Surface Area of Cylinders and Composite Solids

Cut a cylinder open and flatten it and you get two circles and a rectangle. The circles are the top and the bottom, each with area pi r squared. With a radius of three, each is nine pi, so the two together are eighteen pi. The rectangle is the curved surface unrolled. Its height is the height of the cylinder, eight. Its width wrapped all the way around the circle, so it is the circumference, two pi r, which is six pi. Its area is six pi times eight, forty eight pi. The total is eighteen pi plus forty eight pi, which is sixty six pi square centimetres. Leave it in terms of pi: that is the exact answer, and you need no calculator.

Questions often give the diameter instead of the radius. This cylinder is ten centimetres across, so its radius is five. Halve it first, before anything else. The two circles are two times pi times five squared, which is fifty pi. The curved surface is two pi times five times six, sixty pi. The total is one hundred and ten pi square centimetres. If you use ten as the radius you get three hundred and twenty pi, which is far too big.

Real objects are often open. A can with no lid has one circle, not two. With radius four and height ten, the base is sixteen pi and the curved surface is two pi times four times ten, eighty pi, so the can needs ninety six pi square centimetres of metal. A pipe is open at both ends, so it has no circles at all, only the curved surface. With radius two and length fifteen, that is two pi times two times fifteen, sixty pi. Before you add, ask which circles are really there.

A composite solid is made of simpler solids joined together. Here a two centimetre cube sits on a six by six by two block. The block on its own has surface area two times thirty six plus twelve plus twelve, one hundred and twenty. The cube on its own has six faces of four, twenty four. But where they touch, two faces disappear: the bottom of the cube, and the patch of the block's top that it covers. That is four plus four, eight. So the surface area is one hundred and twenty plus twenty four minus eight, which is one hundred and thirty six square centimetres. A quicker way to see it: the top of the cube exactly replaces the patch it covers, so you only add the cube's four sides, sixteen.

Here is a two tier cake to ice. The bottom tier has radius five and height two, so on its own it has two circles of twenty five pi and a curved surface of two pi times five times two, twenty pi: seventy pi in all. The top tier has radius three. Its base is hidden, and so is the nine pi patch of the bottom tier underneath it. But its lid is nine pi, exactly the size of the patch it covers. So the top tier only adds its curved surface, two pi times three times two, twelve pi. The total is seventy pi plus twelve pi, which is eighty two pi square centimetres. Quiz and exercises now.