### Building Tetrahedra

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?

### Areas and Ratios

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

# One or Two

##### Stage: 4 Short Challenge Level:

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.