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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

(a) Let An be the area of a polygon with n equal sides inscribed in a circle with radius r. By dividing the polygon into n congruent triangles with central angle 2π/n, show that

A n = 1 2 n r 2 sin ( 2 π n )

(b) Show that limn → ∞ An = πr2. [Hint: Use Equation 3.3.2 on page 191.]

(a)

To determine

To show:

The expression of area of a polygon as 12nr2sin(2πn).

Explanation

Given information:

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

Polygon has n congruent triangles with central angle 2πn.

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 2πn as shown in Figure 1.

Refer Figure 1.

The angle AOB is divided into 2 right angle triangles with legs of length 12(AB)=rsin(πn)rcos(πn)

Therefore, the triangle AOB has area of 212[rsin(πn)rcos(πn)] (1)

Rearrange Equation (1) as shown below.

AreaofΔAOB=[rsin(πn)rcos(πn)]=r2sin(π

(b)

To determine

To show:

The value of limnAn=πr2

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