1The idea in one paragraph

Degrees split a full turn into 360 parts, a number chosen by ancient astronomers, not by the circle. A radian is measured by the circle itself: it is the angle at the centre that cuts off an arc exactly as long as the radius. A full turn contains 2π of them, because the circumference is 2π radii long, so 2π radians = 360° and π radians = 180°. Measured this way, the arc cut off by an angle θ is simply r times θ, and the sector it cuts off has area ½r²θ. Every formula in this subtopic comes from asking "what fraction of the whole circle is this?", and radians make the answer tidy. They make calculus tidy too, which is why the rest of the course assumes them.

2What a radian is

One radian is the angle at the centre of a circle subtended by an arc equal in length to the radius.

Figure 1 draws it. Take a circle of radius r, walk a distance r around its edge, and join both ends to the centre. The angle between those radii is 1 radian, about 57.3°. It does not depend on the size of the circle: a bigger circle has a bigger radius and a proportionally bigger arc, so the angle stays the same.

How many radians in a full turn? The circumference is 2πr, which is 2π lots of r, so 2π arcs of length r fit around the circle. A full turn is therefore 2π radians, about 6.28.

Radians are pure numbers. An angle in radians is a length divided by a length (arc ÷ radius), so the units cancel. That is why you will often see an angle written 1.2 with no unit at all: no unit means radians. Degrees always carry their ° sign.

3Converting between degrees and radians

Since π rad = 180°:

Degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π.

The guide says radian measure may be written as exact multiples of π or as decimals, and both are correct. On Paper 1 you will usually work with exact multiples, and the angles in Figure 2 are the ones worth knowing on sight. Each is a simple fraction of the half turn π: 30° is one sixth of 180°, so it is π/6.

Calculator mode. On examination papers, radian measure should be assumed unless a question says otherwise. Check the mode before every trigonometric calculation: sin 1.2 in radian mode is 0.932, and in degree mode it is 0.0209. If a question gives angles in degrees, either switch to degree mode or convert.

4Arc length

An arc is part of the circumference. An angle θ at the centre is the fraction θ/2π of a full turn, so its arc is that fraction of the whole circumference 2πr, as Figure 3(a) shows.

Arc length l = rθ and sector area A = ½r²θ, with θ in radians · both in the formula booklet.

This is the pay-off for radians. In degrees the same formula would be l = (θ/360) × 2πr, with an awkward constant in it.

Exact (Paper 1). A circle has radius 9 cm. Find the length of the arc cut off by an angle of 2π/3 at the centre: l = 9 × 2π/3 = 6π cm.

Decimal. Radius 12 cm, angle 1.4 radians: l = 12 × 1.4 = 16.8 cm. No π anywhere, because 1.4 is already a number of radians.

Degrees given. Radius 8 cm, angle 75°. Convert first: 75° = 75π/180 = 5π/12. Then l = 8 × 5π/12 = 10π/3 ≈ 10.5 cm. Putting 75 straight into rθ gives 600 cm, an arc longer than the whole circumference, which a sketch would expose at once.

Backwards. An arc of length 15 cm on a circle of radius 6 cm: θ = l ÷ r = 15 ÷ 6 = 2.5 rad. This is how the definition works in practice: the angle in radians is how many radii fit along the arc.

5Sector area

A sector is the region between two radii and the arc joining them, a slice of pizza. By the same fraction argument, Figure 3(b):

Exact (Paper 1). Radius 9 cm, angle 2π/3: A = ½ × 81 × 2π/3 = 27π cm².

Decimal. Radius 12 cm, angle 1.4 rad: A = ½ × 144 × 1.4 = 100.8 cm².

Backwards. A sector has angle 0.8 rad and area 40 cm². Then ½r²(0.8) = 40, so r² = 100 and r = 10 cm. Reject r = −10: a radius is a length.

Perimeter of a sector. The boundary of a sector is two radii and an arc, so perimeter = 2r + rθ. For radius 9 and angle 2π/3 this is 18 + 6π cm. Forgetting the two radii is the commonest slip in perimeter questions.

Minor and major sectors. Two radii cut a circle into two sectors. The smaller is the minor sector, with angle θ; the larger is the major sector, with angle 2π − θ. Figure 4 shows a circle of radius 6 with θ = π/3.

6Segments: sectors meet triangles

A segment is the region between a chord and its arc. Figure 5 shows why its area is a subtraction: the sector contains the segment plus an isosceles triangle with two sides r and the angle θ between them. The triangle's area comes from 3.2, ½ab sin C with a = b = r.

Worked example (Paper 2). A chord AB of a circle of radius 10 cm subtends an angle of 1.2 radians at the centre O. Find the area of the minor segment and the length of the chord.

The chord can also be found as 2r sin(θ/2) = 20 sin 0.6 = 11.3 cm, by splitting the isosceles triangle into two right-angled triangles. Either route is fine; the cosine rule is the one you already have.

This is where a calculator in degree mode does real damage: sin 1.2 in degrees is 0.0209, which makes the triangle almost nothing and the segment almost the whole sector.

7Working backwards with two unknowns

Examiners like to give two facts about a sector and ask for both r and θ. The method: write each fact as an equation, eliminate θ, solve for r, then check that θ makes sense (it must be positive and less than 2π for a sector).

Worked example (Paper 1). A sector has perimeter 20 cm and area 16 cm². Find its radius and its angle.

So r = 8 cm and θ = 0.5 radians. Check: perimeter 16 + 4 = 20 and area ½ × 64 × 0.5 = 16. The rejected root is the mark examiners are testing: a quadratic gives two candidates, and the context decides which survive. Sometimes both do, and then you give both.

8A context: the windscreen wiper

Real shapes are often a sector with a smaller sector removed. A windscreen wiper blade runs from 12 cm to 55 cm from its pivot and sweeps through 120°. Find the area of glass it cleans. Figure 6 shows the region.

The question gave degrees, so the first line converts. Leaving the answer as 2881π/3 cm² would be the exact form on Paper 1.