IB Maths AA SL Topic 3 — Geometry & Trig Paper 1 & 2 ~7 min read

Radian Measure

Radians are just another way to measure angles, alongside degrees. Once you know that π radians = 180°, switching between the two becomes a one-line calculation. By the end of this note you’ll convert in seconds.

📘 What you need to know

A radian is just another unit for measuring angles — like cm vs inches but for angles.
The single fact you need: π radians = 180°. Everything else flows from this.
Degrees → radians: multiply by π180
Radians → degrees: multiply by 180π
In IB exams, use radians by default unless the question says otherwise.

What is a radian?

You’re already familiar with degrees — a full turn is 360°, half a turn is 180°, and so on. Radians are a different way of measuring the same thing: how much you’ve rotated. The change is just the units.

A radian is the angle you sweep out when the arc you draw is exactly as long as the radius. Take a circle, measure the radius, then bend that length around the curve — the angle you’ve covered is 1 radian.

1 radian on the unit circle

🤔 Why does π radians = 180°?

The full circumference of the unit circle is 2π. Half the circumference (going halfway around) is just π. Since a radian counts arc length on the unit circle, a half-turn must be π radians. And a half-turn is also 180°. That’s it — that’s the entire reason behind every radian conversion you’ll ever do.

There’s a special symbol for radians (a small ^c for “circular”), but most of the time you just leave radians unmarked. If you see an angle written without a degree symbol, assume it’s in radians.

The fundamental fact

Every conversion you’ll ever do comes from this single equation:

The radian–degree bridge π radians = 180°

From this, you can scale up or down to any angle. Half of it: π2 = 90°. A third of it: π3 = 60°. Twice it: 2π = 360°. Same idea every time.

How to convert (the practical bit)

Two directions, two multipliers. Pick whichever you need.

Degrees → Radians

multiply by

π180

e.g. 60° × π180 = π3

Radians → Degrees

multiply by

180π

e.g. π3 × 180π = 60°

Easy way to remember: the one with π on top gets you radians (the answer has π in it). The one with 180 on top gets you degrees (the answer is just a number). Whatever is on top of the multiplier ends up in your answer.

Common angles to memorise

These come up over and over again. Memorise them and you won’t waste time converting in the middle of a trig question.

Degrees

Radians (exact)

Radians (decimal)

30°

π6

≈ 0.524

45°

π4

≈ 0.785

60°

π3

≈ 1.047

90°

π2

≈ 1.571

180°

≈ 3.142

270°

3π2

≈ 4.712

360°

2π

≈ 6.283

🧠 Memory trick — the denominator counts how many fit in 180°

For the small ones: π6 means “6 of these fit in a half-turn”, and 180 ÷ 6 = 30°. Same for π4 → 180 ÷ 4 = 45°, and π3 → 180 ÷ 3 = 60°. The denominator divides 180 to give the degrees.

The common angles on a circle

In your IB exam, leave answers as multiples of π wherever possible — like 5π4 rather than 3.927. Exact answers are preferred and don’t lose marks for rounding.

Worked examples

WE 1

Convert a common angle to radians

Convert 120° to radians, leaving your answer as a multiple of π.

Step 1: Pick the multiplier Degrees → radians: multiply by π180. Step 2: Plug in 120 × π180 = 120π180 Step 3: Simplify Divide top and bottom by 60. 2π3 always simplify the fraction!

WE 2

Convert an “ugly” decimal angle

Convert 43.8° to radians, giving your answer to 3 significant figures.

Step 1: Apply the multiplier 43.8 × π180 = 43.8π180 Step 2: Simplify (optional) = 73π300 Step 3: Round to 3 s.f. ≈ 0.764 radians if the question asks for a decimal, give a decimal

WE 3

Convert a multiple of π to degrees

Convert 5π4 radians to degrees.

Step 1: Pick the multiplier Radians → degrees: multiply by 180π. Step 2: Plug in 5π4 × 180π Step 3: π cancels — compute = 5 × 1804 = 9004 = 225° π cancels every time — no calculator needed

WE 4

Recognise a common angle

Without a calculator, find the value of 7π6 in degrees.

Step 1: Spot the structure The denominator is 6 → each π6 piece is 30°. Step 2: Multiply by the numerator 7 × 30° = 210° 210° when you know π/6 = 30°, you don’t need the formula!

WE 5

Add angles in different units

Find the sum of 75° and π3 radians, giving your answer in radians as a multiple of π.

Step 1: Get both into radians first 75° × π180 = 75π180 = 5π12 Step 2: Add (common denominator) 5π12 + π3 = 5π12 + 4π12 Step 3: Combine 9π12 = 3π4 always convert into ONE unit before adding!

💡 Top tips

Default to radians in IB exams. Unless the question literally says “degrees”, give your answer in radians.
Leave answers as multiples of π wherever possible. 5π4 is a better answer than 3.927.
Check your calculator’s mode. Switch between DEG and RAD as needed — getting a radian answer when the calculator is in degrees is a classic mark-loser.
If a question gives you a mix of degrees and radians, convert everything into one unit first, then do the arithmetic.
Memorise the six common angles (30°, 45°, 60°, 90°, 180°, 360°). They appear in nearly every trig problem.
Use the denominator trick: if you see πk, that angle is 180k degrees.
For decimals, write a small ^c after the number (e.g. 0.764^c) only when there might be ambiguity — usually you don’t need it.

⚠ Common mistakes

Multiplying by the wrong fraction. π/180 takes you to radians; 180/π takes you to degrees. Mixing them up gives an answer 180/π or π/180 times too big.
Wrong calculator mode. Doing trig in degree mode when the question is in radians (or vice versa) gives wildly wrong answers — always check before evaluating sin, cos, tan.
Mixing units within a calculation. Adding 30° + π/4 directly gives nonsense — you must convert first.
Forgetting to simplify. 120π/180 isn’t wrong, but it’s not a clean final answer. Simplify to 2π/3.
Rounding too early. Keep π in your working and only switch to a decimal at the end (and only when asked).
Treating π as a unit. π is a number (≈3.14159), not a unit. “π3 rad” and “π3” mean the same thing.
Forgetting that 0 = 0. 0 degrees and 0 radians are the same angle — no conversion needed.

Now you can convert in either direction. The next note (Arcs & Sectors Using Radians) is going to feel like a freebie — once you’re in radians, the formulas get even simpler.

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