# Surface Area and Volume of Solids

> Surface area and volume of prisms, cylinders, pyramids, cones and spheres, capacity, and composite solids.

From HSC Maths Advanced on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-advanced/surface-area-and-volume/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The ice cream shop
Lily works at an ice cream shop. A waffle cone is 12 cm tall with a 3 cm radius, and the boss wants a scoop of ice cream on top that is half a sphere. How much ice cream fills the cone and the scoop? And how much waffle does each cone need?

The first question is about **volume**, the space inside. The second is about **surface area**, the outside skin. Every package, tank, can and cone in a factory is designed with these two numbers.

**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 $V = \dfrac{4}{3}\pi r^3$.

## 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.

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

### Prisms and cylinders

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

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

$A$ 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.

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

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

*Diagram: A cone with radius r, height h and slant height l forming a right-angled triangle*
*Height, radius and slant height make a right-angled triangle.*

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

### Spheres

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

### Worked example: Lily's cone
Cone: $V = \dfrac{1}{3}\pi(3)^2(12) = 36\pi \approx 113$ cm³.

Half scoop: $V = \dfrac{1}{2} \times \dfrac{4}{3}\pi(3)^3 = 18\pi \approx 57$ cm³.

Total ice cream $54\pi \approx 170$ cm³, about 170 mL.

Waffle (curved surface only): $l = \sqrt{9 + 144} \approx 12.37$ cm, so $\pi rl \approx \pi \times 3 \times 12.37 \approx 117$ cm².

### Key idea: Volume formulas
- Prism $Ah$. Cylinder $\pi r^2 h$.
- Pyramid $\dfrac{1}{3}Ah$. Cone $\dfrac{1}{3}\pi r^2 h$.
- Sphere $\dfrac{4}{3}\pi r^3$.

### Watch out: Slant height or height?
Volume uses the **perpendicular height** $h$. The cone's surface area uses the **slant height** $l$.

### Exam tip: Open or closed?
Read whether a container has a lid. An open cylinder has only one circle: $SA = \pi r^2 + 2\pi rh$.

### Calculator tip: Fractions, powers and cube roots
For $V = \frac{4}{3}\pi r^3$, type `4 ÷ 3 × π × 6 ^ 3` or use the fraction key for $\frac{4}{3}$. To find a radius from a volume, use the cube root key: $r = \sqrt[3]{\dfrac{3V}{4\pi}}$, typed with brackets around the whole fraction.

**Mind map**

```text
Solids
  Prisms
    V = Ah
    Cylinder πr²h
  Pyramids and cones
    V = ⅓Ah
    Slant height √(r² + h²)
    Cone SA πr² + πrl
  Spheres
    V = 4/3 πr³
    SA = 4πr²
  Capacity
    1 cm³ = 1 mL
  Composite
    Add or subtract solids
    Count only outside faces
```

### Year 11: Composite solids and working backwards
For a composite solid, add or subtract volumes. For surface area, count only the faces on the **outside**. Where two solids join, that surface is hidden.

**Working backwards.** A sphere has volume $288\pi$ cm³. Then $\dfrac{4}{3}\pi r^3 = 288\pi$, so $r^3 = 216$ and $r = 6$ cm.

**Similar solids.** If every length is multiplied by $k$, surface area is multiplied by $k^2$ and volume by $k^3$.

### Year 12: Filling a tank and the best can
Water pours into a cylinder of radius 2 m at 3 m³ per minute. Since $V = 4\pi h$, the depth rises at $\dfrac{dh}{dt} = \dfrac{3}{4\pi} \approx 0.24$ m per minute.

The surface area and volume formulas also set up optimisation. A closed can with volume $V$ uses the least metal when its height equals its diameter, a result found by writing $SA$ in terms of $r$ alone and setting $\dfrac{d(SA)}{dr} = 0$.

## 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: $V = \dfrac{1}{3}Ah$. Cone $SA = \pi r^2 + \pi rl$ with $l = \sqrt{r^2 + h^2}$.
- Sphere $V = \dfrac{4}{3}\pi r^3$ and $SA = 4\pi r^2$.
- Composite solids: add or subtract volumes, and count only outside faces.

## Quiz

Try these questions from the chapter quiz.

1. How many litres does a 3000 cm³ container hold?
   - A. 30 L
   - B. 3 L
   - C. 0.3 L
   - D. 300 L
   - Answer: B. 3 L. $1000 \text{ cm}^3 = 1$ L.

2. Find the volume of a box 5 cm by 4 cm by 3 cm.
   - A. 12 cm³
   - B. 60 cm³
   - C. 94 cm³
   - D. 47 cm³
   - Answer: B. 60 cm³. $5 \times 4 \times 3$.

3. A triangular prism has cross-section area 12 cm² and length 10 cm. Its volume is:
   - A. 22 cm³
   - B. 120 cm³
   - C. 60 cm³
   - D. 40 cm³
   - Answer: B. 120 cm³. $V = Ah$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-advanced/surface-area-and-volume/#quiz