🔵 What is π — and where does it come from?

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 — always Bigger diameter → bigger circumference The ratio never changes Irrational — never ends, never repeats

📖 Key Vocabulary

π (pi) — the ratio of a circle's circumference to its diameter.

Diameter — the distance straight across a circle through its centre.

Radius — the distance from the centre to the edge; half the diameter.

Irrational number — a number whose decimal expansion never ends or repeats.

🎯 Drag the diameter — watch the ratio C÷d stay at π

📏 Diameter (d):

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.

📖 Key Vocabulary

Circumference — the distance all the way around a circle.

Diameter — the full width of a circle through its centre.

Radius — half the diameter, from centre to edge.

C = πd = 2πr where d = diameter, r = radius. Both give the same answer since d = 2r.

🎯 Change the diameter — watch the circumference grow proportionally

📏 Diameter:

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.

📖 Key Vocabulary

Area — the amount of surface enclosed by the circle.

Radius — the distance from the centre to the edge of the circle.

Squared — multiplied by itself, e.g. r² = r × r.

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

Radius (r):

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.

📖 Key Vocabulary

Arc — part of the circumference, a curved line.

Sector — a "pie slice" region bounded by two radii and an arc.

Segment — the region between a chord and the arc it cuts off.

Chord — a straight line connecting two points on the circle's edge.

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.

📖 Key Vocabulary

Cross-section — the flat shape you'd see slicing through a 3D solid.

Cylinder — a solid with two circular ends and a constant circular cross-section.

Cone — a solid that tapers from a circular base to a single point.

Sphere — a perfectly round 3D solid; every point on its surface is the same distance from the centre.

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.

📖 Key Vocabulary

Tooth spacing — the fixed distance between gear teeth, set by the gear's circumference.

Cross-sectional area — the area of a circular slice through a pipe, rod or cable.

Tolerance — the allowed variation in a measured dimension.

⚙️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 Level 3 assessments.

📖 Key Vocabulary

Cross-sectional area — the area of a slice cut perpendicular to a pipe or shaft's length.

Flow rate — the volume of fluid passing a point per unit time.

Proportional (∝) — one quantity changes in direct relation to another, e.g. flow ∝ r².

🔩 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.

Gear: 40 teeth, PCD = 160mm
Circumference = π×160 = 502.7mm
Tooth pitch = 502.7/40 = 12.57mm

Module m = PCD/teeth = 160/40 = 4
(standard module m=4)

Mating gear: 20 teeth, m=4
PCD = m×teeth = 4×20 = 80mm
Centre distance = (160+80)/2 = 120mm

🛢️ Tank Volume — Cylindrical Vessel

A storage tank is a cylinder. Engineers calculate volume for capacity and surface area for material cost.

Tank: diameter=2.4m, height=3.6m
r = 1.2m

Volume: V = πr²h = π×1.44×3.6
V = 5.184π = 16.29 m³
= 16,290 litres

Surface area (curved + 2 ends):
SA = 2πrh + 2πr² = 2π×1.2×3.6 + 2π×1.44
= 8.64π + 2.88π = 11.52π = 36.19 m²
Steel at 15 kg/m²: 36.19×15 = 542.9 kg

🏗️ Concrete Column — Circular Section

Structural columns are often circular. Area gives load capacity; volume gives concrete quantity.

Column: d=400mm, height=3.5m
r = 200mm = 0.2m

Cross-section area:
A = π×0.2² = 0.04π = 0.1257 m²

Volume of concrete:
V = A×h = 0.1257×3.5 = 0.440 m³

At 2400 kg/m³: mass = 0.440×2400
= 1056 kg per column

Compressive stress at 500kN load:
σ = F/A = 500000/125700 = 3.98 N/mm²

🔧 Turned Component — Lathe Material Removal

Turning reduces a circular bar's diameter. The material removed is the difference between two circular areas multiplied by the length.

Bar: original d=80mm, turned to d=72mm
r₁=40mm, r₂=36mm, length=200mm

Area removed = π(r₁²−r₂²)
= π(1600−1296) = 304π = 955 mm²

Volume removed = 955×200 = 191,000 mm³
= 191 cm³

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.

📖 Key Vocabulary

Input — the one measurement you already know.

Derived value — a measurement calculated from the input using a circle formula.

◯ 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.

I know the

Diameter

Units

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.

Radius r

Angle

Angle unit

👆 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: 0 Streak: 0 🔥 Q 1/10

Loading...

✏️

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.

[3]

r=8cm, l=17cm
SA = πr² + πrl = π×64 + π×8×17 = 64π+136π = 200π = 628.3 cm²

🔴 Tier 3 — Engineering

Q9

A hydraulic cylinder bore is 80mm diameter. The piston travels 120mm. Find: (a) piston area, (b) swept volume, (c) force at 15 MPa pressure.

[5]

(a) A = π×40² = 1600π = 5027 mm²
(b) V = 5027×120 = 603,200 mm³ = 603.2 cm³
(c) F = P×A = 15×5027 = 75,398 N ≈ 75.4 kN

Q10

A water pipe has bore 60mm. A second pipe is to carry 4× the flow rate at the same velocity. Find the required bore diameter of the second pipe.

[4]

Flow rate ∝ area. Area ∝ r². So 4× flow → 4× area → 2× radius.

A₁ = π×30² = 900π mm²
A₂ = 4×900π = 3600π mm²
r₂ = √3600 = 60mm
d₂ = 2×60 = 120mm bore ✅
(double the diameter gives 4× the area — as expected)

🎯 Score: Q1–5 foundations (18 marks) · Q6–8 problem solving (10 marks) · Q9–10 engineering (9 marks)

🔵 What is π — and where does it come from?

📏 The Rules

📖 Key Vocabulary

🎯 Drag the diameter — watch the ratio C÷d stay at π

Circumference

📏 The Rules

📖 Key Vocabulary

🎯 Change the diameter — watch the circumference grow proportionally

Area of a Circle

📏 The Rules

📖 Key Vocabulary

🎯 Change the radius — see area grow as r²

Parts of a Circle — Arc, Sector, Segment, Chord

📏 The Rules

📖 Key Vocabulary

3D Shapes — Cylinder, Cone, Sphere

📏 The Rules

📖 Key Vocabulary

Applications

📖 Key Vocabulary

Engineering Applications

📏 The Rules

📖 Key Vocabulary

🔩 Pipe Flow — Cross-Sectional Area

⚙️ Gear Wheel — Pitch Circle Circumference

🛢️ Tank Volume — Cylindrical Vessel

🏗️ Concrete Column — Circular Section

🔧 Turned Component — Lathe Material Removal

🌡️ Heat Exchanger — Tube Surface Area

Circle Calculator

📏 The Rules

📖 Key Vocabulary

Quick Fire Quiz

⚡ Circles Blitz

Practice Questions