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Dividing the Field

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Two Circles

Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?

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Star Gazing

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Bobbly Perimeter

Age 14 to 16 Short Challenge Level:
Finding the lengths of the arcs
If the perimeter of the square is 20 cm, then each side must be 5 cm long. 

The sides of the square are the diameters of the semicircles, so the circumferences of the full circles would be $5\times\pi=5\pi$ cm.

As shown in the diagram below, the 4 semicircles make up 2 full circles.

So the total perimeter is $5\pi+5\pi=10\pi$ cm.




Using scale factors
The sides of the square are the diameters of the semicircles, and so the circumferences of the full circles would be $\pi\times\text{diameter}=\pi\times\text{side length}$.

Each semicircle has only half the circumference of a full circle, so its length is $\frac{1}{2}\pi\times\text{side length}$.

So to go from a square to a semicircle, each side length is mutliplied by a scale factor of $\frac{1}{2}\pi$. So the perimeter must also be multiplied by this scale factor. So the perimeter of the new shape will be $20\times\frac{1}{2}\pi=10\pi$ cm.

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