### 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?

### Concrete Calculation

The builders have dug a hole in the ground to be filled with concrete for the foundations of our garage. How many cubic metres of ready-mix concrete should the builders order to fill this hole to make the concrete raft for the foundations?

### In a Spin

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

# Efficient Cutting

##### Stage: 4 Challenge Level:

A cylindrical container, like the tin cans used to package some food, can be made by using two circles for the ends and a rectangle which wraps round to form the body.

To make cylinders of varying sizes, the three pieces can be cut from a single rectangle of flat sheet in several ways. For example:

CHALLENGE:

Your task is to cut out one rectangle and two circles from a single sheet of A4 paper to make a cylinder with the greatest possible volume.

What are its dimensions?

(We will assume that the dimensions of an A4 sheet of paper are $21$cm and $29.6$ cm)