Question

Two circles of radii 5 cm and 12 cm are drawn, partly overlapping. Their centers are 13cm apart. Find the area common to the two circles.

Answer 1

I assume you made a sketch

On mine, I labeled the centre of the smaller circle P and the centre of the larger circle Q
There is a common chord, I called that AB

Adding the areas of the two circles would count the overlap twice, so we have to subtract the overlap.

The overlap consists of a segment in the smaller and a segment in the larger circle with the common chord AB
We need some angles.
Consider triangle APQ
We know we have sides 5, 13 and 12

AHHH, but 5^2 + 12^2 = 13^2, so we have a right angle at A !
tan(APQ) = 13/5
angle APQ = 68.96..°
so central angle P = 137.925°

find the area of sector APQ:

area APQ/137.925.. = 25π/360
area of sector APQ = 30.09...

area of triangle APB
= (1/2)(5)(5)sin 137.925
= 8.376..

area of <b>segment</b> in smaller circle
= 30.09... - 8.376.. = 21.714.... ***

Now find the segment in the larger circle and you have the overlap.
We already know that the central angle at Q
= 2(90-68.96) = 42.075..°
So proceed from there.
Once you have the segment area of ABQ, add that to ***
and finally subtract that sum from the addition of the two circle areas.