These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

If you had 36 cubes, what different cuboids could you make?

The challenge for you is to make a string of six (or more!) graded cubes.

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

A group of children are using measuring cylinders but they lose the labels. Can you help relabel them?

What size square should you cut out of each corner of a 10 x 10 grid to make the box that would hold the greatest number of cubes?

This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal. . . .

How can you change the surface area of a cuboid but keep its volume the same? How can you change the volume but keep the surface area the same?

A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?

What happens to the area and volume of 2D and 3D shapes when you enlarge them?

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.

In the ancient city of Atlantis a solid rectangular object called a Zin was built in honour of the goddess Tina. Your task is to determine on which day of the week the obelisk was completed.

What is the largest cuboid you can wrap in an A3 sheet of paper?

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

Can you sketch graphs to show how the height of water changes in different containers as they are filled?

If I don't have the size of cake tin specified in my recipe, will the size I do have be OK?

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have. . . .

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?