Bobbly perimeter
Find the perimeter of this shape made of semicircles
Age
14 to 16
Challenge level
A square with perimeter 20 cm has a semicircle drawn onto each of its sides, as shown below.
What is the perimeter of the new shape? Give your answer in terms of $\pi$.
This problem is adapted from the World Mathematics Championships
What is the perimeter of the new shape? Give your answer in terms of $\pi$.
Image
This problem is adapted from the World Mathematics Championships
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.
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.
Image
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.