A Geometric Limit Problem

t

28

°

T(θ)

S(θ)

=

sin(θ)(1-cos(θ))

2

1

∫

cos(θ)

1-

2

x

x

= 0.740425

The following constitutes a popular problem in calculus courses. Consider the sector of a unit circle with angle

θ

with the measurements as shown above. Let

T

be the area of triangle

ABC

, while

S

is the area of the curved shape

ABC

. What is the limit of the ratio

T/S

as the angle

θ

approaches 0? The answer is 3/4. The ratio

T/S

takes the value

2/π

when

θ=π/2

and

2(

2

-1)(π-2)

when

θ=π/4

.