One radian is the angle for which:

the arc equals the radius. The definition compares the arc with the radius. The last option describes a different angle — an arc equal to the radius is 1 rad, but a chord equal to the radius is π/3.

3π/4. 135 × π/180 = 135π/180. Dividing top and bottom by 45 gives 3π/4.

An arc of length 15 cm subtends 1.25 rad at the centre. The radius is:

12 cm. Rearranging s = rθ gives r = s/θ = 15 / 1.25 = 12 cm. Multiplying instead of dividing gives 18.75, the tempting wrong answer.

A sector has radius 6 cm and angle 0.5 rad. Its perimeter is:

15 cm. The arc is rθ = 3 cm, but the perimeter also includes both radii: 3 + 6 + 6 = 15 cm. Answering 3 cm is the most common error in the topic.

The area of a sector of radius 4 cm with angle 1.5 rad is:

12 cm². A = ½r²θ = ½ × 16 × 1.5 = 12 cm². Using ½rθ instead of ½r²θ gives 3, and forgetting the ½ gives 24.

Which expression gives the area of a segment?

½r²(θ − sin θ). A segment is the sector minus the triangle: ½r²θ − ½r² sin θ, which factorises to ½r²(θ − sin θ). The order matters — for θ in this range θ > sin θ, so the reversed version would be negative.

Your calculator gives sin 1.2 = 0.0209. This tells you:

the calculator is in degree mode. In radians sin 1.2 ≈ 0.932. Getting 0.0209 means the calculator read 1.2 as 1.2 degrees, so it is in degree mode and every result in the question will be wrong.

Why does s = rθ produce a length even though θ has no unit?

because a length times a pure number is a length. θ is a ratio of two lengths, so it is dimensionless. Multiplying the radius by a pure number leaves the unit of the radius untouched, which is exactly what an arc length should be.

Circular Measure — Mathematics Notes, Worked Examples and Quiz

Why degrees are the awkward unit

Definition

Radian — The angle subtended at the centre of a circle by an arc whose length equals the radius.

There is nothing natural about 360. It reaches us from Babylonian astronomy, where a year was counted as roughly 360 days, and it survives because it divides neatly by so many numbers. It is a convention, and a useful one for navigation and for drawing.

It is a poor unit for mathematics, though, because it has no relationship to the circle it measures. Ask for the length of an arc in degrees and you must first work out what fraction of the full turn you have, then multiply by the full circumference. Two steps, and a fraction to keep track of.

Radians remove that step by defining the angle in terms of the circle itself.

One radian, and why it has no unit

Take a circle, take a piece of string exactly as long as the radius, and bend it round the rim. The angle between the two radii at the ends of that string is one radian. That is the whole definition — no constants, no conversion factor, just a comparison of a curved length against a straight one.

Because a radian is one length divided by another, the units cancel. This is why an angle in radians is a pure number, and why sin θ is happy to accept it while sin 30° quietly needs a degree symbol to be meaningful. It also explains something that confuses people later: in s = rθ the θ contributes no unit at all, so a length times a pure number gives a length, exactly as it should.

Start on One radian: the thick arc is exactly as long as the radius. Then switch to Arc length and Sector area and drag θ — both quantities are just the angle with something multiplied onto it.

Converting, and the values worth knowing cold

A full turn is a circumference of 2πr wrapped onto a radius r, so it is 2π radians. That single fact generates every conversion you will ever need.

Degrees	Radians	Where it turns up
30°	π/6	the 1, 2, √3 triangle
45°	π/4	the isosceles right triangle
60°	π/3	the equilateral triangle
90°	π/2	quarter turn, one asymptote of tan
180°	π	half turn
360°	2π	full turn

2π rad = 360°π rad = 180°1 rad = 180/π ≈ 57.296°degrees → radians: × π/180radians → degrees: × 180/πmultiply by π/180 going in, by 180/π coming out

rad: radiansa ratio of two lengths, so it carries no unit
°: degrees360 to a full turn, purely by convention
× π/180: degrees → radians
× 180/π: radians → degreesthe same fraction, inverted

Set the calculator, then check it

Almost every lost mark in this topic is a calculator left in the wrong mode. Before starting, type sin 1. If the answer is 0.841 you are in radians; if it is 0.0175 you are in degrees. Do this check at the start of the paper, not after an answer looks strange — by then you may have carried the error through several parts.

Arc length

The full circumference 2πr corresponds to the full angle 2π. Arc length is therefore directly proportional to angle, and the constant of proportionality is simply r.

That proportionality is the entire derivation. An angle θ is the fraction θ/2π of a full turn, so the arc is that fraction of 2πr — and the 2π cancels.

s = rθ(θ in radians)arc length = radius × angle

s: arc lengthmeasured along the curve, not across it
r: radius
θ: angle at the centrein radians, always

Worked example

A circle has radius 8 cm. Find the length of the arc subtending an angle of 0.75 radians at the centre.

Check the angle is in radians. It is — the question says so, and there is no degree symbol.The formula is only valid in radians. If the angle had been given in degrees it would need converting first.
s = rθ = 8 × 0.75Straight substitution; no fraction of a turn to work out.
s = 6 cmThe unit is centimetres because θ contributed no unit — a length times a pure number is a length.

s = 6 cm

Sector area

The same proportional argument works for area. A sector is the fraction θ/2π of the disc, and the disc has area πr², so the sector has area (θ/2π) × πr². The π and the 2 tidy up into the standard result.

A = ½r²θ(θ in radians)check: θ = 2π → A = ½r²(2π) = πr²sector area = half the radius squared, times the angle

A: area of the sectortwo radii and an arc — the pizza slice
r: radius
θ: angle at the centrein radians; θ = 2π must return πr²

A useful way to remember it

Sector area is ½ × arc × radius, since ½ × (rθ) × r = ½r²θ. That is the circular version of ½ × base × height for a triangle, which is not a coincidence — a very thin sector is very nearly a triangle with base rθ and height r.

Segments, and the subtraction that defines them

A sector is bounded by two radii and an arc — the slice of pizza. A segment is bounded by a chord and an arc — the slice with the crust, once the triangular part has been cut away.

There is no separate formula worth memorising. A segment is a sector minus the triangle formed by the two radii and the chord, and that triangle has area ½r² sin θ by the ½ab sin C rule, since both enclosing sides are radii.

segment = sector − triangleA = ½r²θ − ½r² sin θ = ½r²(θ − sin θ)the triangle uses ½ab sin C with a = b = r

½r²θ: the sectorbounded by two radii and the arc
½r² sin θ: the triangletwo radii and the chord, by ½ab sin C
θ: angle at the centrein radians throughout, including inside the sine

The three area results together

Sector: ½r²θ — two radii and an arc.
Triangle: ½r² sin θ — two radii and the chord.
Segment: ½r²(θ − sin θ) — the difference between them.
Perimeter of a sector is 2r + rθ: do not forget the two straight sides.
Perimeter of a segment is rθ + 2r sin(θ/2): the arc plus the chord.

Where the marks are actually lost

The mathematics in this topic is short, so examiners test it through the setting up rather than the calculation. Four errors account for most of the lost marks.

Calculator in the wrong mode. Everything downstream is then wrong, and the working looks perfectly correct.
Using a degree angle in a radian formula. s = rθ with θ = 60 rather than π/3 gives an arc nearly sixty times too long. A sanity check helps: an arc cannot exceed the circumference.
Confusing perimeter with arc length. "Find the perimeter of the sector" wants rθ + 2r. Answering rθ alone is the single most common slip in the topic.
Forgetting that the sine also takes radians. In ½r²(θ − sin θ) the θ inside the sine is the same radian value, not a degree conversion.

Read what is being asked for

Area or perimeter? Sector or segment? These questions are worth few marks each and are designed so that the arithmetic is easy — which means the mark is genuinely for identifying the right region. Sketch it and shade the part you want before writing anything down.

Circular Measure