Surface area and volume

Volume, surface area and nets

Two measurements describe a solid.

• Volume is the amount of space inside it, in cubic units: ${\text{cm}}^3$, ${\text{m}}^3$. For a container this is its capacity, and $1000{\text{cm}}^3$ is 1 litre.
• Surface area is the total area of all its faces, in square units: ${\text{cm}}^2$, ${\text{m}}^2$. It is the amount of material needed to make the solid, or the area to be painted.

A net is the flat shape that folds up to make the solid. The surface area of a solid is the area of its net, so if you are unsure which faces to include, picture the net. For a closed solid, include every face; for an open box or a tin with no lid, leave the missing face out.

Cuboids

A cuboid of length $l$, width $w$ and height $h$ holds $h$ layers, each of $l\times w$ unit cubes, so

$$V=lwh$$

It has six rectangular faces in three equal pairs (top and bottom, front and back, the two ends), so

$$\text{surface area}=2lw+2lh+2wh$$

Neither formula is printed on the paper. A cube of side $s$ has $V=s^3$ and surface area $6s^2$.

A cuboid has three pairs of equal rectangular faces.

Prisms

A prism is any solid with the same cross-section all the way along its length: slice it anywhere parallel to an end and you get the same shape. A cuboid is a prism with a rectangular cross-section; a cylinder is a prism with a circular cross-section. A wedge cut from a cylinder (a cylindrical sector, like a piece of cheese) is a prism whose cross-section is a sector.

Because every slice is the same, the volume is the area of the cross-section times the length:

$$V=Al$$

This formula is given. To use it, first find the area $A$ of the cross-section with the methods from the area topics (triangle, trapezium, sector, compound shape), then multiply by the length.

For the surface area of a prism there is no formula: add the two equal end faces and the rectangles round the sides. Each side rectangle has one side equal to the length of the prism and the other equal to a side of the cross-section, so together the side rectangles have area perimeter of the cross-section $\times$ length.

Cylinders

A cylinder is a prism whose cross-section is a circle of area $\pi r^2$, so

$$V=\pi r^{2}h$$

Its curved surface unrolls into a rectangle. One side of the rectangle is the height $h$; the other side wrapped once round the circle, so it is the circumference $2\pi r$. So the curved surface area is

$$2\pi rh$$

A closed cylinder also has two circular ends, so its total surface area is $2\pi rh+2\pi r^2$. A tin with no lid has only one end: $2\pi rh+\pi r^2$. A pipe open at both ends has only the curved surface.

A cylinder of radius $r$ and height $h$.

The net of a closed cylinder: the curved surface is a rectangle $2\pi r$ by $h$.

Pyramids

A pyramid has a flat base (often a square or rectangle) and triangular faces that meet at a single point, the vertex. Its volume is one third of the prism with the same base and height:

$$V=\frac{1}{3}Ah$$

where $A$ is the area of the base and $h$ is the perpendicular height, from the vertex straight down to the base. Three identical pyramids can be fitted together to make a cube, which is one way to see where the third comes from.

The surface area of a pyramid is the base plus the triangular faces. Each triangle needs its own perpendicular height, measured up the face (the slant height of that face), which is not the same as the height of the pyramid. You can find it with Pythagoras' theorem in the right-angled triangle formed by the pyramid's height and half the base.

The perpendicular height $h$ goes from the vertex to the centre of the base.

Cones

A cone is a pyramid with a circular base, so

$$V=\frac{1}{3}\pi r^{2}h$$

Two different lengths go up a cone:

• the perpendicular height $h$, from the vertex straight down to the centre of the base, used for the volume;
• the slant height $l$, down the sloping side, used for the curved surface area: $\pi rl$.

They are linked by Pythagoras' theorem, because $r$, $h$ and $l$ form a right-angled triangle with $l$ as the hypotenuse: $l^2=r^2+h^2$.

The total surface area of a solid cone is the curved surface plus the circular base: $\pi rl+\pi r^2$.

The radius $r$, the height $h$ and the slant height $l$ form a right-angled triangle.

Spheres

For a sphere of radius $r$:

$$V=\frac{4}{3}\pi r^{3}A=4\pi r^2$$

Both are given. Notice the powers: volume is a cube ($r^3$, units ${\text{cm}}^3$) and surface area is a square ($r^2$, units ${\text{cm}}^2$), which helps you remember which is which. The surface area of a sphere is exactly four times the area of a circle with the same radius.

On a calculator, work out $r^3$ first, or type the whole expression with brackets: $4÷3\times \pi \times r^3$.

A sphere is fixed by its radius alone.

Which formulas are given

The List of formulas on the front of the paper gives: the curved surface area of a cylinder ($2\pi rh$) and of a cone ($\pi rl$); the surface area of a sphere ($4\pi r^2$); and the volumes of a prism ($Al$), a pyramid ($\frac{1}{3}Ah$), a cylinder ($\pi r^{2}h$), a cone ($\frac{1}{3}\pi r^{2}h$) and a sphere ($\frac{4}{3}\pi r^3$).

You must know for yourself: the volume and surface area of a cuboid, that a total surface area adds the flat faces (circles, rectangles, triangles) to any curved surface, and the area formulas for the cross-sections. Answers may be asked for in terms of $\pi$, exactly as for circles.

Working backwards and changing units

Questions often give a volume or surface area and ask for a length. Substitute what you know into the formula and solve.

• A cone has volume $500{\text{cm}}^3$ and radius 6 cm: $\frac{1}{3}\pi \times 36\times h=500$, so $12\pi h=500$ and $h=13.3$ cm.
• A sphere has volume $V$: $r^3=\frac{3V}{4\pi }$, then take the cube root.

Keep the units consistent. If a tank is measured in metres, its volume is in ${\text{m}}^3$, and $1{\text{m}}^3=1000$ litres. If it is measured in centimetres, divide the ${\text{cm}}^3$ by 1000 to get litres.