# Area, Perimeter and Accuracy of Measurement

> Length and area units, area of triangles, quadrilaterals, circles and sectors, composite shapes, and limits of accuracy and error.

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

### Story time: Carpet for the classroom
Mr Okafor measures his classroom floor with a tape measure marked in centimetres: 8.45 m by 6.20 m. The carpet shop quotes a price per square metre. But his tape can only be read to the nearest centimetre, so each length could be up to half a centimetre out.

Is that small error worth worrying about? On a big floor, it changes the area by a noticeable amount. Knowing the area formulas, and how accurate a measurement really is, stops him ordering too little carpet.

**In short:** Area is the space inside a flat shape, measured in square units, so you square the conversion factor: $1 \text{ m}^2 = 10\,000 \text{ cm}^2$. Every measurement has an absolute error of half its smallest unit, and percentage error is that error divided by the measurement, times 100%.

## Starting point: Years 9 and 10

### Units

$$1 \text{ km} = 1000 \text{ m} \qquad 1 \text{ m} = 100 \text{ cm} \qquad 1 \text{ cm} = 10 \text{ mm}$$

For area, square the conversion:

$$1 \text{ m}^2 = 100^2 \text{ cm}^2 = 10\,000 \text{ cm}^2 \qquad 1 \text{ ha} = 10\,000 \text{ m}^2 \qquad 1 \text{ km}^2 = 1\,000\,000 \text{ m}^2$$

### Area formulas

| Shape | Area |
|---|---|
| Rectangle | $A = lw$ |
| Triangle | $A = \dfrac{1}{2}bh$ |
| Parallelogram | $A = bh$ |
| Trapezium | $A = \dfrac{h}{2}(a + b)$ |
| Rhombus or kite | $A = \dfrac{1}{2}xy$ (diagonals $x$ and $y$) |
| Circle | $A = \pi r^2$, circumference $C = 2\pi r$ |
| Sector | $A = \dfrac{\theta}{360} \times \pi r^2$ |

The height $h$ is always measured **at right angles** to the base.

### Calculator tip: Use the π key and round once
For a circle of radius 4.5 cm, type `π × 4.5 x²`. Do not use 3.14. Round only the final answer, to the number of decimal places or significant figures the question asks for.

### Composite shapes

Split a shape into simple pieces and add, or take a piece away and subtract.

### Worked example: A running track
A field is a 100 m by 60 m rectangle with a semicircle on each short end. The two semicircles make one circle of radius 30 m.

$$A = 100 \times 60 + \pi(30)^2 \approx 6000 + 2827 = 8827 \text{ m}^2$$

The perimeter is the two straights plus one full circle: $200 + 2\pi(30) \approx 388.5$ m.

### Accuracy of measuring instruments

No measurement is exact. A measurement is only as accurate as the smallest marking on the instrument.

### Key idea: Limits of accuracy
**Absolute error** $= \pm \dfrac{1}{2} \times$ the smallest unit of measurement.

A length of 8.45 m measured to the nearest centimetre has absolute error $\pm 0.005$ m. The true length is between the **lower bound** 8.445 m and the **upper bound** 8.455 m.

**Relative error** and **percentage error** compare the error with the measurement:

$$\text{relative error} = \frac{\text{absolute error}}{\text{measurement}} \qquad \text{percentage error} = \text{relative error} \times 100\%$$

For 8.45 m: $\dfrac{0.005}{8.45} \times 100\% \approx 0.06\%$.

### Watch out: Area units square too
$1 \text{ m}^2$ is not $100 \text{ cm}^2$. It is $100 \times 100 = 10\,000 \text{ cm}^2$.

### Exam tip: Significant figures
Do not give an answer more precisely than your measurements. Lengths to 3 significant figures give an area to about 3 significant figures.

**Mind map**

```text
Area and accuracy
  Units
    Square the length factor
    1 ha = 10 000 m²
  Formulas
    Triangle ½bh
    Trapezium h/2(a + b)
    Circle πr²
    Sector θ/360 × πr²
  Composite
    Add the pieces
    Subtract holes
  Accuracy
    Error ± half the smallest unit
    Lower and upper bounds
    Percentage error
```

### Year 11: Bounds on a calculated area
The classroom's true area is between the smallest and largest possible values:

$$8.445 \times 6.195 \approx 52.32 \text{ m}^2 \quad \text{and} \quad 8.455 \times 6.205 \approx 52.46 \text{ m}^2$$

The measured area $8.45 \times 6.20 = 52.39$ m² sits in the middle. To be safe, Mr Okafor orders for the upper bound.

**Sectors in radians.** With $\theta$ in radians, sector area $= \dfrac{1}{2}r^2\theta$ and arc length $= r\theta$, which is quicker than the degree formulas.

### Year 12: Areas of curved regions
For a shape with a curved edge that is not part of a circle, Year 12 uses integration. If only measurements are known, such as the widths of a lake taken every 10 m, the trapezoidal rule estimates the area:

$$A \approx \frac{h}{2}\left[y_0 + 2(y_1 + \dots + y_{n-1}) + y_n\right]$$

Surveyors use exactly this method, and its accuracy depends on the accuracy of each measured width.

## Quick summary

- Square the conversion factor for area: $1 \text{ m}^2 = 10\,000 \text{ cm}^2$, $1 \text{ ha} = 10\,000 \text{ m}^2$.
- Know the area formulas for triangles, quadrilaterals, circles and sectors. Height is perpendicular to the base.
- Composite shapes: add the pieces, subtract the holes.
- Absolute error is half the smallest unit. Bounds are the measurement plus and minus it.
- Percentage error $= \dfrac{\text{absolute error}}{\text{measurement}} \times 100\%$.

## Quiz

Try these questions from the chapter quiz.

1. How many cm² are in 3 m²?
   - A. 300
   - B. 3000
   - C. 30 000
   - D. 300 000
   - Answer: C. 30 000. $3 \times 10\,000$.

2. Find the area of a triangle with base 12 cm and height 7 cm.
   - A. 84 cm²
   - B. 42 cm²
   - C. 19 cm²
   - D. 38 cm²
   - Answer: B. 42 cm². $\dfrac{1}{2} \times 12 \times 7$.

3. Find the area of a trapezium with parallel sides 5 m and 9 m and height 4 m.
   - A. 180 m²
   - B. 56 m²
   - C. 28 m²
   - D. 18 m²
   - Answer: C. 28 m². $\dfrac{4}{2}(5 + 9)$.

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