Radians, arc length and sector area

Mathematics Analysis and Approaches SL· Geometry & Trigonometry Unit 3· 12 min read

1. Radian Measure and Degree Conversion★★☆☆☆⏱ 3 min

Radians are the standard angle unit for all advanced trigonometry and calculus in IB AA SL. You will be penalized for using degrees in later calculus problems, so mastering conversion is non-negotiable. A full 360° rotation equals 2π radians, so the core conversion identity is $18 0^{\circ} = π$ radians.

📘 Definition

Radian

rad

The angle formed when the arc length of a sector is exactly equal to the radius of the parent circle

Example:

A 90° right angle equals π/2 radians

1\degree = \frac{\pi}{180} \text{ rad}, \quad 1 \text{ rad} = \frac{180^\circ}{\pi}

📐 Worked Example

Convert 150° to radians, and 3π/4 radians to degrees.

1
To convert degrees to radians, multiply the degree value by π/180:
2
$150^\circ \times \frac{\pi}{180} = \frac{150\pi}{180} = \frac{5\pi}{6} \text{ rad}$
3
To convert radians to degrees, multiply the radian value by 180/π:
4
$\frac{3\pi}{4} \times \frac{180^\circ}{\pi} = \frac{540^\circ}{4} = 135^\circ$

✓ Quick check

Test your conversion skills quickly:

What is 60° in radians?
- π/3
- π/6
- π/4
- 2π/3
Reveal answer
π/3 —
60 is 1/3 of 180, so 1/3 π radians.

2. Arc Length Formula★★☆☆☆⏱ 3 min

Arc length is the length of the curved segment between two points on a circle's circumference. The formula is far simpler when angles are measured in radians, and you do not need to use the more complex fraction-based version that uses degrees.

🔬 Derivation

Goal:

Derive the arc length formula

Starting from:

Full circumference of a circle = 2πr, full central angle around the centre = 2π radians

1
The arc length for a full 2π radian rotation is 2πr
2
Scale the circumference proportionally for a smaller angle θ measured in radians
3
Cancel the 2π terms to simplify the expression

Result:

Final simplified arc length formula: $s = r θ$

📐 Worked Example

Find the arc length of a sector with radius 7 cm and central angle 2.3 radians.

1
Identify given values: r = 7 cm, θ = 2.3 rad
2
$s = r\theta = 7 \times 2.3 = 16.1 \text{ cm}$

3. Sector Area Formula★★★☆☆⏱ 3 min

Sector area is the total area enclosed by two radii and the connecting arc. Both standard forms of the formula are provided in your IB formula booklet, but memorizing them will save you valuable time during the exam.

A = \frac{1}{2} r^2 \theta, \quad A = \frac{1}{2} r s

📐 Worked Example

Calculate the area of a sector with radius 12 cm and central angle π/3 radians.

1
Substitute r=12, θ=π/3 into the first sector area formula:
2
$A = \frac{1}{2} \times 12^2 \times \frac{\pi}{3} = \frac{1}{2} \times 144 \times \frac{\pi}{3} = 24\pi \approx 75.4 \text{ cm}^2$

4. Composite Sector Problem Solving★★★★☆⏱ 3 min

Most IB AA SL exam questions on this topic combine sectors with other shapes like triangles, rectangles or other circles to create composite area or perimeter problems. You will need to break the shape down into separate components to calculate the required value.

📐 Worked Example

A sector of radius 5 cm has central angle 0.9 radians. Find the area of the segment formed by the chord connecting the two ends of the arc.

1
The segment area equals total sector area minus the area of the isosceles triangle formed by the two radii and the chord.
2
$Sector area = 0.5 \times 5^2 \times 0.9 = 11.25 \text{ cm}^2$
3
$Triangle area = 0.5 r^2 \sin\theta = 0.5 \times 25 \times \sin(0.9) \approx 9.79 \text{ cm}^2$
4
$Segment area = 11.25 - 9.79 = 1.46 \text{ cm}^2 (3 s.f.)$

5. Common Pitfalls

Wrong move:

Using degree mode on your calculator when θ is given in radians

Why:

The $r θ$ and $sin (θ)$ formulas only work for radians, leading to drastically incorrect answers

Correct move:

Double check your mode setting every time you see a π or decimal angle with no degree symbol

Wrong move:

Forgetting to add the two radii when calculating sector perimeter

Why:

Students only calculate the arc length and miss the straight sides, losing 1-2 method marks

Correct move:

Explicitly write perimeter = 2r + s before substituting values

Wrong move:

Using the radian sector area formula when θ is still in degrees

Why:

The $\frac{1}{2} r^{2} θ$ formula only returns a correct value if θ is measured in radians

Correct move:

Always convert all angles to radians before using any arc or sector formula

Wrong move:

Rounding intermediate steps too early in multi-part problems

Why:

Your final 3 significant figure answer will fall outside the acceptable IB mark tolerance range

Correct move:

Keep full unrounded values in your calculator memory until the very last step

Wrong move:

Confusing segment area with sector area

Why:

The two terms are often used interchangeably incorrectly, leading to full loss of method marks

Correct move:

Draw a quick sketch of the shape to confirm if you are being asked for the full wedge or the curved slice minus the triangle

6. Quick Reference Cheatsheet

Quantity	Formula (θ in radians)	Notes
Arc length s	$s = r θ$	No 2π scaling needed
Sector area A	$A = \frac{1}{2} r^{2} θ$	Given in IB formula booklet
Sector perimeter P	$P = 2 r + r θ$	Includes two straight radii
Segment area	$A_{se g m e n t} = \frac{1}{2} r^{2} (θ - sin θ)$	Sector minus triangle area

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 TZ1
Sector composite perimeter problem
2024 · 2 TZ2
Segment area calculation task

What's Next

Mastering radians, arc length and sector area is a critical prerequisite for all upcoming trigonometry work in IB AA SL, including unit circle definitions of sine and cosine, and radian-based calculus differentiation and integration rules. You will see these formulas reappear in Paper 1 and Paper 2 questions across both short-answer and extended response formats, often combined with triangle trigonometry or area of composite shapes. Practice applying these formulas to mixed problems to build speed and accuracy before your exam.