Math teachers usually have a box of wooden or plastic shapes tucked away in the back of their candy cupboard that they pull out when teaching cone and pyramid theories, but they’re mostly complete, right down to the end. There is a lot of mathematics in a cone or a pyramid, of course, but in everyday life its use is limited. You’ll find cone-shaped ice cream cones and hanging baskets, but the full ones have to be hand-held, attached to string or wire, or propped up on their bases, all of which somewhat restrict their use (unless you want to buy thousands of cones). of road works).
It is the same with the pyramids. Again we are limited to hanging baskets and the like and unless you want to bury a pharaoh, they are of little use standing on their bases.
Enter the truncated cone and pyramid. In simple terms, a truncated cone or pyramid is a complete 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 truncation’, but these shapes have even fewer uses in the construction of physical objects than full shapes.)
But now we find ourselves in a completely different ball game, as truncated cones and pyramids are admirably suited to stacking and we see them everywhere. Have your kids take care of them at garden centers, home improvement stores, DIY stores, etc. A beautiful example is the type of drinking mugs that are sold with a small Easter egg. They are normally well crafted in the shape of a truncated cone with a simple handle added.
I recently saw a metal trash can (the kind used for burning papers, yard waste, etc.) at my local garden center. The main component was an upturned truncated cone with legs. The lid was a short but wide frustum with two handles, one on each side to make room for the chimney that was…you guessed it!
Pointless cones are also used for lampshades, flower pots, fruit bowls, dovecotes, rocket engine nozzles, and fez hats, to name just a few.
Truncated pyramids are found in concrete streetlights (at first glance one might think they are prisms, but they typically reduce in cross-sectional area as height increases), concrete blocks in roadworks, cubes office rubbish bins, lampshades, wheelbarrows and a multitude of items that consist of several conjoined such as birdbaths and fountains.
Finding the volume of a cone or pyramid is a great math exercise and requires a calculator or a good knowledge of multiplication tables. If we imagine a prism surrounding the cone or pyramid and of 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 better to find the volume of the whole shape and then subtract the volume of the piece that has been removed.