### 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.

# Wood Pile Perimeter

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

The centres of the three circles form an equilateral triangle, so each of the sectors marked is $\frac{1}{6}$ of a circle. The shape below is a rectangle, so the sectors are $\frac 14$ of a circle. The curved portion of the perimeters is therefore the same as each of the circles: $2\pi \times 5 = 10\pi \text{cm}$.

The straight part is the same length as two radii, so is $10\text{cm}$ long.

The perimeter of the shaded shapes is therefore $10+10\pi\text{cm}$.

This problem is taken from the UKMT Mathematical Challenges.