Triple Tangent Circles

Geometry Level 3

Three circles, each with a radius of $10$ , are mutually tangent to each other. The area enclosed by the three circles can be written as $a\sqrt{b} - c\pi$ , where $a$ , $b$ and $c$ are positive integers, and $b$ is not divisible by a square of a prime. What is the value of $a + b + c$ ?

Details and assumptions

The enclosed area does not include the area within the circles.

The answer is 153.

1 solution

Arron Kau Staff
May 13, 2014

We connect the center of the circles and form an equilateral triangle with length $10 + 10 = 20$ . The area of the triangle formed is $\frac{1}{2}\cdot 20 \cdot 20 \cdot \sin 60^\circ = 100 \sqrt{3}$ . The area that is within this triangle and within the circles is equal to $3\cdot \left(\frac{\pi \cdot 10 ^2 \cdot 60^\circ}{360^\circ}\right) = 50\pi$ . Thus the area enclosed by the three tangent circles is $100\sqrt{3} - 50\pi$ . Hence $a + b + c = 100 + 3 + 50 = 153$ .

Triple Tangent Circles

The answer is 153.

1 solution

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