Question

What is the area of intersection of 2 circles of radius 6 cm whose centers are 8 cm apart?

Answer 1

Answer:

Label the center of the first circle CC and the center of the second circle C′C′. Label one of the points of intersection of the two circles AA and the other BB. Let the radius of the circles be r>0r>0. It should be clear that the following lengths are all equal to rr. ACAC, AC′AC′, BCBC, BC′BC′, CC′CC′. With a simple application of Pythagoras' Theorem, we get that the length of the line segment ABAB is 3–√r3r.

With some basic trigonometry, we find the angles ∠ACB=∠AC′B=2π3∠ACB=∠AC′B=2π3. So, the area of one half of the intersection is the area of a circular segment with angle θ=2π3θ=2π3 and radius rr, which gives an area of r22(θ−sinθ)=r22(2π3−3–√2)r22(θ−sin⁡θ)=r22(2π3−32) and so the area of the entire intersection is twice this. This gives an area of

r2(2π3−3–√2).

Step-by-step explanation:

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