Surface Area of Prisms

Surface area is the total area of all the faces of a solid. Unfold the box into its net and you can see all six faces at once. They come in three matching pairs. The top and the bottom are each six by four, which is twenty four, so together they make forty eight. The front and the back are each six by three, eighteen, so together thirty six. The two ends are each four by three, twelve, so together twenty four. Add the pairs: forty eight plus thirty six plus twenty four is one hundred and eight square centimetres. As a formula, the surface area is two times l w plus l h plus w h.

A triangular prism has five faces: two triangles at the ends and three rectangles joining them. The triangle here has sides three, four and five, with the right angle between the three and the four. Each triangle has area one half times four times three, which is six, so the two ends make twelve. Each rectangle is one side of the triangle times the length of the prism, ten. That gives forty, thirty and fifty. Do not forget the sloping one: it is the five centimetre side times ten. The total is twelve plus forty plus thirty plus fifty, which is one hundred and thirty two square centimetres.

This prism has a triangle with base twenty four and height five, but the sloping sides are not given. The height cuts the triangle into two right-angled triangles, each with a base of twelve and a height of five. By Pythagoras, the slant squared is twelve squared plus five squared, which is one hundred and sixty nine, so the slant is thirteen. Now count every face. The two triangles give two times one half times twenty four times five, which is one hundred and twenty. The base rectangle is twenty four by twenty, four hundred and eighty. The two sloping rectangles are each thirteen by twenty, so together five hundred and twenty. The total is one thousand one hundred and twenty square centimetres.

An open box has no lid, so it has only five faces. The base is eight by five, forty. The front and back are each eight by four, so together sixty four. The two ends are each five by four, so together forty. The total is one hundred and forty four square centimetres. The missing face is the lid, and the lid is the same size as the base. So a closed box would need forty more, one hundred and eighty four. Always check which face is missing before you leave one out.

Sometimes you know the surface area and need a length. This box is six by five with an unknown height h, and its surface area is one hundred and forty eight. The top and the bottom are known: two times six times five, sixty. The four sides wrap around the box, so together they are the perimeter of the base, twenty two, times the height: twenty two h. So sixty plus twenty two h equals one hundred and forty eight. Take away sixty: twenty two h equals eighty eight, so h is four centimetres. Check: two times thirty plus twenty four plus twenty is one hundred and forty eight. Quiz and exercises now.