Chapter 6.1, Problem 37E

(a)

To determine

The area of the polygon for the n=4(square).

Answer to Problem 37E

The area of the square is 2.

Explanation of Solution

Given information:

The Area of a Circle is inscribed at a regular n-sided polygon inside a circle of radius 1.

Formula used:

  $\text{length of side of square}=\frac{\text{length of diagonal of square}}{\sqrt{2}}$

Calculation:

Since diameter of circle is diagonal of square.

Now,

  $\begin{array}{l}\text{length of side of square}=\frac{\text{length of diagonal of square}}{\sqrt{2}}\\ =\frac{2}{\sqrt{2}}\\ \text{length of side of square}=\sqrt{2}\end{array}$

So,

  $\begin{array}{l}\text{area of square}={\left(\text{length of side of square}\right)}^2\\ ={\left(\sqrt{2}\right)}^2\\ \text{area of square}=2\end{array}$

Conclusion:

The area of the square is 2.

(b)

To determine

The area of the polygon for the n=8(octagon).

Answer to Problem 37E

The total area of octagon is 2.828.

Explanation of Solution

Given information:

The Area of a Circle is inscribed at a regular n-sided polygon inside a circle of radius 1.

Formula used:

  $\text{length of base of traingle = }\sin \frac{\pi }{8}$

  $\text{length of height of traingle = }\cos \frac{\pi }{8}$

Calculation:

Octagon is a collection of 16 right angle triangles with radius of circle as hypotenuse. And one angle of triangle is

  $\frac{2\pi }{16}=\frac{\pi }{8}$

  $\text{length of base of traingle = }\sin \frac{\pi }{8}$

  $\text{length of height of traingle = }\cos \frac{\pi }{8}$

So,

  $\begin{array}{l}\text{Total area of octagon}=16\left(\frac{1}{2}\right)\sin \frac{\pi }{8}\cos \frac{\pi }{8}\\ =8\left(0.3827\right)\left(0.9239\right)\\ \text{Total area of octagon}=2.828\end{array}$

Conclusion:

The total area of octagon is 2.828

(c)

To determine

The area of the polygon for the n=16.

Answer to Problem 37E

The total area of octagon is 3.061.

Explanation of Solution

Given information:

The Area of a Circle is inscribed at a regular n-sided polygon inside a circle of radius 1.

Formula used:

  $\begin{array}{l}\text{length of base of traingle}=\sin \frac{\pi }{16}\\ \text{length of height of traingle}=\cos \frac{\pi }{16}\end{array}$

Calculation:

16-gon is a collection of 32 right angle triangles with radius of circle as hypotenuse. And one angle of triangle is

  $\frac{2\pi }{32}=\frac{\pi }{16}$

  $\begin{array}{l}\text{length of base of traingle}=\sin \frac{\pi }{16}\\ \text{length of height of traingle}=\cos \frac{\pi }{16}\end{array}$

So,

  $\begin{array}{l}\text{Total area of octagon}=32\left(\frac{1}{2}\right)\sin \frac{\pi }{16}\cos \frac{\pi }{16}\\ =8\left(0.1951\right)\left(0.9808\right)\\ \text{Total area of octagon}=3.061\end{array}$

Conclusion:

The total area of octagon is 3.061

(d)

To determine

The comparison of the area for different n values.

Answer to Problem 37E

Each area is smaller than area or circle as n increases towards infinity area approaches to area or circle.

Explanation of Solution

Given information:

The Area of a Circle is inscribed at a regular n-sided polygon inside a circle of radius 1.

Formula used:

  $\text{area}\text{of}\text{circle}=\pi {\left(r\right)}^2$

Calculation:

Area of circle is calculate as

  $\begin{array}{l}\text{area}\text{of}\text{circle}=\pi {\left(1\right)}^2\\ \text{area}\text{of}\text{circle}=3.1415\end{array}$

Each area is smaller than area or circle as n increases towards infinity area approaches to area or circle.

Conclusion:

Each area is smaller than area or circle as n increases towards infinity area approaches to area or circle.