Circular measure word problems are solved in three moves: fix one length unit, put angles in radians, and then use the formulas. Most slips come from a unit that changes halfway through.
This lesson completes circular measure. It builds on shaded regions and arc length and sector area.
How do you keep the units consistent?
Before calculating, write a short units table: the length unit, the angle in radians and the unit of the answer. Then every formula uses the same values.
Worked example 1: a garden fence
A sector-shaped garden has radius 4.5 m and angle 80°. The fence along the whole edge is RM12 for each metre. Find the total for the fence.
- Angle in radians: 80 × π ÷ 180 = 1.3963.
- Arc length: 4.5 × 1.3963 = 6.283 m.
- Perimeter: 6.283 + 2 × 4.5 = 15.283 m.
- Total: 15.283 × 12 = RM183.40 (to the nearest sen).
Worked example 2: a windscreen wiper
The wiper arm is 45 cm long. The rubber blade covers the part from 30 cm to 45 cm along the arm, and the arm sweeps through 110°. Find the area swept by the blade, in cm² and in m².
- Angle: 110 × π ÷ 180 = 1.9199 radians.
- Area: ½ × (45² − 30²) × 1.9199 = ½ × 1 125 × 1.9199 = 1 080 cm² (to 3 significant figures).
- Convert: 1 080 ÷ 10 000 = 0.108 m².
The mistake that costs marks
A common unit slip is to divide an area by 100 when changing cm² to m². Take a sector of radius 30 cm and angle 2 radians.
| Step | Wrong | Right |
|---|---|---|
| Area in cm² | ½ × 900 × 2 = 900 | ½ × 900 × 2 = 900 |
| Convert to m² | 900 ÷ 100 = 9 m² | 900 ÷ 10 000 = 0.09 m² |
| Size check | A 9 m² piece would not fit a 30 cm sector | About the size of a dinner plate |
A sector of radius 30 cm has to be smaller than a circle of radius 0.3 m, which is about 0.28 m². The wrong answer of 9 m² fails that size check.
Check yourself
A sector has radius 0.5 m and angle 1.4 radians. Give the arc length in cm and the area in cm².
Answer
Arc length: s = 0.5 × 1.4 = 0.7 m = 70 cm.
Area: ½ × 0.25 × 1.4 = 0.175 m². Since 1 m² = 10 000 cm², this is 0.175 × 10 000 = 1 750 cm².
Check in cm: radius 50 cm, so ½ × 2 500 × 1.4 = 1 750 cm². The two routes agree.
What to study next
Test the whole chapter with the circular measure practice set. The units and significant figure checker helps you check your rounding and units, and a mistake log shows whether units are your recurring slip.
If you want a teacher to watch you work a whole question, see online one-to-one Additional Mathematics tuition.