Four Tangent Circles

Category: Plane Geometry, Algebra

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Each of the four circles shown in the figure is tangent to the other three. If the radius of each of the smaller circles is x, find the area of the largest circle. If x = 2.71, what is the area of the largest circle?

Photo by Math Principles in Everyday Life

Solution:

When you connect the centers of the three small circles, the figure is an equilateral triangle. The center of an equilateral triangle will be the radius of a big circle as shown below

Photo by Math Principles in Everyday Life

Consider the equilateral triangle, draw the perpendicular lines from each vertex, and label further as follows

Photo by Math Principles in Everyday Life

In a 30° - 60° - 90° triangle, the shortest side is half of the hypotenuse which is ½ y. The other side of a triangle is equal to the shortest side times the square root of three. Therefore, we can solve for the value of y in terms of x as follows

or

The radius of a big circle is

Therefore, the area of a big circle is

If x = 2.71, then the area of a big circle is