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# How big is the circular ring?

2021-10-17T15:46:01.109Z

On a circular ring is a ten centimeter long rod that touches both edges of the ring without protruding beyond it. What is the area of ​​the circular ring? Enlarge image

Photo: DER SPIEGEL

This week's puzzle is comparatively easy, as long as you are not at war with geometry.

A circular ring is given.

We do not know its dimensions.

But we know that a ten centimeter long rod fits exactly on the circular ring.

In such a way that it touches the inner edge of the ring - and that the two ends of the rod touch the outer edge without protruding beyond the ring - see sketch above.

So that we can calculate better, we assume that the rod has a diameter of zero.

How big is the area of ​​the annulus?

The area is Pi * 25 cm2.

The area of ​​the annulus is calculated using the formula Pi * (R2 - r2).

The radii R, r and half the rod length 5 form a right-angled triangle - see the following sketch.

The Pythagorean theorem solves the problem of the two unknowns r and R. The following applies:

52 + r2 = R2

And thus:

R2 - r2 = 25

Accordingly, Pi * (R2 - r2) = Pi * 25.

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