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Trigonometry

(a)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.5: The Binomial Theorem

Problem 15E

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Question

A regular n-gon is inscribed in a circle of radius r.
(a) Derive a formula for the area of the regular n-gon in terms of n and r.
Use a trigonometric function and radian angle measure.
Hint: A regular n-gon inscribed in a circle of radius r can be partitioned into n isosceles triangles.
An inscribed regular 10-gon (decagon) is shown as an example.

Two-dimensional figure measured in terms of radius. It is formed by a set of points that are at a constant or fixed distance from a fixed point in the center of the plane. The parts of the circle are circumference, radius, diameter, chord, tangent, secant, arc of a circle, and segment in a circle.

Expert Solution

Step 1

Observe that complete 360 degree angle at the centre of the circle is divided into n parts. Thus to calculate the altitude, consider just one of the triangles of the n-gon as shown:

360
r, radius
n
В
A
CB altitude rsin(360/n)

Step 2

In triangle OCB, OB is the hypotenuse and CB is the perpendicular, thus:

360°
ВС
= sin
ОВ
п
3600
Altitude BC =r sin
п

Step 3

Area of triangle OAB is:

1
-x OA x BC
2
Area of triangle OAB
360°
1
Exrxrsin
2
360°
-sin
2
п

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