Sec 3 · G3 (Express)

Arc length and sector area

Practise arc length and sector area with step-by-step help. Free account.

1. Radians

Degrees	Radians
$360^\circ$	$2\pi$
$180^\circ$	$\pi$
$90^\circ$	$\dfrac{\pi}{2}$
$60^\circ$	$\dfrac{\pi}{3}$
$45^\circ$	$\dfrac{\pi}{4}$
$30^\circ$	$\dfrac{\pi}{6}$

Degrees to radians

\displaystyle \times \dfrac{\pi}{180}

e.g.

\displaystyle 60 \times \dfrac{\pi}{180}

\displaystyle = \dfrac{\pi}{3}

Radians to degrees

\displaystyle \times \dfrac{180}{\pi}

e.g.

\displaystyle \dfrac{2\pi}{3} \times \dfrac{180}{\pi}

\displaystyle = 120

$\dfrac{2\pi}{3}$ rad $= 120^\circ$

📖 Worked example

Express $2.6$ radians in degrees, correct to 1 decimal place.

Step 1 of 2

π radians = 180°

To change radians to degrees, multiply by

\dfrac{180}{\pi}

Practise this →

2. Arc length

Left: a sector with centre O, radii r, angle θ at the centre and the arc AB. Right: the segment between the chord AB and the arc, which is the sector minus triangle OAB.

	$\theta$ in degrees	$\theta$ in radians
Arc length	$\dfrac{\theta}{360} \times 2\pi r$	$r\theta$
Perimeter of a sector	$2r + \dfrac{\theta}{360} \times 2\pi r$	$2r + r\theta$

Degrees: r = 7 cm, θ = 40°

\displaystyle s = \dfrac{\theta}{360} \times 2\pi r

e.g.

\displaystyle s = \dfrac{40}{360} \times 2\pi(7)

\displaystyle s = \dfrac{14\pi}{9}

\displaystyle s = 4.89\text{ (3 s.f.)}

Radians: r = 6 cm

\displaystyle s = r\theta, \quad \theta = \dfrac{5\pi}{6}

e.g.

\displaystyle s = 6 \times \dfrac{5\pi}{6}

\displaystyle s = 5\pi

\displaystyle s = 15.7\text{ (3 s.f.)}

📖 Worked example

A sector of a circle has radius $6$ cm and angle $\dfrac{2\pi}{5}$ radians. Find its perimeter, correct to 3 significant figures.

Step 1 of 2

First the arc: s = rθ

\mathstrut s = 6 \times \dfrac{2\pi}{5}

s = 7.5398\ldots

Practise this →

3. Sector area

	$\theta$ in degrees	$\theta$ in radians
Sector area	$\dfrac{\theta}{360} \times \pi r^{2}$	$\dfrac{1}{2}r^{2}\theta$

Radians: r = 6 cm

\displaystyle A = \dfrac{1}{2}r^{2}\theta, \quad \theta = \dfrac{2\pi}{3}

e.g.

\displaystyle A = \dfrac{1}{2} \times 6^{2} \times \dfrac{2\pi}{3}

\displaystyle A = 12\pi

\displaystyle A = 37.7\text{ (3 s.f.)}

Angle from the area: r = 6 cm, A = 40 cm²

\displaystyle \theta = \dfrac{2A}{r^{2}}

e.g.

\displaystyle \theta = \dfrac{2(40)}{6^{2}}

\displaystyle \theta = \dfrac{20}{9}

\displaystyle \theta = 2.22\text{ (3 s.f.)}

📖 Worked example

A sector of a circle of radius $6$ cm has an arc of length $15$ cm. Find the area of the sector.

Step 1 of 2

First the angle: θ = s ÷ r

\mathstrut \theta = \dfrac{15}{6}

\theta = 2.5

Practise this →

4. Segment area

	$\theta$ in radians
Segment area	$\dfrac{1}{2}r^{2}\theta - \dfrac{1}{2}r^{2}\sin\theta = \dfrac{1}{2}r^{2}(\theta - \sin\theta)$
Segment perimeter	arc $+$ chord

r = 6 cm

\displaystyle \dfrac{1}{2}r^{2}(\theta - \sin\theta), \quad \theta = \dfrac{\pi}{4}

e.g.

\displaystyle \dfrac{1}{2} \times 6^{2} \times \left(\dfrac{\pi}{4} - \sin\dfrac{\pi}{4}\right)

\displaystyle = 1.41\text{ (3 s.f.)}

📖 Worked example

$AB$ is a chord of a circle with centre $O$ and radius $6$ cm, and angle $AOB = 120^\circ$ . Find the area of the shaded segment, correct to 3 significant figures.

Step 1 of 3

Area of the sector

\mathstrut \text{Sector} = \dfrac{120}{360} \times \pi(6)^2

\text{Sector} = 37.6991\ldots

Practise this →

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Sec 3 · G3 (Express)

Arc length and sector area

Practise arc length and sector area with step-by-step help. Free account.

Start free Worksheet (PDF)

1. Radians

Degrees	Radians
$360^\circ$	$2\pi$
$180^\circ$	$\pi$
$90^\circ$	$\dfrac{\pi}{2}$
$60^\circ$	$\dfrac{\pi}{3}$
$45^\circ$	$\dfrac{\pi}{4}$
$30^\circ$	$\dfrac{\pi}{6}$

Degrees to radians

\displaystyle \times \dfrac{\pi}{180}

e.g.

\displaystyle 60 \times \dfrac{\pi}{180}

\displaystyle = \dfrac{\pi}{3}

Radians to degrees

\displaystyle \times \dfrac{180}{\pi}

e.g.

\displaystyle \dfrac{2\pi}{3} \times \dfrac{180}{\pi}

\displaystyle = 120

$\dfrac{2\pi}{3}$ rad $= 120^\circ$

📖 Worked example

Express $2.6$ radians in degrees, correct to 1 decimal place.

Step 1 of 2

π radians = 180°

To change radians to degrees, multiply by

\dfrac{180}{\pi}

Practise this →

2. Arc length

	$\theta$ in degrees	$\theta$ in radians
Arc length	$\dfrac{\theta}{360} \times 2\pi r$	$r\theta$
Perimeter of a sector	$2r + \dfrac{\theta}{360} \times 2\pi r$	$2r + r\theta$

Degrees: r = 7 cm, θ = 40°

\displaystyle s = \dfrac{\theta}{360} \times 2\pi r

e.g.

\displaystyle s = \dfrac{40}{360} \times 2\pi(7)

\displaystyle s = \dfrac{14\pi}{9}

\displaystyle s = 4.89\text{ (3 s.f.)}

Radians: r = 6 cm

\displaystyle s = r\theta, \quad \theta = \dfrac{5\pi}{6}

e.g.

\displaystyle s = 6 \times \dfrac{5\pi}{6}

\displaystyle s = 5\pi

\displaystyle s = 15.7\text{ (3 s.f.)}

📖 Worked example

A sector of a circle has radius $6$ cm and angle $\dfrac{2\pi}{5}$ radians. Find its perimeter, correct to 3 significant figures.

Step 1 of 2

First the arc: s = rθ

\mathstrut s = 6 \times \dfrac{2\pi}{5}

s = 7.5398\ldots

Practise this →

3. Sector area

	$\theta$ in degrees	$\theta$ in radians
Sector area	$\dfrac{\theta}{360} \times \pi r^{2}$	$\dfrac{1}{2}r^{2}\theta$

Radians: r = 6 cm

\displaystyle A = \dfrac{1}{2}r^{2}\theta, \quad \theta = \dfrac{2\pi}{3}

e.g.

\displaystyle A = \dfrac{1}{2} \times 6^{2} \times \dfrac{2\pi}{3}

\displaystyle A = 12\pi

\displaystyle A = 37.7\text{ (3 s.f.)}

Angle from the area: r = 6 cm, A = 40 cm²

\displaystyle \theta = \dfrac{2A}{r^{2}}

e.g.

\displaystyle \theta = \dfrac{2(40)}{6^{2}}

\displaystyle \theta = \dfrac{20}{9}

\displaystyle \theta = 2.22\text{ (3 s.f.)}

📖 Worked example

A sector of a circle of radius $6$ cm has an arc of length $15$ cm. Find the area of the sector.

Step 1 of 2

First the angle: θ = s ÷ r

\mathstrut \theta = \dfrac{15}{6}

\theta = 2.5

Practise this →

4. Segment area

	$\theta$ in radians
Segment area	$\dfrac{1}{2}r^{2}\theta - \dfrac{1}{2}r^{2}\sin\theta = \dfrac{1}{2}r^{2}(\theta - \sin\theta)$
Segment perimeter	arc $+$ chord

r = 6 cm

\displaystyle \dfrac{1}{2}r^{2}(\theta - \sin\theta), \quad \theta = \dfrac{\pi}{4}

e.g.

\displaystyle \dfrac{1}{2} \times 6^{2} \times \left(\dfrac{\pi}{4} - \sin\dfrac{\pi}{4}\right)

\displaystyle = 1.41\text{ (3 s.f.)}

📖 Worked example

$AB$ is a chord of a circle with centre $O$ and radius $6$ cm, and angle $AOB = 120^\circ$ . Find the area of the shaded segment, correct to 3 significant figures.

Step 1 of 3

Area of the sector

\mathstrut \text{Sector} = \dfrac{120}{360} \times \pi(6)^2

\text{Sector} = 37.6991\ldots

Practise this →

Try a question: Arc length and sector area

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Practise arc length and sector area with step-by-step help

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