Circular Disposition

Geometry Level 1

Each circle in the diagram below has a radius of r = 6 r = 6 . What is the total area of the shaded regions?

By this construction, the distance from the center of one circle to its adjoining circles is equal.

31 π 31\pi 34 π 34\pi 36 π 36\pi 39 π 39\pi

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Rajdeep Ghosh by Rajdeep Ghosh , 18, India

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3 solutions

Rajdeep Ghosh
Aug 26, 2016

Relevant wiki: Composite Figures

It would be a huge mistake to do any advanced calculations to solve this. The three lower oval shapes can be inserted into the three hollow spaces adjacent to the upper shaded region, and together, the four shapes will neatly form a single circle. The circle has r = 6, so area = 36 π \boxed{36\pi} .

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Short and sweet!

Akhash Raja Raam - 4 years, 9 months ago

@Rajdeep Ghosh - Can you justify the fact that the vertical "ovals" have the same area as the horizontal ones?

Eli Ross Staff - 4 years, 9 months ago

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Yes. Since the circles are of equal area and the distance from the centre of one circle to another is equal,their intersection has to be equal.

Rajdeep Ghosh - 4 years, 9 months ago
Mike Holden
Nov 18, 2017

The area of one circle is 36pi. The area of the shaded regions adds up to the area of one circle. So the area of the shaded regions is 36pi.

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Acecace Ace
Sep 11, 2016

Add all the shaded regions together, you will get a perfect area of the circle.

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