A segment of a circle has a 120 arc and a chord of 8 √3 in. Find the area of the segment.

Solution:

Here the area of segment = Area of the sector - Area of triangle

Consider the triangle from the figure to determine the radius.

Since the two right triangles have the same radii, a common side and the same 90° angle, by cpctc they are congruent.

Thus BM = AM = 4√3

From the definition of trigonometric ratios,

We know that cos 30° = adjacent /hypotenuse

√3/2 = 4√3/r

r = 4√3/√3/2

r = 8 units²

Area of a Sector of Circle = (θ/360º) × πr²  unit²

= (120º/360º) × π × 8² 

= 1/3 × 3.14 × 64

Area of the sector = 66.98 unit² 

Area of triangle = ½ × ab × Sin C

Where a and b are the two side lengths and C is the angle included between them.

Here radii are the two sides and 120° is the angle between them.

Thus the area of the triangle = ½ × 64 × Sin 120°

=32× √3/2 = 16√3

Area of the triangle = 27.71 unit² 

Area of the segment = Area of the sector - Area of triangle

Substituting the values

=  66.98 - 27.71

= 39.27 unit²

Therefore, the area of the segment is 39.27 unit².

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A segment of a circle has a 120 arc and a chord of 8 √3 in. Find the area of the segment.

Summary:

A segment of a circle has a 120 arc and a chord of 8 √3 in. The area of the segment is 39.27 unit².