How Do You Calculate the Area of a Shaded Segment of a Circle?

[xyz_1965]

Compute the area of the shaded segment of a circle. A segment of a circle is a region bounded by an arc of the circle and its chord. The radius r is given to be 3 cm and the central angle theta is 120°. Give two forms for the answer: an exact expression and a calculator approximation rounded to two decimal places.

Use: area of segment = (area of sector OPQ) - (area of triangle OPQ).

Area of sector OPQ = (1/2)r^2(theta).

Area of triangle OPQ = (ab/2)(sin (theta)).

Solution:

Central angle is theta.

= 120°

= 2•pi/3 rad

Area of the shaded segment

= (Area of the sector) - (Area of the triangle)

= [(1/2) × 3^2 × (2•pi/3) - (1/2) × 3^2 × sin(120°)] cm^2

= [3•pi - (9/2) × sin(180° - 60°)] cm^2

= [3•pi - (9/2) × sin(60°)] cm^2

= [3•pi - (9/2) × (sqrt{3}/2)] cm^2

= [3•pi - (9/4)sqrt{3}] cm^2

= (3/4)[4•pi - 3sqrt{3}] cm^2

= 5.53 cm^2

You say?

Note: All work is done on paper prior to posting.

 

[jvicens]

Great job on solving the problem and providing both an exact expression and a calculator approximation! Your solution is clear and well-organized, making it easy to follow. Keep up the good work!