Surface Area of Prisms and Cylinders
Theory
The surface area of a solid is the total area of all its faces, the area of its net. This Year 10 maths page revises Year 9: cubes, rectangular and triangular prisms, trapezium prisms, open boxes, cylinders (closed, open at one end and pipes), exact answers in \(\pi\), practical painting and covering problems, and working backwards to a missing length.
A right prism is a solid with two identical, parallel ends joined by rectangles. The shape of the end (the cross-section) names the prism: rectangular, triangular, trapezium and so on.
The surface area (SA) is the total area of all the faces. If you unfold the prism flat you get its net; the surface area is the area of the net.
A rectangular prism has three pairs of matching faces. A triangular prism has two triangular ends and three rectangles. The rectangles around any prism fit together into one long rectangle whose length is the perimeter of the end and whose width is the length of the prism.
In real problems some faces are missing or hidden: a pool or planter has no top, a shed floor is not painted, and where two solids are glued together the touching faces are no longer on the surface.
A cylinder has two identical circular ends joined by a curved surface. Cut the curved surface straight down and unroll it: it is a rectangle whose length is the circumference \(2\pi r\) and whose width is the height \(h\). A closed cylinder has both circles, a cup or open tank has one, and a pipe has none.
| Prism | Faces | Surface area |
|---|---|---|
| Cube, edge \(s\) | 6 equal squares | \(6s^{2}\) |
| Rectangular prism | 3 pairs of rectangles | \(2(lw + lh + wh)\) |
| Any right prism | 2 ends + rectangles | \(2 \times\) end area \(+\) perimeter of end \(\times\) length |
Cylinders:
| Cylinder | Surfaces | Area |
|---|---|---|
| Closed (solid) | 2 circles + curved surface | \(2\pi r^{2} + 2\pi rh\) |
| Open at one end | 1 circle + curved surface | \(\pi r^{2} + 2\pi rh\) |
| Open at both ends (tube, pipe) | curved surface only | \(2\pi rh\) |
How to find the surface area of a prism
- Sketch the prism or its net and list every face. Check which faces are really there (a lid? a floor?).
- Find any missing side lengths, using Pythagoras' theorem if needed. Put all lengths in the same unit.
- Calculate each face area. For matching faces, find one and multiply.
- Add the face areas and give square units.
Problems
- Paint, tiles or material: find the area that is covered, then multiply by the price per square metre.
- Working backwards: write the surface area with a pronumeral for the unknown length, set it equal to the given area and solve.
- Hole through a solid: total \(=\) outside surface \(-\) the two openings \(+\) the inside walls.
- Stacked or glued solids: add both surface areas, then subtract the touching area twice (once from each solid).
- Cylinders: \(\pi r^{2}\) for each end that is there plus \(2\pi rh\) for the curved surface. Halve a diameter first. A painted tank standing on the ground has no painted base.
- Working backwards with a prism: a closed box with a \(6\) cm square base and \(SA = 240\text{ cm}^2\) gives \(2 \times 36 + 4 \times 6h = 240\), so \(24h = 168\) and \(h = 7\) cm.
The rectangles need all three sides of the triangle, so find the sloping side \(x\) first.
| \(x^{2}\) | \(=\) | \(9^{2} + 12^{2} = 225\) |
| \(x\) | \(=\) | \(15\text{ cm}\) |
| \(\text{two ends}\) | \(=\) | \(2 \times \dfrac{1}{2} \times 12 \times 9 = 108\) |
| \(\text{rectangles}\) | \(=\) | \((9 + 12 + 15) \times 20 = 720\) |
| \(SA\) | \(=\) | \(108 + 720 = \mathbf{828}\text{ cm}^2\) |
Change every length to metres: \(50\text{ cm} = 0.5\text{ m}\) and \(40\text{ cm} = 0.4\text{ m}\). There is no top, so count the floor once.
| \(\text{floor}\) | \(=\) | \(1.2 \times 0.5 = 0.6\) |
| \(\text{long sides}\) | \(=\) | \(2 \times 1.2 \times 0.4 = 0.96\) |
| \(\text{short sides}\) | \(=\) | \(2 \times 0.5 \times 0.4 = 0.4\) |
| \(A\) | \(=\) | \(0.6 + 0.96 + 0.4\) |
| \(\) | \(=\) | \(\mathbf{1.96}\text{ m}^2\) |
To cost the timber, multiply by the price per square metre. At \(\$35\) per square metre: \(1.96 \times 35 = \$68.60\).
| \(SA\) | \(=\) | \(2\pi r^{2} + 2\pi rh\) |
| \(\) | \(=\) | \(2\pi \times 5^{2} + 2\pi \times 5 \times 8\) |
| \(\) | \(=\) | \(50\pi + 80\pi\) |
| \(\) | \(=\) | \(\mathbf{130\pi}\text{ cm}^2\) |
| \(\) | \(=\) | \(408.4070\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{408.41}\text{ cm}^2\) |
Collect the \(\pi\) terms before you use the calculator. The two circles give \(50\pi\); the curved surface gives \(80\pi\).
Substitute what you know, divide every term by \(\pi\), then solve.
| \(2\pi \times 4^{2} + 2\pi \times 4 \times h\) | \(=\) | \(120\pi\) |
| \(32\pi + 8\pi h\) | \(=\) | \(120\pi\) |
| \(32 + 8h\) | \(=\) | \(120\) |
| \(8h\) | \(=\) | \(88\) |
| \(h\) | \(=\) | \(\mathbf{11}\text{ cm}\) |
If the area is a decimal (not a multiple of \(\pi\)), work out \(2\pi r^{2}\) and \(2\pi r\) on the calculator and solve the same way.
Common pitfalls
Frequently asked questions
What is the surface area of a prism?
It is the total area of all the faces of the prism, the same as the area of its net. It is measured in square units.
How do you find the surface area of a rectangular prism?
Find the area of the top, the front and the side, add them, then double: \(SA = 2(lw + lh + wh)\).
How do you find the surface area of a triangular prism?
Add the areas of the two triangular ends to the three rectangles. The rectangles together equal the perimeter of the triangle times the length of the prism.
How do you find the surface area of an open box?
Add the areas of the base and the four sides, but leave out the top. That is the closed surface area minus the area of the missing lid.
What is the formula for the surface area of a cylinder?
For a closed cylinder, \(SA = 2\pi r^{2} + 2\pi rh\). The first part is the two circular ends and the second part is the curved surface, which unrolls into a rectangle \(2\pi r\) long and \(h\) wide.
How do you find the surface area of an open cylinder?
Open at one end: \(\pi r^{2} + 2\pi rh\). Open at both ends, like a pipe: \(2\pi rh\) only.