Radians, arc length and sectors

IB Mathematics Analysis & Approaches SL· 25 min read

1. Radian Measure of Angles★☆☆☆☆⏱ 15 min

🚫 No Calculator

Angles do not have to be measured in degrees. The radian is an alternative unit that arises naturally from the geometry of the circle itself, and it is the unit used throughout advanced trigonometry and calculus.

📘 Definition

Radian

$θ$ (in radians)

One radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle. Because the full circumference is $2 π r$ , a complete turn contains $2 π$ radians.

Example:

An arc of length $r$ on a circle of radius $r$ subtends an angle of exactly $1$ radian at the centre.

📐 Worked Example

How many radians are there in a quarter turn (a right angle)?

1
A full turn is $2 π$ radians. A quarter turn is one fourth of a full turn.
2
$\frac{1}{4} \times 2\pi = \frac{\pi}{2}$
3
So a right angle measures $\frac{π}{2}$ radians, which matches $90°$ .

2. Converting Between Degrees and Radians★★☆☆☆⏱ 15 min

🚫 No Calculator

Because $180° = π$ radians, we can convert in either direction by multiplying by the correct fraction. Standard angles convert to exact fractions of $π$ , which are required in non-calculator Paper 1 questions.

📘 Definition

Conversion Rules

To convert degrees to radians, multiply by $\frac{π}{180}$ . To convert radians to degrees, multiply by $\frac{180}{π}$ .

\text{radians} = \text{degrees} \times \frac{\pi}{180}, \qquad \text{degrees} = \text{radians} \times \frac{180}{\pi}

📐 Worked Example

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

1
Multiply the number of degrees by $\frac{π}{180}$ :
2
$150 \times \frac{\pi}{180} = \frac{150\pi}{180}$
3
Simplify the fraction by dividing numerator and denominator by $30$ :
4
$\frac{150\pi}{180} = \frac{5\pi}{6}$

✓ Quick check

Test your conversion skills

What is $\frac{π}{3}$ radians in degrees?
- $30°$
- $45°$
- $60°$
- $90°$
Reveal answer
$60°$ —
Correct! Multiply by $\frac{180}{π}$ : $\frac{π}{3} \times \frac{180}{π} = \frac{180}{3} = 60°$ .

3. Length of an Arc★★☆☆☆⏱ 20 min

✓ Calculator OK

An arc is a fraction of the circumference. A full circle of circumference $2 π r$ corresponds to the angle $2 π$ radians, so an arc for angle $θ$ is the fraction $\frac{θ}{2 π}$ of the whole circumference. This simplifies to a remarkably clean formula.

📘 Definition

Arc Length Formula

$l = r θ$

The length $l$ of an arc that subtends a central angle $θ$ (measured in radians) on a circle of radius $r$ is $l = r θ$ . This formula only works when $θ$ is in radians.

Example:

An arc subtending $2$ radians on a circle of radius $5$ has length $l = 5 \times 2 = 10$ .

📐 Worked Example

A circle has radius $9 cm$ . Find the exact length of the arc that subtends a central angle of $\frac{2 π}{3}$ radians.

1
The angle is already in radians, so apply $l = r θ$ directly:
2
$l = 9 \times \frac{2\pi}{3}$
3
Cancel the factor of $3$ into $9$ :
4
$l = \frac{9 \times 2\pi}{3} = 6\pi \text{ cm}$

4. Area of a Sector★★★☆☆⏱ 20 min

🚫 No Calculator

A sector is the pie-slice region bounded by two radii and an arc. Just like the arc length, a sector is the fraction $\frac{θ}{2 π}$ of the full circle, whose area is $π r^{2}$ . Simplifying $\frac{θ}{2 π} \times π r^{2}$ gives another clean radian formula.

📘 Definition

Sector Area Formula

$A = \frac{1}{2} r^{2} θ$

The area $A$ of a sector with central angle $θ$ (in radians) and radius $r$ is $A = \frac{1}{2} r^{2} θ$ . As with arc length, $θ$ must be in radians.

Example:

A sector of radius $4$ with angle $\frac{π}{2}$ has area $\frac{1}{2} \times 16 \times \frac{π}{2} = 4 π$ .

📐 Worked Example

A sector of a circle has radius $6 cm$ and central angle $\frac{π}{3}$ radians. Find its exact area.

1
Apply the sector area formula $A = \frac{1}{2} r^{2} θ$ with $r = 6$ and $θ = \frac{π}{3}$ :
2
$A = \frac{1}{2} \times 6^2 \times \frac{\pi}{3}$
3
Evaluate $6^{2} = 36$ and simplify step by step:
4
$A = \frac{1}{2} \times 36 \times \frac{\pi}{3} = 18 \times \frac{\pi}{3} = 6\pi \text{ cm}^2$

5. Common Pitfalls

Wrong move:

Using an angle in degrees directly in $l = r θ$ or $A = \frac{1}{2} r^{2} θ$

Why:

Both formulas are derived from the fact that a full circle is $2 π$ radians, so they only give correct results when $θ$ is in radians

Correct move:

Convert any degree angle to radians first by multiplying by $\frac{π}{180}$

Wrong move:

Forgetting the factor of $\frac{1}{2}$ in the sector area formula

Why:

Writing $A = r^{2} θ$ doubles the correct area and mirrors the circumference formula rather than the area formula

Correct move:

Always use $A = \frac{1}{2} r^{2} θ$ , and sanity check that a full circle $(θ = 2 π)$ gives $π r^{2}$

Wrong move:

Confusing the arc length formula with the sector area formula

Why:

$l = r θ$ is linear in $r$ while $A = \frac{1}{2} r^{2} θ$ is quadratic in $r$ ; mixing them gives wrong units

Correct move:

Check units: a length answer should carry cm, an area answer should carry cm², which flags a swapped formula

Wrong move:

Reporting the perimeter of a sector as just the arc length

Why:

The perimeter of a sector also includes the two bounding radii, not only the curved arc

Correct move:

Use $P = r θ + 2 r$ , adding both radii to the arc

6. Quick Reference Cheatsheet

Degrees	Radians
$0°$	$0$
$30°$	$\frac{π}{6}$
$45°$	$\frac{π}{4}$
$60°$	$\frac{π}{3}$
$90°$	$\frac{π}{2}$
$120°$	$\frac{2 π}{3}$
$135°$	$\frac{3 π}{4}$
$150°$	$\frac{5 π}{6}$
$180°$	$π$
$270°$	$\frac{3 π}{2}$
$360°$	$2 π$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · 1
Find the exact area of a sector in terms of π
2024 · 1
Convert an angle to radians and find the arc length

What's Next

Radian measure is the foundation for the rest of IB AA SL trigonometry. Once angles are measured in radians, the arc length and sector area formulas become clean and quick, and the same radian scale is used when you graph trigonometric functions and later differentiate and integrate them. Because Paper 1 is non-calculator, being fluent with exact multiples of π and the two circle formulas will save you time and secure easy marks. Next, you will apply radian angles to the graphs and equations of trigonometric functions.