A geometry problem by Kulsoom Rizvi
Geometry
Level
4
The diagram shows two identical circles of radius 5cm touching each other externally and a larger circle, centre O and radius 15cm, touching internally. calculate the area of the shaded region correct to 1 decimal place.
The answer is 22.1.
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1 solution
The three centers form an equilateral triangle of sides 10 cm. So the sector of interest from big circle has an area of + 3 6 0 6 0 ∗ π ∗ 1 5 2 .
The equilateral triangle is not shaded. So its area − 4 3 ∗ π ∗ 1 0 2
Sectors each of the small circles that is not shaded makes 120^o at their centers. So their areas − 2 ∗ 3 6 0 1 2 0 ∗ π ∗ 5 2 .
Shaded area = + 3 6 0 6 0 ∗ π ∗ 1 5 2 − 4 3 ∗ π ∗ 1 0 2 − 2 ∗ 3 6 0 1 2 0 ∗ π ∗ 5 2 = 2 2 . 1 4 8