IGCSE Add Maths · Circular Measure
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Cambridge IGCSE Additional Mathematics (0606) · Circular Measure

Radians vs Degrees

Radian definition · Conversion · Arc length · Sector area
16 questions · interactive circle · self-marked
📐 Basics 🔄 Convert 📏 Arc & Sector ⭐ Challenge
📐
Section A

What Is a Radian?

One radian is the angle formed at the centre of a circle by an arc whose length equals the radius. A full turn (360°) measures 2π radians (≈ 6.28 rad). Because the arc length of a full circle is its circumference, 2πr, dividing by r gives exactly 2π radians — the definition and the circumference formula are the same idea.
Drag the slider — watch radians, degrees and arc length change together
Radians
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Degrees
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Arc length
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Common angles to memorise
90° (π/2) 180° (π) 270° (3π/2) 0° (0) 360° (2π) 60° (π/3) 45° (π/4) 30° (π/6)
These are the angles you'll meet again and again — worth learning by heart.
M1
A circle has radius 5 cm. An arc along its edge measures 5 cm. What angle does that arc subtend at the centre?
1 mark
A
1 radian
B
5 radians
C
⅕ radian
D
5°
M2
How many radians make a full turn (360°)? Give your answer to 2 decimal places.
1 mark
radians
M3
Using the common-angles diagram above, what is π/2 radians in degrees?
1 mark
A
45°
B
60°
C
90°
D
180°
M4
What is π/3 radians in degrees?
1 mark
A
30°
B
45°
C
60°
D
90°
🔄
Section B

Converting Degrees ⇄ Radians

Degrees to radians: θ(rad) = θ(deg) × π ÷ 180. Radians to degrees: θ(deg) = θ(rad) × 180 ÷ π. Use π ≈ 3.14 (or 3.14159 for more precision) throughout this section.
M5
Convert 90° to radians. Give your answer to 2 decimal places.
1 mark
radians
M6
Convert 45° to radians. Give your answer to 2 decimal places.
1 mark
radians
M7
Convert 3 radians to degrees. Give your answer to 1 decimal place.
1 mark
degrees
M8
What is 270° in radians, expressed as a multiple of π?
1 mark
A
3π/2
B
π
C
π/2
D
2π
M9
Convert 1 radian to degrees. Give your answer to 1 decimal place.
1 mark
degrees
📏
Section C

Arc Length & Sector Area

Arc length: s = rθ. Sector area: A = ½r²θ. Both formulas only work when θ is measured in radians — that's the whole reason radians exist as a unit.
M10
A sector has radius 5 cm and angle 2 radians at the centre. Find the arc length.
2 marks
cm
M11
A sector has radius 8 cm and angle 1.5 radians at the centre. Find the area of the sector.
2 marks
cm²
M12
An arc is 12 cm long and subtends an angle of 3 radians at the centre. Find the radius.
2 marks
cm
M13
Which formula gives the arc length s of a sector with radius r and angle θ (in radians)?
1 mark
A
s = r²θ
B
s = rθ
C
s = ½rθ
D
s = 2rθ
M14
Which formula gives the area A of a sector with radius r and angle θ (in radians)?
1 mark
A
A = r²θ
B
A = ½rθ
C
A = 2r²θ
D
A = ½r²θ
⭐
Section D

Challenge Questions

M15 ⭐
Challenge: A sector has radius 6 cm. The arc length is 15 cm. Find the angle at the centre in radians. Give your answer to 2 decimal places.
3 marks
radians
M16 ⭐
Challenge: A sector has radius 7 cm and the angle at the centre is 2.4 radians. Find the area of the sector. Give your answer to 1 decimal place.
3 marks
cm²
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