Chapter 11.4, Problem 34PPS

a)

To determine

Area of different polygons inscribed in a Circle.

Answer to Problem 34PPS

  $\begin{array}{l}\text{n = number of sides of polygon}\\ \text{(1) Polygon Side : a = }2rSin\frac{\pi }{2}\\ {\text{(2) Polygon Area : S}}_p=\frac{1}{2}n{r}^{2}Sin\frac{2\pi }{n}\\ {\text{(3) Circle Area : S}}_c=\pi r^2\\ \text{(4) Interior Angle of Polygon : }\frac{180(n-2)}{n}\\ \text{where}\\ \text{n is the number of sides}\end{array}$

Explanation of Solution

Given:

• Square inscribed in a Circle.
• Pentagon inscribed in a Circle.
• Hexagon inscribed in a Circle.

Calculation:

Formula/Relationship

  $\begin{array}{l}\text{n = number of sides of polygon}\\ \text{(1) Polygon Side : a = }2rSin\frac{\pi }{2}\\ {\text{(2) Polygon Area : S}}_p=\frac{1}{2}n{r}^{2}Sin\frac{2\pi }{n}\\ {\text{(3) Circle Area : S}}_c=\pi r^2\\ \text{(4) Interior Angle of Polygon : }\frac{180(n-2)}{n}\\ \text{where}\\ \text{n is the number of sides}\end{array}$

b)

To determine

To find the value of interior angles of polygons.

Answer to Problem 34PPS

Number of Sides (n)	Interior Angle (x)	Area of Inscribed Polygon
4	90	200
5	108	237.76
6	120	259.81
8	135	282.84
10	144	293.89
20	162	309.01
50	172.8	313.33
100	176.4	313.95

Explanation of Solution

Given:

• Square inscribed in a Circle.
• Pentagon inscribed in a Circle.
• Hexagon inscribed in a Circle.
• Let the Radius = 10

Calculation:

  $\begin{array}{l}\text{n = number of sides of polygon}\\ \text{(1) Polygon Side : a = }2rSin\frac{\pi }{2}\\ {\text{(2) Polygon Area : S}}_p=\frac{1}{2}n{r}^{2}Sin\frac{2\pi }{n}\\ {\text{(3) Circle Area : S}}_c=\pi r^2\\ \text{(4) Interior Angle of Polygon : }\frac{180(n-2)}{n}\\ \text{where}\\ \text{n is the number of sides}\end{array}$

Conclusion:

Number of Sides (n)	Interior Angle (x)	Area of Inscribed Polygon
4	90	200
5	108	237.76
6	120	259.81
8	135	282.84
10	144	293.89
20	162	309.01
50	172.8	313.33
100	176.4	313.95