March 14, 2005
Problem of the Month

Three Tangent Circles

In honor of PI DAY (3/14), you will need to use pi in your solution of this week's problem.

There are three circles, each tangent to the other two.

The radius of circle A is 6 inches.

The radius of circle B is 4 inches.

The radius of circle C is 2 inches.

What is the area of the shaded region in the diagram below?

Give your answer correct to three decimal places.

Solution to the Problem:

The shaded area = 1.857 square inches.

The following facts were used to solve this problem:

• The Pythagorean Theorem.
• The area of a sector of a circle equals the number of degrees
  in the central angle divided by 360° times the area of the circle
  (Area of a circle = pi * the square of the radius).
• The area of a triangle is 1/2 times the base times the height.
• The acute angle of a right triangle is equal to the inverse tangent of the ratio of the leg opposite the angle and the leg adjacent to the angle.

First, draw segments connecting the centers of the circles.
Then AB = 10 inches, BC = 6 inches, and AC = 8 inches.
This forms a right triangle since 62 + 82 = 102.
Therefore, angle C is the right angle.

Observe that the shaded area (in red in the diagram above) is equal to the area of triangle ABC minus the areas of the three sectors (these are shown as tan, grey, and cyan in the diagram below).

The area of triangle ABC = 1/2 * 6 * 8 = 24 square inches.

The measure of angle C has been determined to be 90°,
so sector C is 1/4 of the area of circle C (since 90 / 360 = 1/4).
The area of sector C = 1/4 * pi * 22 = pi = 3.1419 square inches.

To determine angle A: A = Tan-1 (6/8) = 36.8699°.
The area of sector A = 36.8699/360 * pi * 6 2 = 11.583 square inches.

To determine angle B: B = Tan-1 (8/6) = 53.130°.
The area of sector B = 53.130/360 * pi * 4 2 = 7.41836 square inches.

Hence, the shaded area = 24 - 3.14159 - 11.582 - 7.41836 = 1.857 square inches.

Correctly solved by:

1. Keith Mealy	Cincinnati, Ohio
2. Jim Arrison	Norristown, Pennsylvania
3. Arsalan Heydarian	Harrisonburg, Virginia
4. William Proffitt	Winchester, Virginia
5. James Alarie	University of Michigan -- Flint, Flint, Michigan
6. Jeffrey Gaither	Winchester, Virginia
7. Nathan Seifert	Harrisonburg, Virginia
8. Wajih Ansari	Harrisonburg, Virginia
9. Jaime Garcia	Virginia Tech, Blacksburg, Virginia
10. James Simmons	Harrisonburg, Virginia
11. Walt Arrison	Philadelphia, Pennsylvania
12. Rich Murray	Ridgetown, Ontario, Canada
13. Michael Singer	Winchester, Virginia
14. António Baptista	Lisboa, Portugal
15. Tom Robb	Winchester, Virginia
16. Carrie Wilson	Ticonderoga, New York
17. Daniel Surber	Winchester, Virginia
18. Trey Mason	Winchester, Virginia
19. Tristan Collins	Winchester, Virginia
20. Misty Carlisle	Winchester, Virginia
21. Kailen Ko	Winchester, Virginia

Send any comments or questions to: David Pleacher