Volume of Composite Solids

A composite solid is made of simpler solids. This shed is a box with a triangular prism on top as its roof. The box part is four metres wide, three metres high and six metres long, so its volume is seventy two cubic metres. The roof is a triangle with base four and height two, area four square metres, running for six metres, so twenty four cubic metres. Add them: ninety six cubic metres. You could also see the whole shed as one prism whose cross-section is the house shape: twelve plus four is sixteen square metres, times six is ninety six again.

An L-shaped solid is still a prism, because its cross-section, the L, is the same all the way through. So find the area of the L first. One way is to split it: a tall strip two by seven, fourteen, and a short piece three by three, nine, making twenty three square centimetres. Another way is to take the cut-out away from the full rectangle: five by seven is thirty five, the missing corner is three by four, twelve, and thirty five minus twelve is twenty three again. Then multiply by the length, four: ninety two cubic centimetres.

When a solid has a hole, find the volume of the whole solid and subtract the hole. The block is ten by ten by six, six hundred cubic centimetres. The hole is a cylinder of radius two running right through the block, so it is six centimetres long, the same as the block. Its volume is pi times two squared times six, twenty four pi. The volume left is six hundred minus twenty four pi cubic centimetres. That is the exact answer: you cannot combine the six hundred and the twenty four pi.

A pipe is a cylinder with a smaller cylinder taken out of the middle. The outer cylinder has radius five and length ten, so its volume is pi times twenty five times ten, two hundred and fifty pi. The hole has radius three, so it is pi times nine times ten, ninety pi. The material in the pipe is two hundred and fifty pi minus ninety pi, which is one hundred and sixty pi cubic centimetres. In one line: pi times twenty five minus nine, times ten.

To find how many small containers a big one fills, put both volumes in the same unit and divide. The tank is fifty by forty by twenty five, fifty thousand cubic centimetres, which is fifty thousand millilitres. Each cup holds two hundred and fifty millilitres, so the tank fills fifty thousand divided by two hundred and fifty, two hundred cups. When the division is not exact, think about what the question asks. The number of full cups rounds down. The number of trips needed to fill something rounds up, because a part trip is still a trip. Quiz and exercises now.