# Perimeter, Area and Volume

> Perimeter and area of rectangles and triangles, shaded fractions of a grid, and the volume of a box.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/area-and-volume/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: Painting the cubby house
Ali and his dad are painting a cubby house. The front wall is a rectangle 3 m wide and 2 m tall, with a triangle roof on top that is 3 m wide and 1 m high.

"How much paint?" asks Ali. The tin says 1 litre covers 4 square metres. The rectangle is $3 \times 2 = 6$ square metres. The triangle is half of a $3 \times 1$ rectangle, so $1.5$ square metres. Total $7.5$ square metres, so 2 litres will be plenty.

**Area** is the amount of flat surface. **Perimeter** is the distance around the edge. **Volume** is the space inside a solid. Knowing which one a question wants is half the battle.

**In short:** Perimeter is the distance around a shape, area is the space inside it, and volume is the space inside a solid. A rectangle's area is length × width, a triangle's area is half of base × height, and a box's volume is length × width × height.

## Before you start

- Multiplying decimals, like $6.5 \times 4.8$ (see Working with Decimals).
- Metric units of length and capacity (see Metric Units and Conversions).
- **Square units:** a square 1 cm by 1 cm has an area of 1 cm². **Cubic units:** a cube 1 cm on each edge has a volume of 1 cm³.

## Perimeter

Add up the lengths of all the sides.

A rectangle 7 cm by 4 cm: $7 + 4 + 7 + 4 = 22$ cm, or $2 \times (7 + 4) = 22$ cm.

## Area of a rectangle

### Key idea: Length times width
$$A = l \times w$$

A rectangle 7 cm by 4 cm has area $28$ cm². A square with side 4 cm has area $16$ cm².

### Watch out: Units are squared
Area is measured in **square** units (cm², m²). Perimeter is in plain units (cm, m). Writing "28 cm" for an area loses marks.

## Area of a triangle

*Diagram: A triangle inside a rectangle with the same base and height; the triangle fills exactly half the rectangle*
*The triangle takes up exactly half of the rectangle around it.*

### Key idea: Half of base times height
$$A = \frac{1}{2} \times b \times h$$

The **height** must be at right angles to the base. It is the straight up-and-down distance from the base to the top corner, not the slanting side.

### Worked example: Using the formula
$b = 5$ cm and $h = 2.5$ cm: $A = \frac{1}{2} \times 5 \times 2.5 = 6.25$ cm².

$b = 32$ mm and $h = 9$ mm: $A = \frac{1}{2} \times 32 \times 9 = 144$ mm².

$b = 7.2$ cm and $h = 5.6$ cm: $A = 3.6 \times 5.6 = 20.16$ cm².

### Exam tip: Halve the easy number first
For $\frac{1}{2} \times 32 \times 9$, halve 32 to get 16, then $16 \times 9 = 144$. Halving first keeps the numbers whole.

## Shaded fractions of a grid

To find what fraction of a grid is shaded, find the shaded area and divide by the total area.

### Worked example: Triangles in a grid
A grid is 8 squares wide and 4 squares tall, so it has 32 squares. Two shaded triangles each have a base of 4 squares and a height of 4 squares.

Each triangle: $\frac{1}{2} \times 4 \times 4 = 8$ squares. Both: 16 squares.

Fraction shaded: $\frac{16}{32} = \frac{1}{2}$.

## Volume of a box

### Key idea: Length times width times height
$$V = l \times w \times h$$

A box 5 cm by 7 cm by 10 cm has volume $350$ cm³. Since 1 cm³ holds 1 mL, it holds 350 mL.

### Challenge: Selective level: filling a tank
A tank is 30 cm by 40 cm by 50 cm. A hose fills it at 30 mL per second. How long does it take?

Volume: $30 \times 40 \times 50 = 60\,000$ cm³ $= 60\,000$ mL.

Time: $60\,000 \div 30 = 2000$ seconds.

$2000 \div 60 = 33$ remainder 20, so it takes 33 minutes 20 seconds.

A square and a rectangle have the same perimeter. The rectangle is 10 cm by 6 cm, so its perimeter is 32 cm. The square has side $32 \div 4 = 8$ cm and area 64 cm², which is more than the rectangle's 60 cm².

**Mind map**

```text
Perimeter, area, volume
  Perimeter
    Add all sides
    cm, m
  Rectangle area
    l x w
    cm², m²
  Triangle area
    ½ x b x h
    Height at right angles
  Grid fractions
    Shaded ÷ total
  Volume
    l x w x h
    1 cm³ holds 1 mL
```

## Quick summary

- Perimeter is the distance around a shape. Add all the sides.
- Area of a rectangle $= l \times w$. Area of a triangle $= \frac{1}{2} \times b \times h$.
- The height of a triangle is at right angles to the base.
- Area is in square units, volume in cubic units.
- Volume of a box $= l \times w \times h$, and 1 cm³ holds 1 mL.

## Quiz

Try these questions from the chapter quiz.

1. What is the area of a rectangle 7 cm by 4 cm?
   - A. 22 cm²
   - B. 28 cm²
   - C. 11 cm²
   - D. 28 cm
   - Answer: B. 28 cm². $7 \times 4 = 28$, in square centimetres.

2. What is the perimeter of a rectangle 7 cm by 4 cm?
   - A. 22 cm
   - B. 28 cm
   - C. 11 cm
   - D. 14 cm
   - Answer: A. 22 cm. $2 \times (7 + 4) = 22$ cm.

3. A triangle has base 8 mm and height 2 mm. What is its area?
   - A. 16 mm²
   - B. 8 mm²
   - C. 10 mm²
   - D. 4 mm²
   - Answer: B. 8 mm². $\frac{1}{2} \times 8 \times 2 = 8$.

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