Question

In the figure, the $3$ circles have equal radii and are tangent to each other and to the sides of the rectangle. The width of the rectangle is $20$ feet long. What is the area of the shaded (red) region outside and enclosed by all three circles?

• A. $\frac{50\sqrt{3}}{2}$
• B. $25\pi$
• C. $2\sqrt{3}+25\pi$
• D. $50\sqrt{3}-25\pi$
• E. $\frac{25(2\sqrt{3}-\pi )}{2}$

Answer 1

Answer: (E)

$\frac{25(2\sqrt{3}-\pi )}{2}$

The above diagram is reproduced below where the centers of the circles are connected to make an isosceles triangle as shown below.

The radius $r$ of each circle is given by

$4r=20$ that is $r=5$

The area of the enclosed region (red) is equal to the area of the equilateral triangle of side $10$ minus $3$ times the area of one sector.

Area of triangle $=\left(\frac{1}{2}\right)\sin \left(t\right)\times 10\times 10=50\sin \left(60\right)=25\sqrt{3}$

Area of one sector $=\left(\frac{1}{2}\right)\times r^2\times t=\frac{1}{2}\times 25\times \frac{\pi }{3}$

The area of the enclosed region (red) is equal to

$25\sqrt{3}-3(\frac{1}{2}\times 25\times \frac{\pi }{3})=\frac{25(2\sqrt{3}-\pi )}{2}$

Hence, the area of the shaded region outside and enclosed by all three circles is $\frac{25(2\sqrt{3}-\pi )}{2}$.