Surface Area and Volume of Solids

In short: Surface area is the total area of all the faces of a solid, and volume is the space inside it. A prism has $V=Ah$, a cylinder $V=\pi r^{2}h$, a pyramid or cone one third of its prism, and a sphere $V=\frac{4}{3}\pi r^3$.

Starting point: Years 9 and 10

Volume and capacity

Volume is the space a solid takes up, in cubic units. Capacity is how much liquid it holds.

$$1{\text{ cm}}^3=1\text{ mL}1000{\text{ cm}}^3=1\text{ L}1{\text{ m}}^3=1000\text{ L}$$

Prisms and cylinders

A prism has the same cross-section all the way through.

$$V=Ah\text{cylinder: }V=\pi r^{2}h$$

$A$ is the area of the cross-section and $h$ is the length or height.

Surface area is the total area of all the faces. Imagine unfolding the solid into a flat net.

$$\text{cylinder: }SA=2\pi r^2+2\pi rh$$

The two circles are the ends. The curved part unrolls into a rectangle of length $2\pi r$ and height $h$.

Pyramids and cones

A pyramid or cone holds exactly one third of the prism or cylinder with the same base and height.

$$V=\frac{1}{3}Ah\text{cone: }V=\frac{1}{3}\pi r^{2}h$$

The slant height $l$ of a cone runs from the tip down the side to the edge of the base. By Pythagoras, $l=\sqrt{r^2+h^2}$.

$$\text{cone: }SA=\pi r^2+\pi rl\text{pyramid: }SA=\text{base}+\text{all triangular faces}$$

Spheres

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

Quick summary

• $1{\text{ cm}}^3=1\text{ mL}$ and $1{\text{ m}}^3=1000\text{ L}$.
• Prism $V=Ah$, cylinder $V=\pi r^{2}h$, cylinder $SA=2\pi r^2+2\pi rh$.
• Pyramid and cone hold one third: $V=\frac{1}{3}Ah$. Cone $SA=\pi r^2+\pi rl$ with $l=\sqrt{r^2+h^2}$.
• Sphere $V=\frac{4}{3}\pi r^3$ and $SA=4\pi r^2$.
• Composite solids: add or subtract volumes, and count only outside faces.

Quiz