A radian is the angle you get when you wrap one radius around the edge of a circle. Measure angles this way and the arc length and sector area formulas become \(s = r\theta\) and \(A = \frac{1}{2}r^{2}\theta\), with no 360 in sight. This lesson converts between degrees and radians, uses both formulas and the sector perimeter, and works backwards to a radius or an angle.
One radian is the angle at the centre of a circle subtended by an arc equal in length to the radius, so \(\pi\) rad \(= 180^{\circ}\). With \(\theta\) in radians, a sector of radius \(r\) has arc length \(s = r\theta\), area \(A = \frac{1}{2}r^{2}\theta\) and perimeter \(2r + r\theta\). These formulas are not given in the exam: learn them.
Fig. 1: a sector OAB of a circle with centre O and radius r. The angle at the centre is θ radians and the arc AB has length s. With θ in radians, s = rθ, the shaded area is ½r²θ and the perimeter is r + r + s
Section 01
What is a radian?
Take a circle of radius \(r\) and mark off an arc whose length is exactly \(r\). The angle this arc makes at the centre is one radian. It does not depend on the size of the circle: a bigger circle has a longer radius and a longer arc in the same proportion. The whole circumference is \(2\pi r\), which is \(2\pi\) radii, so a full turn is \(2\pi\) radians. That gives the one fact you need for every conversion: \(\pi\) rad \(= 180^{\circ}\), so 1 rad is about \(57.3^{\circ}\).
Degrees to radians: multiply by \(\dfrac{\pi}{180}\). Leave the answer as a fraction of \(\pi\) when asked for an exact value.
Radians to degrees: multiply by \(\dfrac{180}{\pi}\). An angle written with \(\pi\) converts by replacing \(\pi\) with \(180^{\circ}\).
\(\pi\) rad is \(180^{\circ}\), not \(360^{\circ}\), and 1 rad is NOT \(180^{\circ}\). When you use a calculator for trigonometry in radians, switch it to radian mode first; a degree-mode calculator gives \(\sin 1.2\) as 0.0209 instead of 0.932.
Section 02
Arc length and sector area.
In the lab, drag the red point to change the angle of the sector and watch the arc length and area. The lab measures the angle in degrees, as in IGCSE Mathematics, so it writes the arc as a fraction of 360 of the circumference. Convert its angle to radians and check that \(r\theta\) gives the same arc.
A sector with angle \(\theta\) radians is the fraction \(\dfrac{\theta}{2\pi}\) of the whole circle, because a full turn is \(2\pi\). Take that fraction of the circumference and of the area:
\[ s = \frac{\theta}{2\pi} \times 2\pi r = r\theta \]
\[ A = \frac{\theta}{2\pi} \times \pi r^{2} = \frac{1}{2}r^{2}\theta \]
The \(\pi\) cancels: that is why radians make the formulas so short. The perimeter of the sector is the arc plus two radii, \(2r + r\theta\). On Paper 1 (no calculator), an angle such as \(\frac{2\pi}{3}\) gives exact answers in terms of \(\pi\); leave them that way.
Worked example Arc, area and perimeter
A sector has radius 8 cm and angle 0.75 radians. Find its arc length, area and perimeter.
\(s = r\theta\) and \(A = \frac{1}{2}r^{2}\theta\) only work with \(\theta\) in RADIANS. Putting \(\theta = 60\) (degrees) into \(s = r\theta\) with \(r = 5\) gives an arc of 300 cm round a circle whose whole circumference is about 31 cm. Convert first: \(60^{\circ} = \frac{\pi}{3}\).
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
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Write \(135^{\circ}\) in radians.
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Convert \(2.5\) radians to degrees, to 1 decimal place.
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A sector has radius 4 cm and angle 1.5 radians. What is its arc length?
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A sector has radius 10 cm and angle 0.6 radians. What is its area?
Section 03
Working backwards.
Many questions give you the arc, the area or the perimeter and ask for the radius or the angle. Write down the formula for what you are given, substitute what you know and solve. When two facts are given and both \(r\) and \(\theta\) are unknown, write two equations and eliminate \(r\theta\) or \(\theta\).
Area and radius known: \(\theta = \dfrac{2A}{r^{2}}\).
Arc and radius known: \(\theta = \dfrac{s}{r}\).
Area and arc known: \(A = \frac{1}{2}r(r\theta) = \frac{1}{2}rs\), so \(r = \dfrac{2A}{s}\).
Fixed perimeter \(P\): \(r\theta = P - 2r\), so the area is \(A = \frac{1}{2}r(P - 2r)\), a quadratic in \(r\). Finding the largest such area is an idea you will meet later.
Check every angle you find is between 0 and \(2\pi\) (about 6.28) before you accept it.
The next lesson builds on this: a segment is a sector minus a triangle, and compound shapes are made of sectors, triangles and rectangles.
Worked example Angle from the area
A sector of radius 6 cm has area 30 cm2. Find the angle of the sector and its arc length.
Both angles are less than \(2\pi\), so both sectors exist.
Answer: \(r = 3\) cm, \(\theta = \frac{10}{3}\) rad, or \(r = 5\) cm, \(\theta = 1.2\) rad
Examiner trap
The perimeter of a sector has TWO radii: \(2r + r\theta\), not \(r + r\theta\) and not just the arc \(r\theta\). Leaving out a radius is the most common way to lose every mark in a perimeter question.
Stage 2 · Exam
Exam-style questions
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A sector of a circle has radius 9 cm and angle \(\dfrac{2\pi}{9}\) radians. Find the exact perimeter of the sector.
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A sector has area 48 cm2 and arc length 12 cm. Find the angle of the sector in radians.
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A sector has angle 1 radian and perimeter 30 cm. Find its area.
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A sector of radius \(r\) cm has a perimeter of 20 cm. Which expression gives its area in cm2?
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Arcs and sectors measured in radians.
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