WOLFRAM|DEMONSTRATIONS PROJECT

Minimum Area between a Semicircle and a Rectangle

angle (radians)

1.0472

answer

Calculate the shaded area.

60.°

0.0

0.5

1.0

1.5

angle

0.0

0.5

1.0

1.5

shaded

area

The figure shows a semicircle of radius one and a horizontal line parallel to the diameter. What central angle corresponds to the minimum shaded area? This problem can be solved by either integration or trigonometry. The shaded area is equal to

0.5(π-4θ+4sin(θ)-2sin(2θ)

, where

θ

is the central angle. The minimum shaded area occurs when the central angle is

π/3

.