Areas and Tangent Circles

After many questions about trigonometry, calculus, and parabolas, we finally get a fresh new question about geometry. This question deals with one of the most hated figures in the world of geometry: circles.

Two circles with radii of 5 cm each are externally tangent to each other. Each of these two circles is internally tangent to a circle with a radius of 15 cm. What is the perimeter and area of the shape enclosed between the three circles?

To get the idea of what it looks like, try to draw it. You will eventually see what this whole question is about, and that will make the question much easier for you.

First, we need to get the perimeter. When you connect all the centers of the circles, you get an equilateral triangle. This happens because the side lengths are 5 + 5 and 15 - 5 (externally tangent circles and internally tangent circles, respectively). This triangle is the key to the whole question.

There is a theorem saying that the centers of two tangent circles and the point of tangency all lie on the same line. This means you can extend two sides of the equilateral triangle to create a sector of the big circle. This sector's area and perimeter are 60°/360° = 1/6 of the area and the perimeter of the circle.

Since these extensions go through the centers of the small circles as well, they split them up to 60° and 120° angles (the 60° angle is formed by the equilateral triangle, and the 120° angle is its supplement). Now we can calculate both the perimeter and the area of the shape between the three circles.

The perimeter is given by adding the three arcs which form the shape. Each small circle gives an arc of 120°, and the big circle gives an arc of 60°. Let's find these arc lengths.

Small circles:
2π * 5 * 120°/360° = 10π/3

Large circle:
2π * 15 * 60°/360° = 30π/6 = 5π

There are two small circles, so the total perimeter is:
5π + 2 * 10π/3 = 15π/3 + 20π/3 = 35π/3 cm
This equals about 36.65 cm.

As for the area, we can calculate the area of the sector of the large circle, subract the area of the equilateral triangle, and subtract the area of the two sectors of the small circles. This will give us the area of the shape we want.

Area of the big sector:
π * 15² / 6 = 225π / 6 = 37.5π

Area of the equilateral triangle:
10²√3 / 4 = 25√3

Area of a small sector:
π * 5² / 3 = 25π/3

Remember there are two small sectors, so we can now calculate:
37.5π - 25√3 - 2 * 25π/3 = 225π/6 - 25√3 - 100π/6 = 125π/6 - 25√3 cm²
This equals about 22.15 cm²

That's it for today. Hope you liked it. Visit super math tips for more cool information like that.

Yours,
Nadav

nadavs