of the circle.

That is,

Center is $R$.

Radius is $RK\text{ or }RL$.

It can be observed from the figure that KL is one side of the hexagon and RS is the perpendicular that origins from the center of the hexagon and meets the side of the hexagon. Therefore, the segment $\stackrel{\rightharpoonup }{RS}$ is the apothem.

Apothem is $RS$.

The vertex of a central angle lies at the center of the hexagon. The central angle is such that the sides of the angle passes through consecutive vertices of the given hexagon.

The $\angle KRL$ fulfils all these conditions.

Hence, the central angle is $\angle KRL$.

Calculate the measure of the central angle.

The given polygon is a hexagon. Therefore, total number of sides is 6.

The measure of the central angle can be obtained by dividing ${360}^{\circ }$ by the number of sides, that is, 6 in this case.

  $\begin{array}{c}\angle KRL=\frac{{360}^{\circ }}{6}\\ ={60}^{\circ }\end{array}$

Thus, the measure of $\angle KRL={60}^{\circ }$.