Q.335. Common area of overlapping circles under given conditions.

Question 335.
Two circles of radii 5cm and 12cm are drawn partly overlapping. Centres 13cm apart. Find common area.

Answer 335.
5^2 + 12^2 = 13^2
=> The centers of the circle and their point of intersection form a right triangle.

Angle subtended by the arc of smaller circle at its center
= 2 arccos (5/13)
=> area of the sector formed by the arc subtending this angle at the center
= (1/2) * [2 arccos (5/13)] * 5^2
= 29.40 sq. units

Angle subtended by the arc of larger circle at its center
= 2 arccos (12/13)
=> area of the sector formed by the arc subtending this angle at the center
= (1/2) * [2 arccos (12/13)] * (12)^2
= 56.85 sq. units

Area of the kite formed by the centers and the intersecting points
= 5 * 12
= 60 sq. units

=> common area
= 29.40 + 56.85 - 60
= 26.25 sq. units.

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