Surface Area of Prisms and Cylinders

What counts as a prism

A prism has the same cross-section all the way through. Imagine slicing it like a loaf of bread: every slice is identical.

This matters because the prism formulas only work for prisms.

Surface area means every face

Total surface area is exactly what it sounds like: add up the area of every face on the outside.

Two ways to do it:

The careful way — list every face, work out each area, add them up. Slower, but you can see what you have done and check nothing is missing.

The formula way — two ends plus a wrapper:

$\text{TSA} = 2 \times \text{(cross-section area)} + \text{(cross-section perimeter)} \times \text{length}$

Both give the same answer. Use the formula once you trust it; use the careful way when the shape is unusual.

Nets: the trick that stops you missing a face

Sketch the solid unfolded flat. Once every face is drawn side by side, it is very hard to forget one — and forgetting a face is the most common error in this topic.

For a cuboid you get six rectangles in a cross shape. For a triangular prism, two triangles and three rectangles. Count the faces on your net and check the number is right before you start calculating.

Why the cylinder formula looks like that

Take a label off a tin and flatten it out. You get a rectangle.

• Its height is the height of the cylinder, $h$.
• Its width is the distance round the tin — the circumference, $2\pi r$.

So the curved surface area is $2\pi r \times h$. Add the two circular ends, $2 \times \pi r^2$, and you have

$\text{TSA} = 2\pi r^2 + 2\pi r h$

Once you have seen the label unroll, the formula stops being something to memorise.

Watch for open shapes. A tin without a lid has only one circle, not two. A pipe has none. Read the question and count the faces the object actually has.