When a question gives a sector’s perimeter together with its area or another length, write each condition as an equation and eliminate θ. This lesson in radians, arcs and sectors uses both the arc-length and the sector-area formulas at once, and usually ends in a quadratic.
How do you turn the words into equations?
For a sector with radius r and angle θ in radians:
- Perimeter: P = 2r + rθ (two radii and the arc).
- Area: A = ½r²θ.
- Key link: the arc is s = rθ, so A = ½ r s.
The trick is to use the perimeter to write the arc in terms of r: rθ = P − 2r. Then the area becomes A = ½ r (P − 2r), which has only one unknown.
How do you solve it, step by step?
- Write the perimeter equation and isolate the arc: rθ = P − 2r.
- Write the area as ½ r × (arc) and substitute the arc from step 1.
- Form a quadratic in r and solve it by factorising or the formula. The quadratic structure explorer lets you check the roots.
- Find θ for each r using θ = arc ÷ r.
- Check each pair against the original conditions, and that 0 < θ ≤ 2π.
Worked example
A sector has perimeter 24 cm and area 32 cm². Find the possible radii and the matching angles.
Step 1, perimeter: 2r + rθ = 24, so the arc rθ = 24 − 2r.
Step 2, area: ½ r (24 − 2r) = 32, so r(12 − r) = 32.
Step 3, quadratic: 12r − r² = 32, so r² − 12r + 32 = 0. Factorise: (r − 4)(r − 8) = 0, so r = 4 or r = 8.
Step 4, angles: if r = 4, arc = 24 − 8 = 16, so θ = 16/4 = 4 rad. If r = 8, arc = 24 − 16 = 8, so θ = 8/8 = 1 rad.
Step 5, check: r = 4, θ = 4: perimeter 8 + 16 = 24 ✓, area ½ × 16 × 4 = 32 ✓. r = 8, θ = 1: perimeter 16 + 8 = 24 ✓, area ½ × 64 × 1 = 32 ✓. Both angles are below 2π, so both are valid.
The answers are r = 4 cm with θ = 4 rad, or r = 8 cm with θ = 1 rad.
The mistake to watch for
A common slip is to forget the two radii in the perimeter.
Mistaken working: perimeter = rθ = 24, so the arc is 24 cm.
That treats the arc as the whole perimeter. The straight edges add 2r, so the arc is 24 − 2r, which changes every later line.
The correction is to sketch the sector and trace the boundary with a finger: radius, arc, radius. Write “2r + arc” before any substitution.
Check yourself
1. A sector has radius 9 cm and perimeter 30 cm. Find the angle in radians.
Show answer
Arc = 30 − 2 × 9 = 12. θ = arc ÷ r = 12/9 = 4/3.
θ = 4/3 rad (1.33 to 3 significant figures).
2. A sector has angle 1 rad and perimeter 18 cm. Find its radius.
Show answer
P = 2r + r × 1 = 3r = 18, so r = 6.
6 cm. Check: arc = 6, perimeter = 12 + 6 = 18. ✓
3. A sector has perimeter 20 cm and area 25 cm². Find the radius and the angle, and explain why there is only one answer.
Show answer
Arc = 20 − 2r, so ½ r (20 − 2r) = 25, giving r(10 − r) = 25, so r² − 10r + 25 = 0 and (r − 5)² = 0. Then r = 5 and θ = (20 − 10)/5 = 2.
r = 5 cm, θ = 2 rad. The quadratic has a repeated root, so only one radius works. Check: area ½ × 25 × 2 = 25. ✓
Where this leads next
Test the whole module with the radians, arcs and sectors practice set. The non-calculator working trainer helps with the exact arithmetic.
If the algebra in the middle of these questions is where you slow down, our teachers can work through it with you in online one-to-one Additional Mathematics tuition.