Mathematics teachers normally have a box of wooden or plastic shapes tucked deep in their cupboard of goodies that they bring out when teaching the theories of cones and pyramids, but mostly these are complete – right up to the sharp end. There is a lot of mathematics in a cone or pyramid, of course, but in everyday life their use is limited. You will find ice cream cones and hanging baskets in the shape of a cone, but complete ones have to either be held in the hand, supported by strings or wires or stood on their bases, all of which rather restricts their use (unless you wish to purchase thousands of road works cones).
It is the same with pyramids. Again we are limited to hanging baskets and the like and unless you want to bury a pharaoh, they do not have a lot of use stood on their bases.
Enter the truncated cone and pyramid. In simple terms, a truncated cone or pyramid is a full one with the top removed. If the cut is made parallel to the base, the shape is simply called ‘truncated’. (If the cut is not parallel to the base it is called ‘oblique truncated’, but these shapes have even fewer uses in the construction of physical objects than the full shapes.)
But now we are into a wholly different ball game as truncated cones and pyramids are admirably suite to stacking and we see them everywhere. Get your children to watch out for them in garden centres, crockery shops, DIY stores and so on. A beautiful example is the type of drinking mugs that are sold with a small Easter egg. They are normally well crafted in a truncated cone shape with a simple handle added.
I recently saw a metal dustbin (the sort used for burning papers, garden waste etc) in my local garden centre. The main component was an upturned truncated cone with legs. The lid was a short but wide truncated cone with two handles, one on each side to allow space for the chimney which was…you’ve guessed it!
Cones with no points are also used for lamp shades, flower pots, fruit bowls, dove cotes, rocket engine nozzles and fez hats, to name but a few.
Truncated pyramids are found in concrete lampposts (a first glance may lead you to think these are prisms, but they normally reduce in cross sectional area as the height increases), concrete blocks at road works, office rubbish bins, lamp shades, wheelbarrows and a multitude of objects which consist of several joined together such as bird baths and fountains.
Finding the volume of a cone or pyramid is a great mathematical exercise and requires a calculator or a good knowledge of times tables. If we imagine a prism surrounding the cone or pyramid and the same height, the volume is always one third of the volume of the prism, which gives us the formula V = Base Area x Height ÷3.
It is possible to find the volume of a truncated cone or pyramid using a more complex formula, but at GCSE level it is best to find the volume of the full shape and then subtract the volume of the piece that has been removed.