Radius times the angle (in radians): The arc length of a sector is s = rθ, where θ is the central angle in radians. (It's a fraction θ/2π of the whole circumference 2πr.)
Interactive: change the sector angle and watch the arc length (s = rθ) and sector area (½r²θ) update.
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IB-style question — arc length
A sector has radius 8 and central angle π/3. Find the arc length.
Step by step
- Use s = rθ.
- Evaluate.
Final answer
Arc length = 8π/3 ≈ 8.38.
θ MUST be in radians: s = rθ only works with θ in radians — convert a degree angle first.
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Half r-squared times the angle: The area of a sector is A = ½r²θ, with θ in radians (a fraction θ/2π of the whole circle area πr²).
IB-style question — sector area
A sector has radius 6 and central angle 1.5 radians. Find its area.
Step by step
- Use A = ½r²θ.
- Evaluate.
Final answer
Area = 27.
The sector from the question: radius 6, angle 1.5 rad. Area = ½r²θ = ½ × 36 × 1.5 = 27 (θ must be in radians).
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Square the radius first: Work out r² before multiplying by the angle and the ½ — keeps the arithmetic clean.
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Rearrange for r or θ: Given an arc length or sector area, substitute and solve for the unknown radius or angle. From s = rθ: θ = s/r. From A = ½r²θ: solve for whichever is missing.
IB-style question — find the angle
A sector of radius 5 has arc length 15. Find the central angle θ.
Step by step
- Use s = rθ.
- Solve.
Final answer
θ = 3 radians.
Working backwards: radius 5, arc length 15, so θ = s/r = 15/5 = 3 rad — a wide sector (3 rad ≈ 172°, nearly a half-turn).
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Cancel π if it appears: If both sides carry π (e.g. area = 6π), divide it out before solving — the numbers simplify.
Perimeter = arc + two radii; segment = sector − triangle: The perimeter of a sector is the arc plus the two straight radii (s + 2r). The area of a segment (between a chord and the arc) is the sector area minus the triangle area: ½r²θ − ½r²sinθ.
IB-style question — perimeter then segment
A sector has radius 10 and angle π/2. Find (a) its perimeter and (b) the area of the segment cut off by the chord.
Step by step
- (a) Perimeter = arc + 2r.
- (b) Segment = sector − triangle.
Final answer
(a) 5π + 20 ≈ 35.7; (b) ≈ 28.5.
Radius 10, angle π/2 (90°). Perimeter = arc + 2r = 5π + 20 ≈ 35.7; the shaded segment (sector − triangle) ≈ 78.5 − 50 = 28.5.
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Triangle uses ½r²sinθ: The triangle between the two radii has area ½r²sinθ (two sides r with included angle θ) — subtract it from the sector for the segment.