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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 5.1, Problem 28E

(a)

To determine

**To show:**

The expression of area of a polygon as

Expert Solution

The area of a polygon

**Given information:**

The area of the polygon is *n*-equal sides).

Polygon has *n* congruent triangles with central angle

**Calculation:**

Procedure to draw one of the *n* congruent triangles in a circle is explained below:

- Let
*O*be the center of the circle and*AB*be the one of the sides of the polygon. - Draw the radius
*OC*, so as to bisect the angle*AOB*. Also*OC*intersects*AB*at right angles and bisects*AB*.

Sketch one of the *n* congruent triangles in a circle with central angle

Refer Figure 1.

The angle *AOB* is divided into 2 right angle triangles with legs of length

Therefore, the triangle

Rearrange Equation (1) as shown below.

The area of the polygon with *n* equal sides

Substitute

Therefore, the area of a polygon with *n* equal sides

Therefore, The expression of area of a polygon as

(b)

To determine

**To show:**

The value of

Expert Solution

The value of

Refer part (a).

The area of a polygon with *n* equal sides is

Find the value of

Substitute

Let

As

Therefore, the value of