Name and identify all key parts of a circle (centre, radius, diameter, circumference, chord, tangent)
Understand what pi (π) is and its approximate value
Use the formulas C = πd and C = 2πr to calculate circumference
Find diameter or radius when given the circumference
Calculate the perimeter of a semicircle
Find arc lengths using a fraction of the full circumference
Solve real-world problems involving circumference
Real-World Applications
Engineering: The circumference of a wheel or pipe determines how far a vehicle travels per rotation and what length of material is needed to wrap around cylindrical components.
Sports: Running track distances are calculated using the circumference of circular bends, ensuring the total lane length meets official competition standards.
Manufacturing: Circular products such as tyres, lids, and gaskets require accurate circumference measurements so that materials are cut to exactly the right length.
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Before we can calculate the circumference, we need to know the names of the parts of a circle.
Centre
The middle point of the circle. Every point on the circumference is the same distance from the centre.
Radius (r)
The distance from the centre to the edge. All radii of the same circle are equal.
Diameter (d)
A straight line through the centre from one side to the other. d = 2 × r
Circumference (C)
The perimeter of the circle — the total distance around the outside edge.
Chord
A straight line joining two points on the circumference. The diameter is the longest chord.
Tangent
A line that touches the circle at one point, perpendicular to the radius at that point.
Key relationship: d = 2r and r = d ÷ 2
What is Pi (π)?
No matter how big or small a circle is, if you divide its circumference by its diameter you always get the same answer: approximately 3.14159... This never-ending, never-repeating decimal is called pi (π).
💡 Using π in calculations:
• Calculator: Use the π button (stores more decimal places automatically).
• Non-calculator: Use π ≈ 3.14 (or π ≈ 22/7 if the question asks).
• Exact answers: Leave in terms of π (e.g. 6π cm) unless asked to evaluate.
C = π × d OR C = 2 × π × r Both are identical since d = 2r → 2πr = πd ✓
✓ Quick Check 1 — Circumference Formulas
Question 1 of 10
◯ Calculating Circumference
Follow this four-step method every time.
Step 1
Identify whether you are given the radius or the diameter.
Step 2
Choose the correct formula: given d → use C = πd. Given r → use C = 2πr.
Step 3
Substitute the value and calculate (use the π button on a calculator, or 3.14 without one).
Step 4
Round to the required number of decimal places and always state the unit.
📄 Worked Example A — Given diameter
A circle has diameter 10 cm. Find its circumference to 1 d.p.
Formula
C = π × d = π × 10 = 10π
Calculate
C ≈ 31.4 cm
📄 Worked Example B — Given radius
A circle has radius 7 m. Find its circumference to 1 d.p.
Formula
C = 2 × π × 7 = 14π ≈ 44.0 m
📄 Worked Example C — Exact answer
Circle with diameter 6 cm. Give an exact answer.
Do not evaluate π
C = π × 6 = 6π cm
An exact answer means leaving π in the answer. 6π ≈ 18.85 cm, but 6π cm is exact.
📄 Worked Example D — Finding diameter from circumference
C = 50 cm. Find the diameter to 2 d.p.
Rearrange C = πd
d = C ÷ π = 50 ÷ π ≈ 15.92 cm
📄 Worked Example E — Perimeter of a semicircle
A semicircle has radius 5 m. Find its total perimeter to 1 d.p.
Curved part (half circumference)
πr = π × 5 ≈ 15.7 m
Straight part (diameter)
2r = 2 × 5 = 10 m
Total perimeter
P = πr + 2r ≈ 15.7 + 10 = 25.7 m
⚠ Don't forget the straight diameter edge — a very common mistake is to only include the curved half.
✓ Quick Check 2 — Finding Circumference and Diameter
Question 1 of 10
◯ Real Problems — Arc Length and Problem Solving
An arc is a portion of the circumference. If you know the angle at the centre, you can find what fraction of the full circumference the arc represents.
Arc length = (angle ÷ 360) × 2πr
📄 Worked Example A — Arc length
Arc of 90° at centre, radius 6 cm. Find arc length to 2 d.p.
Fraction of circle
90 ÷ 360 = ¼
Full circumference
2π × 6 = 12π
Arc length
¼ × 12π = 3π ≈ 9.42 cm
📄 Worked Example B — The wheel problem
A wheel has diameter 50 cm. How many complete rotations to travel 1 km?
Convert distance
1 km = 100,000 cm
Circumference
C = π × 50 ≈ 157.08 cm
Rotations
100,000 ÷ 157.08 ≈ 637 rotations
🎯 Exam tips:
1. Convert all measurements to the same unit before calculating.
2. For arc length, find the fraction of 360° first.
3. If given diameter, do not halve it unless using C = 2πr.
4. Round only at the very end to avoid rounding errors.
✓ Quick Check 3 — Real Life Problems
Question 1 of 10
Practice: 8 Interactive Calculations
Work through all 8 circumference calculations. Give answers to 2 decimal places unless finding radius/diameter (give exact where possible).
Practice: Circumference Calculations
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Question
Answer
Result
⚠ Things to Watch Out For
The formula C = πd uses the diameter — if you're given the radius, double it before multiplying by π.
Don't confuse radius and diameter — the diameter is twice the radius.
Don't use the area formula (πr²) when the question asks for circumference.
Always use the π button on your calculator — don't round π to 3 or 3.14 mid-calculation.
For arc length, multiply the full circumference by the fraction of the circle (angle ÷ 360).