Problems

Today we’ll look at 3-dimensional shapes, including their volumes and surfaces areas. One special kind are the Platonic Solids - the tetrahedron, cube, octahedron, dodecahedron and icosahedron.

The volume of a pyramid is $\frac{1}{3}Bh$, where $B$ is the area of the base and $h$ is the perpendicular height. What’s the volume of a regular tetrahedron with side length $1$?

The Great Pyramid of Giza is the largest pyramid in Egypt. For the purposes of this problem, assume that it’s a perfect square-based pyramid, with perpendicular height $140$m and the square has side length $230$m.

What is its volume in cubic metres?

A regular octahedron is a solid with eight faces, all of which are equilateral triangles. It can be formed by placing together two square based pyramids at their bases.

What is the volume of an octahedron with side length $1$?

In the picture below, there are the $12$ pentominoes. Is it possible to tile a $6\times 10$ rectangle with them, using each pentominoe exactly once?

Show how to tile a $5\times 12$ rectangle with the twelve pentominoes.

Show how to tile a $4\times 15$ rectangle with the twelve pentominoes.

Is it possible to tile a $3\times 20$ rectangle with the twelve pentominoes?

How can we tile the plane with this cube net?

How can we tile the plane with this cube net?