IB Maths AA HL Topic 3 — Geometry & Trigonometry Paper 1 & 2 ~6 min read

Radian Measure

Radians are the natural unit for angles in higher maths. Instead of dividing the circle into 360 equal parts, radians measure an angle by the arc length on the unit circle that the angle cuts off — so a full turn is 2π radians, not 360°. This single change unlocks much cleaner formulas in trigonometry, calculus, and circular motion. From this point on, your default in IB AA HL should be radians unless degrees are explicitly asked for.

📘 What you need to know

Definition: 1 radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.
Full turn: 2π radians = 360°. Half turn: π radians = 180°.
The key conversion: π radians = 180°. Everything else follows.
Degrees → radians: multiply by π/180.
Radians → degrees: multiply by 180/π.
No unit symbol needed — an angle written without ° is assumed to be in radians. The superscript ^c exists but is rarely used.
Forms: radians are typically written as exact multiples of π (like 3π/4) for “nice” angles, or as decimal approximations otherwise.
Calculator mode matters: if your calculator is in degree mode and you give it a radian input (or vice versa), every trig answer will be wrong.

What are radians?

Imagine a unit circle (radius 1). Take an angle θ at the centre, and look at the arc it cuts on the circumference. The radian measure of θ is simply the length of that arc. Because the full circumference of the unit circle is 2π, a complete revolution is 2π radians.

The fundamental conversion π radians = 180°

Everything else flows from this single equation. Halve both sides: π/2 = 90°. Divide by 3: π/3 = 60°. Divide by 6: π/6 = 30°. The “nice” angles all have radian measures involving simple fractions of π.

Why bother switching? In radians, the arc length on a circle of radius r is just l = rθ — no factors of 360 floating around. Differentiation rules in trigonometry only work in radians: d/dx(sin x) = cos x only when x is in radians. The whole machinery is cleaner.

Converting between radians and degrees

Degrees → Radians

multiply by π/180

e.g. 90 × π/180 = π/2

Radians → Degrees

multiply by 180/π

e.g. π/3 × 180/π = 60°

The two factors π/180 and 180/π are reciprocals of each other — applying both takes you back to where you started, which is a useful sanity check for any conversion.

Common conversions to memorise

These come up over and over again — knowing them by heart saves time in every trig and calculus question that follows.

Degrees	Radians (exact)	Radians (decimal, 3 s.f.)
0°	0	0
30°	π/6	0.524
45°	π/4	0.785
60°	π/3	1.05
90°	π/2	1.57
120°	2π/3	2.09
135°	3π/4	2.36
150°	5π/6	2.62
180°	π	3.14
270°	3π/2	4.71
360°	2π	6.28

A useful pattern: if the degree measure is a “nice” multiple of 30 or 45, the radian form involves either π/6 or π/4 as a building block. Quick mental check — count how many 30° (or 45°) blocks fit in your angle, and that’s the multiplier.

🧭 Recipe — converting angles between units

Identify the starting unit: degrees usually have the ° symbol; radians often appear as multiples of π or as decimals without symbols.
Pick the conversion factor: π/180 going to radians, or 180/π going to degrees.
Multiply and simplify. Cancel π or factor out common terms wherever possible.
For “nice” angles: leave answers as exact fractions of π (e.g., 5π/4 not 3.927).
For arbitrary angles: round to 3 s.f. unless told otherwise.
Sanity check: 1 radian ≈ 57.3°. If your radian answer is bigger than 7 or so, the angle is bigger than a full turn — possibly an error or something to reduce.

Worked examples

WE 1

Convert a standard degree angle to radians

Convert 135° to radians, giving your answer as an exact multiple of π.

Step 1: Multiply by π/180 135 × π/180 = 135π/180 Step 2: Simplify (divide top and bottom by 45) 135 ÷ 45 = 3; 180 ÷ 45 = 4 135° = 3π/4 check: 3π/4 × 180/π = 540/4 = 135 ✓

WE 2

Convert a standard radian angle to degrees

Convert 7π/6 to degrees.

Step 1: Multiply by 180/π 7π/6 × 180/π Step 2: Cancel the π 7/6 × 180 = 7 × 30 7π/6 = 210° a quick anchor: π/6 = 30°, so 7π/6 = 7 × 30° = 210°

WE 3

Convert a non-standard degree angle to radians

Convert 67.5° to radians, giving your answer (a) as an exact multiple of π, and (b) as a decimal correct to 3 s.f.

Step 1: Multiply by π/180 67.5 × π/180 = 67.5π/180 Step 2: Simplify (multiply top and bottom by 2 to clear the decimal) 67.5π/180 = 135π/360 = 3π/8 (a) 67.5° = 3π/8 Step 3: Decimal value 3π/8 ≈ 9.4248/8 ≈ 1.1781… (b) 67.5° ≈ 1.18 rad (3 s.f.) 67.5 = 1.5 × 45, so the answer is 1.5 × π/4 = 3π/8 — a faster mental route

WE 4

Convert a decimal radian value to degrees

Convert 2.5 radians to degrees, giving your answer correct to 3 significant figures.

Step 1: Multiply by 180/π 2.5 × 180/π = 450/π Step 2: Calculate 450/π ≈ 450/3.14159… ≈ 143.239… 2.5 rad ≈ 143° (3 s.f.) 2.5 rad isn’t a “nice” angle, so an exact form would just be 450/π — decimal is the natural answer

WE 5

Reduce an angle to the range [0, 2π)

An angle of 7π/2 radians is given. Find the equivalent angle in the range [0, 2π) and state the equivalent in degrees.

Step 1: Subtract full revolutions (2π) until the angle lies in [0, 2π) 7π/2 − 2π = 7π/2 − 4π/2 = 3π/2 3π/2 lies in [0, 2π)? Yes — since 3π/2 < 2π Step 2: Convert 3π/2 to degrees 3π/2 × 180/π = 3 × 90 = 270° 3π/2 radians, equivalent to 270° 7π/2 = 3.5π is one and three-quarter full turns — landing at the same position as 3π/2 (three-quarter turn)

WE 6

Mixed conversion problem

Convert each of the following: (a) 4π/9 to degrees; (b) 225° to radians (exact); (c) 1.2 radians to degrees (3 s.f.).

(a) 4π/9 to degrees: multiply by 180/π 4π/9 × 180/π = 4 × 180/9 = 4 × 20 (a) 4π/9 = 80° (b) 225° to radians: multiply by π/180 225 × π/180 = 225π/180 simplify: 225/180 = 5/4 (b) 225° = 5π/4 (c) 1.2 rad to degrees: multiply by 180/π 1.2 × 180/π = 216/π ≈ 68.755… (c) 1.2 rad ≈ 68.8° (3 s.f.) three quick conversions — each uses the same two-factor toolkit

💡 Top tips

Default to radians in IB AA HL — it’s the expected unit unless degrees are specified.
Check your calculator mode before every trig calculation. Most calculators show RAD or DEG in a corner of the display.
Memorise the common multiples of π/6, π/4, π/3. Faster than reaching for the conversion formula every time.
1 radian ≈ 57.3°. Useful as a sanity check — if a degree-to-radian conversion gives a number bigger than 7, you’ve probably gone wrong somewhere.
Leave exact answers as multiples of π unless decimals are requested. 5π/4 is more precise (and often shorter to write) than 3.927.
Reduce angles to [0, 2π) by subtracting 2π until they land in range — useful for trig identities and unit-circle problems.
The two factors are reciprocals: π/180 and 180/π. If you forget which way around, multiply your input by both — one cancels neatly, the other doesn’t.

⚠ Common mistakes

Wrong calculator mode. Trig values change drastically between degree and radian mode — sin(30°) = 0.5 but sin(30 rad) ≈ −0.988.
Using the wrong conversion factor. Going to radians needs π/180; going to degrees needs 180/π. Swapping them gives an answer off by a factor of (180/π)².
Forgetting to cancel π. When converting π/3 to degrees: π/3 × 180/π = 60°, not 60π°.
Treating π as 3 in mental approximations. π ≈ 3.14159, not 3 — being too rough loses noticeable accuracy.
Mixing units within one expression. Don’t write things like “π/4 + 30°” — convert to a single unit first.
Using 360° in radian formulas. Once you’ve switched to radians, the “fraction of the circle” is θ/(2π), not θ/360.
Forgetting that radians have no unit symbol. An expression like “x = π/6″ already says x is in radians — no ^c needed.

With radians under your belt, the next note — Arcs & Sectors Using Radians — finally pays off the investment. The arc length formula collapses from (θ/360) × 2πr down to just l = rθ, and the sector area becomes A = ½r²θ. Both of those are in the formula booklet — and both are dramatically cleaner than their degree counterparts.

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