π (pi) is not just a button on your calculator. It is a ratio that exists in every circle ever drawn. If you measure the circumference of any circle and divide it by its diameter, you always get the same answer: π = 3.14159… No matter the size.
π = C ÷ d — alwaysBigger diameter → bigger circumferenceThe ratio never changesIrrational — never ends, never repeats
🎯 Drag the diameter — watch the ratio C÷d stay at π
ℹ️ How to use: Drag the slider to change the circle diameter. Watch the circumference bar below grow and see how C ÷ d always equals π ≈ 3.1416, no matter what size the circle is.
120 mm
Circumference unwrapped into diameter lengths:
The circumference always fits exactly π = 3.14159… diameters — never more, never less.
376.99
Circumference (mm)
120
Diameter (mm)
3.1416
C ÷ d = π
Change the diameter — the ratio is always 3.14159…
Small circle, huge circle, it doesn't matter. π is built into the shape of a circle itself.
The definition of π
π = Circumference ÷ Diameter
π = C / d → C = π × d
π = 3.14159265358979… (never ends!)
π is irrational — it cannot be written as a fraction (22/7 is just a rough approximation). Its decimal expansion goes on forever without repeating.
💡 Use π = 3.14159 for 5 s.f. answers, or press the π button on your calculator for maximum accuracy.
Circle vocabulary
Radius (r) — centre to edge. r = d/2
Diameter (d) — across the full circle. d = 2r
Circumference (C) — distance around the edge
Area (A) — space inside the circle
📌 The diameter is the longest chord — it passes through the centre. Any other chord is shorter.
⭕
Circumference
The circumference is the distance all the way around a circle. Two equivalent formulas — use whichever matches what you're given.
C = πd = 2πr where d = diameter, r = radius. Both give the same answer since d = 2r.
🎯 Change the diameter — watch the circumference grow proportionally
ℹ️ How to use: Drag the slider to change the diameter. The circumference (C = πd) and radius update live below the diagram.
100 mm
314.2
Circumference (mm)
50.0
Radius (mm)
100
Diameter (mm)
Finding circumference
Diameter = 14 cm. Find C.
Formula: C = πd
Substitute: C = π × 14
Answer:C = 14π = 43.98 cm ✅
Radius = 8.5 m. Find C.
Formula: C = 2πr
Substitute: C = 2 × π × 8.5
Answer:C = 17π = 53.41 m ✅
Finding diameter or radius from C
d = C / π
r = C / (2π)
C = 88 cm. Find d and r.
Diameter: d = 88/π = 28.01 cm
Radius: r = 88/(2π) = 14.01 cm
Answer:d≈28.0cm, r≈14.0cm ✅
📐
Area of a Circle
The area of a circle is πr². This comes from rearranging sectors of the circle into a rectangle — a beautiful geometric proof.
A = πr² where r = radius. If given diameter: r = d/2, so A = π(d/2)² = πd²/4
🎯 Change the radius — see area grow as r²
ℹ️ How to use: Drag the slider to change the radius. The area (A = πr²) updates live and the diagram shows the radius and area together.
70 mm
15394
Area (mm²)
439.8
Circumference (mm)
70
Radius (mm)
⚠️ Double the radius → area multiplies by 4 (not 2!). Area grows as r² — this is why large pipes carry so much more flow than small ones.
Finding area
r = 6 cm. Find A.
Formula: A = πr²
Substitute: A = π × 36
Answer:A = 36π = 113.1 cm² ✅
d = 10 m. Find A.
Find r: r = d/2 = 5 m
Formula: A = π × 5² = 25π
Answer:A = 78.54 m² ✅
Finding r or d from area
r = √(A / π)
d = 2√(A / π)
A = 200 cm². Find r.
Rearrange: r² = A/π = 200/π
Calculate: r² = 63.66
Answer:r = √63.66 = 7.98 cm ✅
🥧
Parts of a Circle — Arc, Sector, Segment, Chord
Four important regions and lines. All use π in their area or length formulas.
Worked example — sector
r=10cm, θ=120°. Find arc, sector area, perimeter.
Convert: θ = 120×π/180 = 2π/3 rad
Arc: s = 10 × 2π/3 = 20π/3 = 20.94 cm
Area: A = ½×100×2π/3 = 100π/3 = 104.7 cm²
Perimeter:P = 20.94 + 20 = 40.94 cm ✅
Worked example — segment
r=8m, θ=π/3 rad. Find segment area.
Sector area: ½×64×π/3 = 32π/3 = 33.51 m²
Triangle area: ½r²sin θ = ½×64×sin(60°)
Triangle: = 32×0.866 = 27.71 m²
Segment:33.51−27.71 = 5.80 m² ✅
📦
3D Shapes — Cylinder, Cone, Sphere
All three 3D shapes involve π — their circular cross-sections bring the circle formulas into three dimensions.
Cylinder — worked example
r=5cm, h=12cm. Find V and SA.
Volume: V = πr²h = π×25×12 = 300π
V =942.5 cm³ ✅
SA: 2πr²+2πrh = 2π×25+2π×5×12
SA =50π+120π = 170π = 534.1 cm² ✅
Cone and Sphere examples
Cone: r=6cm, h=8cm. Find V and slant l.
Slant: l = √(r²+h²) = √(36+64) = 10 cm
Volume: V = ⅓π×36×8 = 96π = 301.6 cm³
SA:π×36+π×6×10 = 96π = 301.6 cm² ✅
Sphere: r=7cm. Find V and SA.
Volume: V = 4π×343/3 = 1372π/3
V =1436.8 cm³ ✅
SA =4π×49 = 196π = 615.8 cm² ✅
⚙️
Applications
Where this topic is used in engineering, manufacturing, maintenance and daily life.
⚙️Mechanical:Belt length = πd. Gear circumference for tooth spacing.
🔧Piping:Pipe area A=πr² for flow rate Q=Av.
🚗Daily Life:Tyre: distance per revolution = πd. Revs per km.
🏭Manufacturing:Ring area = π(R²−r²). Washer, flange, annular section.
🔨
Engineering Applications
Circles and cylinders are everywhere in engineering. These are the most tested contexts in T Level / Level 3 assessments.
🔩 Pipe Flow — Cross-Sectional Area
The flow rate through a pipe depends on its cross-sectional area A = πr². Doubling the radius quadruples the flow.
Pipe bore (internal diameter) = 50mm r = 25mm A = π×25² = 625π = 1963 mm²
At flow velocity v = 2 m/s: Q = A×v = 1963×10⁻⁶×2 = 3.93×10⁻³ m³/s = 3.93 litres/second
New pipe d=100mm (double): A = π×50² = 7854mm² = 4× original ✓
⚙️ Gear Wheel — Pitch Circle Circumference
Gears are sized by their Pitch Circle Diameter (PCD). The circumference of the PCD × number of teeth gives tooth pitch.
Steel density 7.85g/cm³: Mass removed = 191×7.85 = 1499g ≈ 1.5kg
🌡️ Heat Exchanger — Tube Surface Area
Heat transfer depends on surface area. For a bank of tubes, total area = circumference × length × number of tubes.
Tube: OD=25mm, length=1.5m, qty=40
Surface area per tube: SA = πdL = π×0.025×1.5 = 0.1178 m²
Total area = 40×0.1178 = 4.71 m²
Heat transfer: Q = U×A×ΔT U=500 W/m²K, ΔT=30°C: Q = 500×4.71×30 = 70,650 W = 70.7 kW heat transfer rate
🧮
Circle Calculator
Enter any one measurement and instantly find all the others — with every formula and working step shown clearly.
◯ Full Circle CalculatorC, A, r, d from any input
ℹ️ How to use: Use the dropdown to choose what you know, type the value into the box, and all other measurements appear instantly with full working. All results update as you type.
Enter a value above to see the full working.
👆
Try: d = 10 cm ·
r = 7 mm ·
C = 31.416 ·
A = 78.54
⏜ Arc Length & Sector Areas = rθ A = ½r²θ
ℹ️ How to use: Enter the radius and the angle (choose degrees or radians). Arc length, sector area and chord length all calculate instantly with full step-by-step working.
👆r=6, 60° ·
r=5, 1.2 rad ·
r=10, 180° (semicircle)
All circle formulas
Circumference: C = πd = 2πr
Area: A = πr²
Arc length: s = rθ (θ in radians)
Sector area: A = ½r²θ
Chord: c = 2r sin(θ/2)
Common conversions
r = d/2 d = 2r
r from C: r = C / (2π)
r from A: r = √(A/π)
Degrees to radians: × π/180
💡 π ≈ 3.14159 — use the π key on your calculator for accuracy.
🎮
Quick Fire Quiz
Test your knowledge — 10 questions, instant feedback.
⚡ Circles Blitz
ℹ️ How to play: A circles question appears. Click the correct answer from the four options. Your score and streak update after each answer — click Next to continue.
Score: 0Streak: 0 🔥Q 1/10
Loading...
Score: 0/10
✏️
Practice Questions
Leave answers in terms of π where possible, then give decimal to 3 s.f.
🟢 Tier 1 — Foundations
ℹ️ How to use: Work through each question and write down your answer. When ready, click Show Answer for the full worked solution.
Q1
Find the circumference and area of a circle with radius 9 cm.
[4]
C = 2πr = 2π×9 = 18π = 56.55 cm A = πr² = π×81 = 81π = 254.5 cm²
Q2
A circle has circumference 44 cm. Find its diameter and area.
[4]
d = C/π = 44/π = 14.0 cm r = 7.0 cm A = π×49 = 49π = 154.0 cm²
Q3
Find the volume of a cylinder with r=4cm and h=10cm.
[2]
V = πr²h = π×16×10 = 160π = 502.7 cm³
Q4
A sector has r=12m and θ=π/4 rad. Find (a) arc length, (b) sector area.
[4]
(a) s = rθ = 12×π/4 = 3π = 9.42 m (b) A = ½r²θ = ½×144×π/4 = 18π = 56.55 m²
Q5
A sphere has radius 5 cm. Find its volume and surface area.
[4]
V = 4πr³/3 = 4π×125/3 = 500π/3 = 523.6 cm³ SA = 4πr² = 4π×25 = 100π = 314.2 cm²
🟡 Tier 2 — Problem Solving
Q6
A circular lawn has area 200 m². Find the cost of fencing its perimeter at £8.50 per metre.
[4]
r = √(200/π) = √63.66 = 7.979 m C = 2πr = 2π×7.979 = 50.13 m Cost = 50.13×8.50 = £426.10
Q7
An annulus (ring) has outer radius 15cm and inner radius 9cm. Find its area.
[3]
Annulus area = π(R² − r²)
A = π(15² − 9²) = π(225−81) = 144π = 452.4 cm²
Q8
A cone has base diameter 16cm and slant height 17cm. Find its total surface area.