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Perfectly arranged semicircles

2022-09-25T05:40:18.738Z

Three semicircles touch each other. The size of two semicircles is known - how big is the yellow area?



Photo:

THE MIRROR

A semicircle has the area of ​​2*Pi.

To his right is a semicircle with an area of ​​Pi/2.

A third semicircle touches both semicircles from the top right – see drawing.

What is the area of ​​the semicircle colored yellow in the drawing?

The yellow semicircle has an

area of ​​Pi

.

The blue semicircle has a diameter of 2, and the orange semicircle has a diameter of 1. (A semicircle has an area of ​​Pi*r2/2, where r is the radius.)

We connect the midpoints of the two lower semicircles to the points where the yellow semicircle touches them - see sketch below.

These connections are perpendicular to the straight outer edge of the yellow semicircle, which is its diameter.

Therefore both connections are parallel to each other.

If we connect the center of the left link to the center of the orange semicircle, we get a right triangle.

We know two sides of this triangle.

The length of the leg on the left is 1, the length of the hypotenuse is 3. The second leg corresponds to the diameter of the yellow semicircle whose size is being sought.

Using the Pythagorean theorem, we can easily calculate the diameter - it is sqrt(9 - 1) = sqrt(8).

The radius is then root(8)/2.

The area of ​​the yellow semicircle is therefore Pi*8/4*1/2 = Pi.

I discovered this easy geometry puzzle on Twitter from the Swiss mathematician Diego Rattaggi's account.

In case you missed a mystery from the past few weeks, here are the most recent episodes:

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  • water march!

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  • The card trick

  • How many elevators are needed?

  • Rook versus bishop

  • Beetles in love

  • John Conway's super number

Source: spiegel

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