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Lunes

Problem

A square is inscribed in a circle with diameter 2. Four smaller circles are then constructed with their diameters on each of the sides of the square.

Find the shaded area.


Solution

Consider the diagram:

This is best tackled by working towards the area of a single lune, but this needs to be done in a series of careful additive and subtractive steps.

The area of the right angle triangle is 1/2.

The area of the large circle is π times 12 = π, so the area of the (shaded) quarter circle is π/4.

Therefore the area of the shaded segment is π/4 minus 1/2.

Using the Pythagorean Theorem, the length of the square's diagonal, d = radical2. So the radius of the smaller circle is radical2/2 = 1/radical2.

Therefore the area of one small circle is π times (1/radical2)2 = π/2 and so the area of the shaded semi-circle will be π/4.

Hence the area of one lune is π/4 minus (π/4 minus 1/2) = 1/2 (surprising eh?)

So the shaded area of the original shape is 2.

What would be the shaded area if the same construction was performed on the edges of an inscribed equilateral triangle?

What about other inscribed regular polygons?

Problem ID: 20 (Oct 2000)     Difficulty: 3 Star

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