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Circular Measure

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  Cambridge (CIE) IGCSE — Additional Maths

Circular Measure

Big idea: Radians are just a different — and honestly smarter — way to measure angles, one that makes arc lengths and sector areas fall out of dead-simple formulas instead of messy fraction-of-360° calculations.

Summary

  • Radians measure angles using arc length instead of "360 slices" — 1 radian is the angle made when the arc equals the radius.
  • The golden conversion fact: π radians = 180°. Everything else builds from this.
  • An arc is a curved piece of the circumference; a sector is the pizza-slice shape (two radii + an arc).
  • In degrees: arc length l = (θ/360) × 2πr, sector area A = (θ/360) × πr².
  • In radians, these collapse beautifully to: l = rθ and A = ½r²θ.
  • Minor arc/sector = angle < 180°. Major arc/sector = angle > 180° (the "rest of the pizza").
  • Perimeters of sectors/compound shapes = arc length + straight radii (don't forget the straight bits!).
  • Always check your calculator is in the correct mode (degrees vs radians) before computing.

Topic 1 — Radian Measure

1What actually IS a radian?

Degrees are a completely arbitrary unit — someone decided a full turn is 360° a very long time ago (probably because 360 has loads of nice divisors). Radians, on the other hand, are built directly out of the geometry of the circle itself, which is why they make so many formulas simpler later on.

Analogy: Imagine you take a piece of string exactly as long as the radius of a circle, and you lay it along the circumference, starting from one point. Walk your finger along that string-length of arc. The angle you've swept out at the centre, from the start to where your string ends — that angle is defined as 1 radian. You're literally measuring the angle in "radius-lengths of arc."

Formally: 1 radian is the angle in a sector of radius 1 and arc length 1. A circle with radius 1 is called a unit circle, and it's the cleanest way to picture this — because in a unit circle, the arc length and the angle (in radians) are literally the same number.

In the unit circle, an arc of length 1 sweeps out an angle of exactly 1 radian (≈ 57.3°).

Notation note The symbol for radians is technically ᶜ (a small c), but in practice you'll almost always see it written as rad, or with no symbol at all when π is involved (since it's "obviously" radians). You must never drop the ° symbol for degrees though — that one's non-negotiable, since 3 and 3° mean very different things.

2Converting between radians and degrees

This is the single most important fact in the whole chapter — memorise it so hard you could recite it half-asleep:

The golden equation
πc = 180°
π radians is exactly the same angle as 180° (a straight line / half a turn)

From this one fact, you can derive absolutely everything else. Want to convert radians → degrees? Multiply by 180/π. Want degrees → radians? Multiply by π/180.

Conversion rules
radians → degrees:   × 180/π    |    degrees → radians:   × π/180

It's genuinely worth memorising these common conversions outright, because they show up constantly and let you sanity-check your work instantly:

RadiansDegrees
2π360°
π180°
π/290°
π/360°
π/445°
π/630°
How to remember these fast: Notice the denominator of the radian fraction basically tells you "how many of these fit into 180°." π/6 → 180 ÷ 6 = 30°. π/4 → 180 ÷ 4 = 45°. π/3 → 180 ÷ 3 = 60°. Once this clicks, you never need to "look up" a conversion again — you can rebuild it in two seconds.
Worked Example

a) Convert 43.8° to radians.

Step 1 — divide by 180°:
43.8° ÷ 180° = 73/300
Step 2 — multiply by π:
73/300 × π = 0.764 (3 s.f.)

b) Convert 5π/4 to degrees.

Step 1 — divide by π:
(5π/4) ÷ π = 5/4
Step 2 — multiply by 180°:
5/4 × 180° = 225°
Practice Question 1

Convert 150° into radians, giving your answer as an exact multiple of π.

Practice Question 2

Convert 2.4 radians into degrees, giving your answer to 1 decimal place.

Calculator check Always double-check your calculator's mode (DEG vs RAD) before hitting trig or angle calculations. This is one of the most common silent errors students make — the calculator won't warn you, it'll just quietly give you the wrong number.

Topic 2 — Arcs & Sectors

3What are arcs, minor arcs, and major arcs?

An arc is simply a curved piece of a circle's circumference — think of it as the crust of a slice of pizza.

Minor arc → angle < 180° (the small slice's crust) Major arc → angle > 180° (the rest of the pizza's crust)

Whenever an angle at the centre is given, ask yourself: is this describing the small piece, or is it describing everything except the small piece? If a question gives you a minor arc's angle but asks about the major arc, remember the major arc's angle is simply 2π − θ (in radians) or 360° − θ (in degrees).

4Length of an arc

An arc length is just a fraction of the whole circumference — the same fraction that the angle at the centre is of a full turn.

In Degrees
l = (θ/360) × 2πr
θ is the angle in degrees, r is the radius

Now here's where radians earn their keep. Since a full turn in radians is 2π (not 360), the fraction of the circle becomes θ/2π instead of θ/360. Substitute that in:

In Radians — the elegant version
l = rθ
θ MUST be in radians here. r is the radius. That's it — no messy fractions, no 360, no π needed in the formula itself.
Why this matters This is the reason radians exist in the eyes of an exam board: l = rθ is dramatically simpler than the degrees version. Anywhere you see π floating around in a question, working in radians usually lets you cancel it out cleanly — that's a strong hint to switch modes.
Worked Example

A circular pizza has had a slice cut from it — the angle of the slice cut was π/6 rad. The radius of the pizza is 12 cm.

i) Find the length of the outside crust of the slice (the minor arc).

Use l = rθ:
l = 12 × π/6 = 2π cm

ii) Find the perimeter of the remaining pizza (after the slice is removed).

Step 1 — find the major arc's angle:
2π − π/6 = 11π/6
Step 2 — find the major arc length:
M = 12 × 11π/6 = 22π
Step 3 — add the two straight radii cut edges:
22π + 12 + 12 = 22π + 24 cm

Unless told otherwise, leave your answer in exact form (with π) rather than rounding to a decimal.

Watch this trap: The perimeter of a sector or "remaining pizza" shape is NOT just the arc length. It also includes the straight radii forming the two edges. Students very commonly forget the "+ 12 + 12" step above and lose easy marks.
Practice Question 1

A sector has radius 8 cm and an angle of 1.5 radians at the centre. Find the arc length.

Practice Question 2

Find the perimeter of a sector with radius 5 cm and angle π/3 radians (include the straight edges).

5Area of a sector

A sector is the pizza-slice shape itself (the region, not just the crust) — bounded by two radii and an arc. Just like arc length, the area of a sector is a fraction of the area of the whole circle.

In Degrees
A = (θ/360) × πr²
θ is the angle in degrees, r is the radius

Swapping the fraction to θ/2π (since a full turn = 2π radians) and simplifying gives us the clean radian version:

In Radians — the elegant version
A = ½r²θ
θ MUST be in radians. r is the radius.
These are NOT given to you Unlike many formulas in exams, l = rθ and A = ½r²θ are not printed on a formula sheet for this syllabus — you're expected to know them from memory. Drill them until they're automatic.
Worked Example

A sector of radius 6 cm has an area of 30 cm². Find the angle at the centre of the sector in radians.

Use A = ½r²θ:
30 = ½ × 6² × θ
Solve for θ:
θ = (30 × 2) / 6² = 5/3 radians
Practice Question 1

Find the area of a sector with radius 10 cm and angle 0.8 radians.

Practice Question 2

A sector has an angle of π/3 radians and an area of 24π cm². Find the radius.

What to Memorise

Term / FormulaMeaning
πc = 180°The master conversion fact — everything else derives from this
rad → degMultiply by 180/π
deg → radMultiply by π/180
Unit circleA circle with radius 1, used to define what a radian is
1 radianThe angle in a sector of radius 1 with arc length 1 (≈ 57.3°)
Minor arc/sectorAngle at centre < 180°
Major arc/sectorAngle at centre > 180° (= 2π − θ or 360° − θ)
l = rθArc length (θ in radians only)
A = ½r²θSector area (θ in radians only)
l = (θ/360)×2πrArc length (θ in degrees)
A = (θ/360)×πr²Sector area (θ in degrees)
Sector perimeterArc length + 2 × radius (don't forget the straight edges!)

Concepts Checklist

Exam Tips

Wrong calculator modeUsing DEG when the question means radians (or vice versa) is the single most common error in this topic. Get in the habit of checking mode before every calculation.
Forgetting the straight edgesWhen asked for a sector's or compound shape's perimeter, students often calculate the arc length and stop there — forgetting to add the two straight radii.
Give exact answers where possibleUnless told to round, leave answers containing π in exact form (e.g. 22π + 24 cm) rather than converting to a decimal — this is what mark schemes usually expect.
Don't drop the degree symbolYou can omit the radian symbol when π is clearly involved, but the ° symbol for degrees should never be dropped — 45 and 45° are not interchangeable in notation.
Memorise the radian formulasl = rθ and A = ½r²θ aren't provided in the exam — you need these cold, along with πc = 180° to derive any conversion on the spot.
Read carefully: minor vs majorQuestions love to give you the angle of the minor arc/sector and then ask about the major one (or vice versa). Always double check which one is actually being asked for before you calculate.
Also in the full note
  • Topic 2 — Arcs & Sectors
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