Maths › Trigonometry › Radians, arcs and small angles
Radians, arcs and small angles
Degrees are a historical accident. Radians are what circles actually want. Measure angles by arc length and the sector formulae collapse into two short products, calculus starts working properly, and near zero the trig functions flatten into polynomials you can compute in your head.
Builds on Trigonometric graphs and equations.
Where it earns its keep: Circular motion on InkPhysics.
IN THIS TOPIC
- Convert between degrees and radians, and know the standard angles in both.
- Use s = rθ and A = ½r²θ for arcs and sectors.
- Find a segment area as sector minus triangle.
- Apply the small-angle approximations, in radians only.
COMMON MISCONCEPTION
The small-angle approximations work in degrees too.
The natural unit
A radian is the angle whose arc equals the radius. A full turn has circumference 2πr, so it holds exactly 2π radians, and π radians = 180°. The familiar angles become fractions of π. 30° is π/6, 45° is π/4, 60° is π/3, 90° is π/2, and the exact-value table survives the move without a single change.
Arcs, sectors and segments
The reward for the new unit arrives immediately. Arc length and sector area, both on the must-learn list, become
with θ in radians and not a factor of 360 in sight. A segment is then whatever the sector has left over once the triangle joining the two radii is removed, so its area is ½r²θ − ½r² sin θ, and every segment question you will ever be set comes down to that one subtraction.
WORKED EXAMPLE
A sector, measured completely
A sector has radius 6 cm and angle 1.2 radians. Find its arc length, its perimeter, and its area.
Arc: s = 6 × 1.2 = 7.2 cm.
Perimeter: arc plus two radii, 7.2 + 12 = 19.2 cm.
Area: ½ × 36 × 1.2 = 21.6 cm².
Forgetting the two radii in the perimeter is the standard slip. A sector's boundary is an arc and two straight edges, and the word perimeter means all of it.
GUIDED PRACTICE
A segment, by subtraction
Find the area of the minor segment cut off by a chord subtending π/3 at the centre of a circle of radius 5 cm, before opening the working.
Show the working
Sector area: ½ × 25 × π/3 = 13.09 cm².
Triangle area: ½ × 25 × sin (π/3) = 10.83 cm².
Segment = sector − triangle = 2.26 cm².
The triangle uses the same θ the sector used, which is exactly why radians keep this working so short. In degrees you would be juggling two different angle formats in one calculation.
Small angles
Zoom in near θ = 0, in radians, and the trig curves straighten out. All three approximations are printed in the formulae booklet,
so there is nothing here to memorise beyond the condition attached to them. That condition is the entire lesson. Radians only. In degrees the approximation fails at once, since sin 1° is 0.0175 and nowhere near 1. A degree is not the unit these were built in.
Cosine gets a squared term where the other two get nothing, because cos flattens out at its peak while sin and tan cut through zero at 45 degrees of slope. Keep that in mind when a question tells you how many terms to use.
INDEPENDENT PRACTICE
A limit by approximation
For small θ, find the approximate value of (1 − cos 2θ)/(θ sin θ).
Show the working
Replace each piece. cos 2θ ≈ 1 − (2θ)2/2 = 1 − 2θ2, and sin θ ≈ θ.
The numerator becomes 2θ2 and the denominator θ2, so the whole expression is approximately 2.
A numeric check at θ = 0.05 gives 1.9992, and the approximation sharpens as θ shrinks. Notice that the doubled angle went into the cos approximation whole, as 2θ. Substituting the angle exactly as it appears is where these questions are won and lost.
ASSESSMENT FOCUS
- Check the angle mode before anything else. Radian questions say so, and a degree answer to a radian question is simply wrong.
- s = rθ and A = ½r²θ demand θ in radians. Convert first, never after.
- A sector's perimeter includes the two radii, and a segment's area is the sector minus a triangle on the same angle.
- The small-angle approximations are in the booklet, but their radian condition is not. Saying “θ in radians” in words is often worth a mark.
- Substitute multiples like 2θ into the approximations whole, brackets and all, then simplify.
CHECK YOURSELF
An arc of length 10 cm subtends an angle θ at the centre of a circle of radius 8 cm. Find θ and the area of the sector.
Show a hint
Rearrange s = rθ first.
Show the answer
θ = s/r = 10/8 = 1.25 radians.
Area = ½ × 64 × 1.25 = 40 cm².
No degrees appeared anywhere, which is the sign that the formulae were being used as designed.
A radian is one radius of arc; π of them make 180°.
Arc rθ, sector ½r²θ, and for tiny radian angles sine and tan are θ itself.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the radians, arcs and small angles questions page.
CHECK YOUR PROGRESS
Rate how confident you feel with each objective for this lesson. Ratings are saved in this browser, on this device, unless you sign in.
- Convert between degrees and radians, and know the standard angles in both.
- Use s = rθ and A = ½r²θ for arcs and sectors.
- Find a segment area as sector minus triangle.
- Apply the small-angle approximations, in radians only.
Open the full revision checklist to see every objective in the course in one place.