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Summary

Arc length

​​In a nutshell

An arc length is a section of the circumference of a circle. Now that the angle subtending the arc is measured in radians, the formula to calculate the arc length is simple and very easy to use.



Arc length formula

It has been previously stated that an angle θ\thetaθ​ measured in radians is given by θ=lr\theta = \dfrac {l}{r}θ=rl​, where lll is the arc length and rrr is the radius. Rearranging this formula for lll gives you the formula for arc length, which is useful if the angle is given in radians.

​l=rθ\boxed{l = r \theta}l=rθ​​​
  • lll is the arc length.
  • rrr is the radius of the circle.
  • θ\thetaθ is the angle measured in radians.​
Maths; Radians; KS5 Year 13; Arc length


Example 1

The shape given has a radius of 8 cm8 \ cm8 cm. Calculate the perimeter of the shape.

Maths; Radians; KS5 Year 13; Arc length


First find the angle in the major sector in radians. This can be found by converting from degrees:

​θ=270°=3π2 rads\theta = 270 \degree = \dfrac {3 \pi}{2} \ radsθ=270°=23π​ rads​​


Alternatively this is calculated by

​2π−π2=3π22\pi-\frac{\pi}2=\frac{3\pi}22π−2π​=23π​​​


Find the arc length using the formula:

l=rθ=8×3π2=12πl = r \theta = 8 \times \dfrac{3 \pi}{2} = 12 \pil=rθ=8×23π​=12π


Find the perimeter:

​P=12π+8+8=53.7 cm (3 s.f.)‾P = 12 \pi + 8 + 8 = \underline{53.7 \ cm \ (3 \ s.f.)}P=12π+8+8=53.7 cm (3 s.f.)​​​


Example 2

In the sector shown, the perimeter is five times bigger that the arc length. Find the angle ∠AOB\angle AOB∠AOB​.​

Maths; Radians; KS5 Year 13; Arc length


Write a formula for the perimeter which consists of two radii and the arc length:

P=r+r+rθP = r + r + r\thetaP=r+r+rθ​​


The perimeter is five times bigger than the arc length:

5×rθ=r+r+rθ5 \times r \theta = r + r + r\theta5×rθ=r+r+rθ​​


Solve for θ\thetaθ:

5rθ=r+r+rθ4rθ=2rθ=2r4rθ=0.5 rads‾\begin{aligned}5r \theta &= r + r + r\theta \\4r \theta &= 2r \\\theta &= \dfrac {2r}{4r} \\\theta &= \underline{0.5 \ rads}\end{aligned}5rθ4rθθθ​=r+r+rθ=2r=4r2r​=0.5 rads​​​​



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Area and circumference of circles: Formulae

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FAQs - Frequently Asked Questions

What is the formula for arc length?

The formula for arc length is l = r times theta, where r is the radius and theta is the angle given in radians.

What is an arc length?

An arc length is a section of the circumference of a circle.

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