Surface Area of Prisms

Surface area is the total area of every face of a solid added together. The way to picture it is to cut the solid open and flatten it out — that flat shape is called the net. Once you can see the net, the job is simply to find the area of each piece and add them up. There is no other idea in this whole lesson. Two things to keep straight. The units are squared, like square centimetres, because this is an area rather than a volume. And surface area is not volume: volume is the space inside, measured in cubic units, while surface area is the skin around it. Wrapping paper is surface area; what fits in the box is volume.

A rectangular prism has six faces in three matching pairs. For a box 8 by 5 by 3: top and bottom are each 8 times 5, giving 80 for the pair. Front and back are each 8 times 3, giving 48. The two ends are each 5 times 3, giving 30. Adding those gives 158 square centimetres. The formula written compactly is 2 times the bracket lw plus lh plus wh, which is 2 times 79, again 158. A cube is easier still, because all six faces are identical squares, so the surface area is 6 times the side squared — a cube of side 4 gives 6 times 16, which is 96 square centimetres.

A triangular prism has five faces: two triangular ends and three rectangles. With a 3-4-5 triangle and a length of 10, the two triangles are each half of 3 times 4, so 12 for the pair. Then a rectangle sits on each side of the triangle: 3 by 10 is 30, 4 by 10 is 40, and 5 by 10 is 50. The total is 132 square centimetres. There is a neat shortcut — twice the area of the end plus the perimeter of the end times the length, so 12 plus 12 times 10, again 132. The trap is forgetting the slanted rectangle on the hypotenuse. There are five faces, not four, so count them on the net before you start calculating.

A cylinder has three pieces: two circular ends and a curved side. Unroll that curved side and it becomes a rectangle — think of peeling the label off a tin. Its height is the cylinder's height, and its width is the circumference of the circle, because that is what it wrapped around. So for a radius of 3 and a height of 10: the two circles give 2 pi times 9, which is 18 pi. The rectangle gives 2 pi times 3 times 10, which is 60 pi. Together that is 78 pi, or about 245 square centimetres. The formula is 2 pi r squared plus 2 pi r h — two circles plus one rectangle whose width is the circumference.

Here is a method that never fails. Step one: count the faces — six for a rectangular prism, five for a triangular prism, three for a cylinder. Step two: sketch the net or at least list every face, rather than working from the 3D picture alone. Step three: convert all lengths to the same unit before any multiplying. Step four: find each face area separately, written in a column. Step five: add them and check the count, asking whether you used every face exactly once. Step six: answer in squared units. Step one is the one that matters most, because a missed face causes nearly every error in this topic.