You may also like

problem icon

Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

problem icon

Ladder and Cube

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?

problem icon

Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

One or Two

Stage: 4 Short Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
 
Let the radius of each semicircle be $r$.
In the top diagram, let the side-length of the square be $2x$. By Pythagoras' Theorem, 
$x^2 + (2x)^2 = r^2$ and so $5x^2 = r^2$. So this shaded area is $(2x)^2 = 4x^2 = \frac{4r^2}{5}$.

In the bottom diagram, let the side-length of each square be $y$. Then by Pythagoras' Theorem, $y^2 + y^2 = r^2$ and so $2y^2 = r^2$. So this shaded area is $r^2$.

Therefore the ratio of the two shaded areas is $\frac{4}{5} : 1 = 4 : 5$.

This problem is taken from the UKMT Mathematical Challenges.
View the archive of all weekly problems grouped by curriculum topic