Volume of a Cone: Formula, Derivation, Examples & GCSE Questions

If you’ve ever scooped a perfect snowball or poured sand into a party hat, you’ve already touched the geometry of a cone — a shape that holds a clean mathematical secret: the volume of a cone is exactly one-third the volume of a cylinder with the same base and height. We’ll break down the formula V = 1/3 π r² h, explore why that third is no coincidence, work through step-by-step examples you’ll see on GCSE papers, and compare cones with cylinders and spheres.

Formula: V = 1/3 × π × r² × h ·
Variables: r = base radius, h = perpendicular height ·
Base shape: Circle ·
Relation to cylinder: Cone volume = 1/3 cylinder volume (same base & height) ·
Units: Cubic units (e.g., cm³, m³, in³)

Six key facts summarise what you need to remember when working with cone volume problems.

The takeaway: Only two measurements are needed — radius and perpendicular height — and the formula is compact enough to be given on exam sheets.

What is the volume of a cone?

Cone volume formula

• Standard formula: V = 1/3 × π × r² × h (Math is Fun)
• Also written as: V = 1/3 × (base area) × height, where base area = πr² (BYJU’S (Indian learning platform))
• In terms of diameter d: V = 1/12 π d² h (Cuemath)

Variables in the formula

• r = radius of the circular base (BBC Bitesize)
• h = perpendicular height from base centre to apex — never the slant height (Cuemath)

Units of volume

• Volume is expressed in cubic units: cm³, m³, in³, ft³ (Math is Fun)
• For real‑world applications, convert cubic centimetres to litres (1 L = 1000 cm³) (BBC Bitesize). For more on converting cubic units, see our dedicated guide.

The implication: Once you know the radius and height, applying the formula is a straightforward substitution — no additional geometry required.

Why is the volume of a cone one third of the volume of a cylinder?

Geometric intuition using stacking circles

• Imagine a cylinder filled with three identical cones — they pack perfectly because the cone’s cross‑section shrinks linearly from base to apex (Cuemath)
• A simple experiment: fill a cone with water and pour it into a cylinder of the same base and height; exactly three cones fill the cylinder (BYJU’S)

Cavalieri’s principle

• Cavalieri’s principle states that two solids have equal volume if every horizontal cross‑section has the same area at the same height (Brilliant)
• The cross‑sectional area of a cone at height x from the apex is A(x) = π (r·x/h)², which integrates to 1/3 of the cylinder’s volume (Queen’s Online School (online education))

Integration proof

• Integrate circular disks from apex (x=0) to base (x=h): V = ∫₀ʰ π (r·x/h)² dx = 1/3 π r² h (Queen’s Online School)
• The factor 1/3 emerges from integrating x², not from any arbitrary choice

Why not a half or quarter?

• If the cross‑sections shrank linearly (like a pyramid), the volume would be 1/3 of the prism — squaring the radius in the cone gives the same mathematical structure (Math is Fun)
• Ancient Greeks (Eudoxus and Archimedes) used the method of exhaustion to confirm this ratio around 300 BCE (Cuemath)

The pattern: The 1/3 factor holds for any cone, regardless of its proportions, because the scaling of cross-sectional areas is always quadratic.

How do you find the volume of a cone?

Step 1: Identify radius and height

1. Measure the radius of the circular base (or diameter ÷ 2) (BBC Bitesize)
2. Measure the perpendicular height — use a ruler vertically from base centre to apex (Cuemath)

Step 2: Apply formula

1. Substitute r and h into V = 1/3 π r² h (Math is Fun)
2. If only slant height l is given, first find h = √(l² − r²) (Cuemath)

Step 3: Calculate and round

1. Simplify numerically or leave as a multiple of π (common in GCSE non‑calculator papers) (The Knowledge Academy (professional training))

Example: cone with radius 5 cm, height 12 cm

• Substitute: V = 1/3 × π × 5² × 12 = 1/3 × π × 25 × 12 = (25×12)/3 π = 100π cm³ (The Knowledge Academy)
• Numerical approximation: 100 × 3.14159 ≈ 314.16 cm³

The catch: The most common source of error is mistaking slant height for perpendicular height — a distinction that costs marks in exams.

What is the volume of a cone GCSE?

GCSE formula sheet

• The formula V = 1/3 π r² h is printed on the GCSE maths formula sheet for all exam boards in England and Wales (BBC Bitesize). Familiarity with metric conversions is also essential for GCSE maths.
• IGCSE (Cambridge/Edexcel) also includes this formula in the list of given equations

Common question types

• Non‑calculator: leave answer in terms of π (e.g., 100π cm³) (The Knowledge Academy)
• Calculator: compute to 3 significant figures
• Given diameter instead of radius — convert by dividing by 2 (BBC Bitesize)

Worked example (non‑calculator)

• A cone has base radius 4 cm and height 6 cm. Work out the volume in terms of π. V = 1/3 × π × 4² × 6 = 1/3 × π × 16 × 6 = (16×6)/3 π = 32π cm³ (The Knowledge Academy)

Common mistakes

• Using slant height instead of perpendicular height — a frequent error in GCSE exams (Cuemath)
• Forgetting to square the radius before multiplying by height (Math is Fun)
• Confusing the cone formula with the sphere formula (4/3 π r³) (BBC Bitesize)

The takeaway: GCSE examiners consistently reward students who show clear substitution steps and correct rounding.

What is the volume of a cone, a sphere and a cylinder?

Volume of a sphere

• Sphere volume: V = 4/3 π r³ (BBC Bitesize)
• Only depends on radius — no height needed

Volume of a cylinder

• Cylinder volume: V = π r² h (Math is Fun)
• Same base area formula as cone but without the 1/3 factor

Relationship between the three volumes

• For a cone and cylinder with identical radius and height: cylinder = 3 × cone (Cuemath)
• If a sphere has diameter equal to the cylinder’s height, the volumes relate as cylinder : cone : sphere = 3 : 1 : 2 (Math is Fun)

Three shapes, one radius (or diameter) — their volume ratios reveal a neat numerical pattern.

The pattern: Cylinder volume is three times the cone volume; the sphere (with diameter equal to height) sits right in between at twice the cone volume. This 3:1:2 ratio is a memorable shortcut for comparative questions.

“The volume of a cone is given by V = 1/3 π r² h, where r is the radius and h is the perpendicular height.”

— BBC Bitesize (UK curriculum authority) (source)

“The volume of a cone is one‑third pi times radius squared times height.”

— Brilliant (mathematical education platform) (source)

Understanding the cone volume formula — and why it includes that crucial 1/3 factor — equips you for GCSE and IGCSE exams and for real‑world tasks like calculating the capacity of a conical tank. For students aiming for grades 8‑10, the core formula is only the start: mastering derivation and comparing shapes turns a simple equation into a powerful tool. The next time you reach for a cone‑shaped object, you’ll know its volume is just one‑third of the cylinder it could fill.

For a more detailed breakdown of the derivation and additional practice problems, see this guide on the volume of a cone formula.

Frequently asked questions