Atypical Yin-Yang

Geometry Level 2

The image above consists of intertwining semi-circles of increasing radii with equal spacing, which are all enclosed by a big circle.

Which colored region, red or blue, has a larger area?

Blue Both have the same area Not enough information Red

Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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1 solution

To make the image easier to perceive, let us split the figure into two halves as the following:

Now suppose the radius of the smallest semi-circle = r. We can see that the blue half donut-shaped region on the left is the difference of semi-circles of radius 3 r 3r and r r = π 2 [ ( 3 r ) 2 − r 2 ] = 4 π r 2 \dfrac{\pi}{2}[(3r)^2 - r^2] = 4\pi r^2 .

Meanwhile, on the right, the blue region = π 2 [ ( 4 r ) 2 − ( 3 r ) 2 + ( 2 r ) 2 − r 2 ] = π 2 ( r 2 ) ( 16 − 9 + 4 − 1 ) = 5 π r 2 \dfrac{\pi}{2}[(4r)^2 - (3r)^2 + (2r)^2 -r^2] = \dfrac{\pi}{2}(r^2)(16 - 9 + 4 - 1) = 5\pi r^2 .

Thus, the total blue area = 4 π r 2 + 5 π r 2 = 9 π r 2 4\pi r^2 + 5\pi r^2 = 9\pi r^2 .

For the red region, the area is simply the difference of the circles of radii 5 r 5r and 4 r 4r = π [ ( 5 r ) 2 − ( 4 r ) 2 ] = 9 π r 2 \pi[(5r)^2 - (4r)^2] = 9\pi r^2

Therefore, both regions have the same area.

6 Helpful 1 Interesting 1 Brilliant 1 Confused

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