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Can you make a tetrahedron whose faces all have the same perimeter? A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground? Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

# Bobbly Perimeter

##### Age 14 to 16 ShortChallenge 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.

You can find more short problems, arranged by curriculum topic, in our short problems collection.