Radian measure, arc length, sectors and segments
Master radian measure, arc length, sectors and segments in Year 11 VCE Specialist Mathematics. A radian measures an angle by arc length — one radian is the angle a circle's radius subtends as an arc — and with \(\theta\) in radians the arc length is \(s=r\theta\) and the sector area is \(A=\tfrac12 r^2\theta\). It sits in the Space and measurement area of study of the VCE Mathematics Study Design (VCAA), within the Trigonometry topic of Unit 2.
You will learn to convert between degrees and radians using \(\pi\text{ rad}=180^\circ\), find the arc length \(s=r\theta\), the sector area \(A=\tfrac12 r^2\theta\), the perimeter of a sector \(2r+r\theta\) and the segment area \(\tfrac12 r^2(\theta-\sin\theta)\), giving exact answers in terms of \(\pi\) where possible — core measurement skills for circular geometry and later trigonometry.
Theory
A radian measures an angle by arc length: one radian is the angle a circle's radius subtends as an arc. In Year 11 Specialist Mathematics you convert between degrees and radians using \(\pi\text{ rad}=180^\circ\), then use \(\theta\) in radians to find the arc length \(s=r\theta\), the sector area \(A=\tfrac12 r^2\theta\), and the segment area \(\tfrac12 r^2(\theta-\sin\theta)\), giving exact answers in terms of \(\pi\) where possible. This page shows how, with fully worked examples.
A radian is the angle subtended at the centre of a circle by an arc whose length equals the radius. Because the full circumference is \(2\pi r\), a full turn is \(2\pi\) radians. A half turn is therefore \(\pi\) radians, which gives the key link between the two units: \(\pi\text{ rad}=180^\circ\).
Rearranging \(\pi\text{ rad}=180^\circ\) gives the two conversions. To go from degrees to radians multiply by \(\dfrac{\pi}{180}\); to go from radians to degrees multiply by \(\dfrac{180}{\pi}\). Angles that are simple fractions of a half turn become neat multiples of \(\pi\) — for example \(90^\circ=\dfrac{\pi}{2}\), \(60^\circ=\dfrac{\pi}{3}\) and \(45^\circ=\dfrac{\pi}{4}\).
A sector is the "pie slice" region bounded by two radii and the arc between them. The arc is the curved part of the circumference; the angle between the two radii is the central angle \(\theta\). A chord is a straight line joining two points on the circle, and a segment is the region between a chord and the arc it cuts off.
The three measurement formulas — arc length \(s=r\theta\), sector area \(A=\tfrac12 r^2\theta\), and segment area \(\tfrac12 r^2(\theta-\sin\theta)\) — all require the central angle \(\theta\) to be in radians. A segment is a sector with its triangle removed, which is where the \(-\sin\theta\) comes from.
The degree–radian link comes from a half turn being \(\pi\) radians:
So convert by multiplying: degrees \(\to\) radians uses \(\dfrac{\pi}{180}\), radians \(\to\) degrees uses \(\dfrac{180}{\pi}\):
With the central angle \(\theta\) in radians, the arc length and sector area are:
The perimeter of a sector is the two radii plus the arc:
A segment is the sector with its triangle (\(\tfrac12 r^2\sin\theta\)) removed, so the segment area is:
Working with radians, arcs, sectors and segments
- Get the angle into radians. If \(\theta\) is in degrees, multiply by \(\dfrac{\pi}{180}\). Leave it as an exact multiple of \(\pi\) when an exact answer is wanted.
- Arc length: substitute \(r\) and \(\theta\) into \(s=r\theta\).
- Sector area: substitute into \(A=\tfrac12 r^2\theta\) — square \(r\) first, then halve.
- Perimeter of a sector: add the two straight radii to the arc, \(P=2r+r\theta\).
- Segment area: take the sector and subtract the triangle, \(\tfrac12 r^2(\theta-\sin\theta)\), keeping the calculator in radian mode for \(\sin\theta\).
- Rearrange when needed: from \(s=r\theta\) you can find a radius \(r=\dfrac{s}{\theta}\) or an angle \(\theta=\dfrac{s}{r}\).
Keep answers exact (in terms of \(\pi\)) when the angle is a multiple of \(\pi\); round to the requested number of decimal places only when the angle is a plain decimal such as \(1.2\) radians.
Multiply the degrees by \(\dfrac{\pi}{180}\) and simplify:
| \(60^\circ\) | \(=\) | \(60 \times \dfrac{\pi}{180}\) |
| \(=\) | \(\dfrac{60\pi}{180}\) | |
| \(=\) | \(\dfrac{\pi}{3}\) |
So \(60^\circ = \dfrac{\pi}{3}\) radians.
Use the arc-length formula \(s = r\theta\), leaving the answer in terms of \(\pi\):
| \(s\) | \(=\) | \(r\theta\) |
| \(=\) | \(15 \times \dfrac{\pi}{6}\) | |
| \(=\) | \(\dfrac{15\pi}{6}\) | |
| \(=\) | \(\dfrac{5\pi}{2}\) |
The exact arc length is \(\dfrac{5\pi}{2}\) cm.
(i) Arc length \(s = r\theta\):
| \(s\) | \(=\) | \(10 \times 1.2\) |
| \(=\) | \(12\) |
(ii) Sector area \(A = \tfrac12 r^2\theta\) — square \(r\) first:
| \(A\) | \(=\) | \(\tfrac12 \times 10^2 \times 1.2\) |
| \(=\) | \(\tfrac12 \times 100 \times 1.2\) | |
| \(=\) | \(50 \times 1.2\) | |
| \(=\) | \(60\) |
(iii) Perimeter \(=\) two radii \(+\) arc \(= 2r + s\):
| \(P\) | \(=\) | \(2 \times 10 + 12\) |
| \(=\) | \(20 + 12\) | |
| \(=\) | \(32\) |
The arc is \(12\) cm, the area is \(60\) square cm, and the perimeter is \(32\) cm.
A segment is the sector minus its triangle: \(\tfrac12 r^2(\theta - \sin\theta)\):
| \(A\) | \(=\) | \(\tfrac12 r^2\big(\theta - \sin\theta\big)\) |
| \(=\) | \(\tfrac12 \times 10^2 \times \left(\dfrac{\pi}{2} - \sin\dfrac{\pi}{2}\right)\) | |
| \(=\) | \(50\left(\dfrac{\pi}{2} - 1\right)\) | |
| \(=\) | \(25\pi - 50\) |
The exact segment area is \((25\pi - 50)\) square cm.
Common pitfalls
Frequently asked questions
What is a radian?
A radian is the angle subtended at the centre of a circle by an arc whose length equals the radius. A full turn is \(2\pi\) radians and a half turn is \(\pi\) radians, so \(\pi\text{ rad}=180^\circ\).
How do I convert between degrees and radians?
Multiply degrees by \(\dfrac{\pi}{180}\) to get radians, and multiply radians by \(\dfrac{180}{\pi}\) to get degrees. For example \(60^\circ=60\times\dfrac{\pi}{180}=\dfrac{\pi}{3}\).
What is the formula for arc length?
The arc length is \(s=r\theta\), where \(r\) is the radius and \(\theta\) is the central angle in radians. For instance a radius of \(15\) with \(\theta=\dfrac{\pi}{6}\) gives \(s=\dfrac{5\pi}{2}\).
What is the formula for the area of a sector?
The area of a sector is \(A=\tfrac12 r^2\theta\), with \(\theta\) in radians. Square the radius, multiply by the angle, then halve.
How do I find the area of a segment?
A segment is a sector with its triangle removed, so its area is \(\tfrac12 r^2\theta-\tfrac12 r^2\sin\theta=\tfrac12 r^2(\theta-\sin\theta)\). Keep the calculator in radian mode when evaluating \(\sin\theta\).
Why must the angle be in radians?
The formulas \(s=r\theta\) and \(A=\tfrac12 r^2\theta\) come from taking the fraction \(\dfrac{\theta}{2\pi}\) of the whole circle, which only simplifies to these forms when \(\theta\) is measured in radians. In degrees the constant \(\dfrac{\pi}{180}\) would have to be carried through.