To find the area of a sector, use A = ½r²θ with θ in radians. It is the companion of the arc-length formula in radians, arcs and sectors, and it comes from the same idea: a sector is a fraction of a circle.
Why is there a half?
A sector with angle θ is the fraction θ/2π of the whole circle. The circle has area πr², so the sector has area (θ/2π) × πr². The π cancels and the 2 stays in the denominator, which gives ½r²θ.
Substituting s = rθ gives a second form, A = ½rs. This looks like the area of a triangle with base s and height r, which is a useful way to remember it.
How do you use it, step by step?
- Write the angle in radians, converting if necessary.
- Choose the form: ½r²θ if you have r and θ, or ½rs if you have r and the arc length.
- Substitute and simplify, keeping π if an exact value is wanted.
- Attach square units.
- Check the size: the sector area must be less than the full circle area πr².
Worked example
A sector has radius 6 cm and angle 2π/3. Find its area in terms of π, then to 3 significant figures.
Step 1, angle: already in radians, 2π/3.
Step 2, substitute: A = ½ × 6² × 2π/3 = ½ × 36 × 2π/3.
Step 3, simplify: ½ × 36 = 18, and 18 × 2π/3 = 12π.
Step 4, evaluate: 12π = 37.699… so A = 37.7 cm² to 3 significant figures.
Check: the full circle has area 36π, and 2π/3 is one third of 2π. One third of 36π is 12π. ✓
The mistake to watch for
A common slip is to forget the half, or to put the half on the wrong part.
Mistaken working: A = r²θ = 36 × 2π/3 = 24π
This is exactly twice the correct area. It is two thirds of the whole circle, but a sector with angle 2π/3 covers only one third.
The correction is to compare with the full circle: a sector of one third of the circle cannot have an area greater than 36π/3 = 12π. Whenever your answer is more than the fraction of the circle suggests, look for a missing half.
Check yourself
1. Find the area of a sector with radius 4 cm and angle 1.5 rad.
Show answer
A = ½ × 4² × 1.5 = ½ × 16 × 1.5 = 12.
12 cm²
2. A sector has radius 9 cm and angle π/3. Find its area in terms of π and to 3 significant figures.
Show answer
A = ½ × 81 × π/3 = 27π/2 = 13.5π = 42.411…
27π/2 cm², which is 42.4 cm² to 3 significant figures. Check: π/3 is one sixth of 2π, and 81π ÷ 6 = 13.5π. ✓
3. A sector has area 50 cm² and radius 8 cm. Find the angle in radians.
Show answer
50 = ½ × 64 × θ = 32θ, so θ = 50/32 = 1.5625.
1.5625 rad (1.56 to 3 significant figures). Check: ½ × 64 × 1.5625 = 50. ✓
Where this leads next
Next, see how a sector relates to the triangle inside it in comparing a sector with an enclosed triangle. The non-calculator working trainer helps you keep exact π answers tidy.
Some students can do each formula alone but freeze when the question does not say which to use. A teacher in online one-to-one Additional Mathematics tuition can practise that choice with you.