# Radians, arc length and sectors

> IB Mathematics Analysis & Approaches SL · Geometry & Trigonometry
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u3-radians-arc-length-and-sectors/

This sub-topic introduces the radian, the natural unit for measuring angles in higher mathematics. You will learn to convert between degrees and radians, and then use radian measure in the two key circle formulas: the length of an arc and the area of a sector.

**Prerequisites:** Circumference and area of a circle; [Right triangle trigonometry and the unit circle](https://www.owlsprep.com/study/ib-math-aa-sl-u3-right-triangle-trigonometry-and-the/)

## Learning objectives

- Define the radian as a measure of angle and relate it to a full circle
- Convert angles between degrees and radians, including exact values
- Use the formula l = rθ to find the length of an arc
- Use the formula A = ½r²θ to find the area of a sector

## Radian Measure of Angles

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.

**Radian** — 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 $2\pi r$, a complete turn contains $2\pi$ radians.

*Notation:* $\theta$ (in radians)

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

> **The key relationship**
>
> A full turn is $360° = 2\pi$ radians, so a half turn gives the conversion anchor you should memorise:  
> $$180° = \pi \text{ radians}$$

**Worked example:** How many radians are there in a quarter turn (a right angle)?

1. A full turn is $2\pi$ 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 $\frac{\pi}{2}$ radians, which matches $90°$.

*Calculator:* forbidden

## Converting Between Degrees and Radians

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

**Conversion Rules** — To convert degrees to radians, multiply by $\frac{\pi}{180}$. To convert radians to degrees, multiply by $\frac{180}{\pi}$.

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

**Worked example:** Convert $150°$ to radians, giving your answer as an exact multiple of $\pi$.

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

**Check your understanding**

Test your conversion skills

1. What is $\frac{\pi}{3}$ radians in degrees?

   - $30°$
   - $45°$
   - $60°$
   - $90°$

   *Why:* Correct! Multiply by $\frac{180}{\pi}$: $\frac{\pi}{3} \times \frac{180}{\pi} = \frac{180}{3} = 60°$.

*Calculator:* forbidden

## Length of an Arc

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

**Arc Length Formula** — The length $l$ of an arc that subtends a central angle $\theta$ (measured in radians) on a circle of radius $r$ is $l = r\theta$. This formula only works when $\theta$ is in radians.

*Notation:* $l = r\theta$

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

> **warning**
>
> The formula $l = r\theta$ requires $\theta$ in **radians**. If an angle is given in degrees, convert it to radians before substituting.

**Worked example:** A circle has radius $9 \text{ cm}$. Find the exact length of the arc that subtends a central angle of $\frac{2\pi}{3}$ radians.

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

*Calculator:* allowed

## Area of a Sector

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

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

*Notation:* $A = \frac{1}{2}r^2\theta$

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

**Worked example:** A sector of a circle has radius $6 \text{ cm}$ and central angle $\frac{\pi}{3}$ radians. Find its exact area.

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

> **Perimeter of a sector**
>
> The perimeter of a sector is the arc plus the two straight radii: $P = l + 2r = r\theta + 2r$. Do not forget the two radii — a common lost mark.

*Calculator:* forbidden

## Common pitfalls

- **Wrong:** Using an angle in degrees directly in $l = r\theta$ or $A = \frac{1}{2}r^2\theta$
  - Why it fails: Both formulas are derived from the fact that a full circle is $2\pi$ radians, so they only give correct results when $\theta$ is in radians
  - Correct: Convert any degree angle to radians first by multiplying by $\frac{\pi}{180}$
- **Wrong:** Forgetting the factor of $\frac{1}{2}$ in the sector area formula
  - Why it fails: Writing $A = r^2\theta$ doubles the correct area and mirrors the circumference formula rather than the area formula
  - Correct: Always use $A = \frac{1}{2}r^2\theta$, and sanity check that a full circle $(\theta = 2\pi)$ gives $\pi r^2$
- **Wrong:** Confusing the arc length formula with the sector area formula
  - Why it fails: $l = r\theta$ is linear in $r$ while $A = \frac{1}{2}r^2\theta$ is quadratic in $r$; mixing them gives wrong units
  - Correct: Check units: a length answer should carry cm, an area answer should carry cm², which flags a swapped formula
- **Wrong:** Reporting the perimeter of a sector as just the arc length
  - Why it fails: The perimeter of a sector also includes the two bounding radii, not only the curved arc
  - Correct: Use $P = r\theta + 2r$, adding both radii to the arc

## Cheatsheet

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

## 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.

- [Right triangle trigonometry and the unit circle](https://www.owlsprep.com/study/ib-math-aa-sl-u3-right-triangle-trigonometry-and-the/)
- [Graphs of trigonometric functions](https://www.owlsprep.com/study/ib-math-aa-sl-u3-graphs-of-trigonometric-functions/)
- [Solving trigonometric equations](https://www.owlsprep.com/study/ib-math-aa-sl-u3-solving-trigonometric-equations/)

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