Which circle, square, and triangle, have the closest area to the blue doughnut shape?

The drawings are to scale, so you might be able to judge by eye, or you can work out the actual areas.

Puzzle Copyright © Kevin Stone

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Answers

Circle: 35.

Square: 31.

Triangle: 47.

Doughnut

The area of a circle is π x Radius^{2}.

The larger circle has diameter = 40, therefore the radius is 20, and the area is π x 20^{2} = 400π.

The smaller circle has diameter = 20, therefore the radius is 10, and the area is π x 10^{2} = 100π.

Therefore the shaded area is 400π – 100π = 300π ≈ 942.

Circle

The area of a circle is π x Radius^{2}.

The circle with diameter 31 has a radius of 15.5 and an area of π x 15.5^{2} ≈ 755.

The circle with diameter 35 has a radius of 17.5 and an area of π x 17.5^{2} ≈ 962 (closest match).

The circle with diameter 39 has a radius of 19.5 and an area of π x 19.5^{2} ≈ 1195.

Square

The area of a square is Side x Side.

The square with side 31 has an area of 31 x 31 = 961 (closest match).

The square with side 35 has an area of 35 x 35 = 1225.

The square with side 39 has an area of 39 x 39 = 1521.

Triangle

The area of a triangle is ^{1}/_{2} x Base x Height. Using Pythagoras' theorem it can be shown that the area of an equilateral triangle is √3 x Base^{2} ÷ 4.

The triangle with side 39 has an area of √3 x 39 x 39 ÷ 4 ≈ 659.

The triangle with side 43 has an area of √3 x 43 x 43 ÷ 4 ≈ 801.

The triangle with side 47 has an area of √3 x 47 x 47 ÷ 4 ≈ 957 (closest match).

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