### Qqq..cubed

It is known that the area of the largest equilateral triangular section of a cube is 140sq cm. What is the side length of the cube? The distances between the centres of two adjacent faces of another cube is 8cms. What is the side length of this cube? Another cube has an edge length of 12cm. At each vertex a tetrahedron with three mutually perpendicular edges of length 4cm is sliced away. What is the surface area and volume of the remaining solid?

### When the Angles of a Triangle Don't Add up to 180 Degrees

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the triangle.

### The Spider and the Fly

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

# Changing Areas, Changing Volumes

##### Stage: 4 Challenge Level:
This problem follows on from Changing Areas, Changing Perimeters.

Here are the dimensions of nine cuboids. You can download a set of cards here.

 1 by 2 by 28 cuboid 4 by 4 by 4 cube 2 by 4 by 7 cuboid 1 by 2 by 26 cuboid 2 by 4 by 6 cuboid 4 by 5 by 6 cuboid 4 by 5 by 7 cuboid 1 by 2 by 24 cuboid 1 by 4 by 14 cuboid

The challenge is to arrange them in a 3 by 3 grid like the one below:

As you go from left to right, the surface area of the shapes must increase.
As you go from top to bottom, the volume of the shapes must increase.
All the cuboids in the middle column must have the same surface area.
All the cuboids on the middle row must have the same volume.

What reasoning can you use to help you to decide where each cuboid must go?

Once you've placed the nine cards, take a look at the extended grid below:

The ticks represent the nine cards you've already placed. Can you create cards with dimensions for cuboids that could go in the four blank spaces that satisfy the same criteria?

Can you design a set of cards of your own with a different cuboid in the centre?