Arc Lengths
Every question with a fully worked solution.
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Theory
An arc is part of the circumference, and the angle \(\theta\) at the centre says what part: \(l = \dfrac{\theta}{360} \times 2\pi r\). With \(r = 6\) cm and \(\theta = 120^\circ\), \(l = \dfrac{1}{3} \times 12\pi = 4\pi\) cm. This Year 8 maths page shows how.
A full turn is \(360^\circ\), so an arc with angle \(\theta\) at the centre is \(\dfrac{\theta}{360}\) of the circumference \(2\pi r\): \(l = \dfrac{\theta}{360} \times 2\pi r\). Simplify \(\dfrac{\theta}{360}\) first: \(\dfrac{144}{360} = \dfrac{2}{5}\).
Exact or rounded. An exact answer keeps \(\pi\); otherwise use the \(\pi\) key and round as asked. Use \(\pi = \dfrac{22}{7}\) only when the question says so.
Minor and major arcs. If the marked angle is the minor one, the major arc uses \(360 - \theta\) (or take the minor arc off the circumference). If the given angle is already reflex, substitute it as it stands.
Perimeter of a sector \(=\) arc \(+\) two radii.
Working backwards. If \(l\) is given, substitute into the same formula and solve for \(\theta\) or \(r\).
Arc length:
How to find an arc length
- Write the fraction \(\dfrac{\theta}{360}\) and simplify it.
- Multiply by the circumference \(2\pi r\).
- Give an exact answer with \(\pi\), or round at the end.
| \(l\) | \(=\) | \(\dfrac{120}{360} \times 2\pi \times 6\) |
| \(\) | \(=\) | \(\dfrac{1}{3} \times 12\pi = 4\pi\) |
\(4\pi\) cm
| \(l\) | \(=\) | \(\dfrac{1}{6} \times 8\pi = \dfrac{4\pi}{3}\) |
| \(P\) | \(=\) | \(4 + 4 + \dfrac{4\pi}{3}\) |
\(\left(8 + \dfrac{4\pi}{3}\right)\) cm
| \(\dfrac{108}{360}\) | \(=\) | \(\dfrac{3}{10}\) |
| \(l\) | \(=\) | \(\dfrac{3}{10} \times 20\pi = 6\pi\) |
\(6\pi\) cm
| \(l\) | \(=\) | \(\dfrac{250}{360} \times 2\pi \times 6\) |
| \(\) | \(=\) | \(26.17\ldots\) |
\(26.2\) cm. The reflex angle is the major arc’s own angle, so it goes straight in.
| \(l\) | \(=\) | \(\dfrac{110}{360} \times 2\pi \times 2.5\) |
| \(\) | \(=\) | \(4.79\ldots\) |
\(4.8\) m
(i) Find \(\theta\).
(ii) Find the major arc, in terms of \(\pi\).
| \(C\) | \(=\) | \(24\pi\) |
| \(\dfrac{\theta}{360}\) | \(=\) | \(\dfrac{5\pi}{24\pi} = \dfrac{5}{24}\) |
| \(\theta\) | \(=\) | \(75\) |
| \(24\pi - 5\pi\) | \(=\) | \(19\pi\) |
(i) \(75^\circ\)
(ii) \(19\pi\) cm
Common pitfalls
Frequently asked questions
What is the formula for arc length?
\(l = \dfrac{\theta}{360} \times 2\pi r\).
What is a major arc?
The longer arc between two points, using \(360 - \theta\) when \(\theta\) is the minor angle.
How do I find the angle from an arc length?
Divide the arc by the circumference, then multiply by \(360\).
What is the perimeter of a sector?
The arc length plus the two radii.
When do I use 22/7?
Only when the question tells you to; otherwise use the \(\pi\) key or leave \(\pi\) in.