Measures and solids · Ages 14–15
Arc lengths & sectors
From “Arc length” to “Liu Hui's method for π”, step by step, each skill with its methods, worked examples and answers.
Skills and methods in this topic, easy to hard
- Arc lengthArc = θ360 × 2πr
Methodr = 6, θ = 60°: 12π ÷ 6 = 2π cm
Another way: Cancel first: θr/18060 × 6 ÷ 180 = 2 → 2π
- Area of a sectorSector area = θ360 × πr² = 12 lr
Methodr = 6, θ = 60°: 36π ÷ 6 = 6π cm²
Another way: From the arc: 1/2 lr12 × 2π × 6 = 6π
- Perimeter of a sectorPerimeter = arc + 2r
Methodr = 6, θ = 60°: 2π + 12 cm
Another way: Walk right round6 + 2π + 6
- Segments and composite figuresSegment = sector − triangle
Method90°, r = 6: 9π − 18 ≈ 10.26 cm²
Another way: Two quarters − the square2 × 9π − 36 = 2 × (9π − 18)
- Liu Hui's method for πPerimeter ÷ diameter ≈ π, better with more sides
MethodHexagon: 6r ÷ 2r = 3
Another way: By area: the 12-gon's 3r²6 × (1 × 1 ÷ 2) = 3
Worked examples
From different skills in this topic. Have a go first, then open each one for the answer and the working.
Example 1Arc length →
A sector has an angle of 60° at the centre. What fraction of the whole circumference is its arc?
- A. 35
- B. 16
- C. 56
- D. 13
Choose the right answer
Show the answer and working
Answer: B. 16
60° is 60360 = 16 of the whole 360°, so the arc is 16 of the circumference.
Example 2Perimeter of a sector →
The sector in the picture has a radius of 18 cm and an angle of 240°. Its arc is π cm, and its perimeter (π + ) cm.
Fill in each blank
Show the answer and working
Answer: 24, 24, 36
Arc 240360 × 2π × 18 = 24π; the perimeter adds two radii, 2 × 18 = 36.
Example 3Liu Hui's method for π →
The circle in the picture has a radius of 19 cm; the regular hexagon inside it has a perimeter of 114 cm, shorter than the circumference. What does that say about π?
- A. π is less than 3
- B. π is more than 6
- C. π is more than 3
- D. π is exactly 3
Choose the right answer
Show the answer and working
Answer: C. π is more than 3
The circumference 2π × 19 is longer than 114: π × 38 > 114, so π > 114 ÷ 38 = 3.
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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