Review question

# What's the shared area for these two circles? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6838

## Solution

Two coplanar circles, each of radius $\quantity{5}{cm}$, have their centres $\quantity{6}{cm}$ apart. Calculate the area of the region common to the interiors of both the circles, giving your answer in $\mathrm{cm}^2$ correct to two significant figures.

To calculate the shaded area, we’ll compute the area of the circle sector $ACE$, before subtracting the area of the triangle $ACD$, and then multiplying by $4$.

The length $AD$ must be $3$, and so by Pythagoras, the length of $CD$ must be $4$. Thus $\angle CAD$ is $\arctan \dfrac{4}{3}$.

Thus the area of the sector $ACE$ (working in radians) is $0.5 \times 5^2 \times \arctan \dfrac{4}{3}$, while the area of triangle $ACD$ is $0.5 \times 4 \times 3 = 6$.

So the shaded area is $4(12.5 \arctan \dfrac{4}{3}-6) = 22.36\dots$, which is $\quantity{22}{cm^2}$ when given to two significant figures.