Mathematics

Problem

89. Limits and the Area of a Circle Consider an

n

-sided regular polygon made up of

n

congruent isosceles triangles, each with height

h

and base

b

. The figure shows an 8 -sided regular polygon. ![](https://storage.googleapis.com/diraq-contents/question-bank/question-images/0afb62dc8150/2025_11_21_d3353d45a3ce506cf2ecg-10.jpg) (a) Show that the area of an 8-sided regular polygon is

A=4 h b

and the area of the

n

-sided regular polygon is

A=(1 / 2) n h b

. (b) Show that the base

b

of the

n

-sided regular polygon is

b=2 h \tan (180 / n)^{\circ}

. (c) Show that the area

A

of the

n

-sided regular polygon is

A=n h^{2} \tan (180 / n)^{\circ}

. (d) Let

h=1

. Construct a table of values for

n

and

A

for

n=4,8,16,100,500,1000,5000,10000,100000

. Does

A

have a limit as

n \rightarrow \infty

? (e) Repeat part (d) with

h=3

. (f) Give a convincing argument that

\lim _{n \rightarrow \infty} A=\pi h^{2}

, the area of a circle of radius

h

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