Cone Calculator
Compute volume, surface area, and slant height of a cone — or solve for a missing dimension.
Dimensions
Updates as you typeFormula
- V
- Volume of the cone
- SA
- Total surface area (base + lateral)
- r
- Base radius
- h
- Perpendicular height
- ℓ
- Slant height = √(r² + h²)
- π
- Pi ≈ 3.14159
- r² = —
- Slant ℓ = √(r² + h²) = —
- Base area = πr² = —
- Lateral area = πrℓ = —
- Volume V = ⅓πr²h = —
- Surface area SA = πr(r + ℓ) = —
A cone holds exactly one third the volume of a cylinder with the same base and height. The slant height is the distance from the apex straight down the side to the rim — useful for cutting cone-shaped material from a flat sheet.
Examples
How It Works
The slant height (ℓ) is the straight-line distance from the apex to the rim of the base, measured along the surface. It comes from the Pythagorean theorem: ℓ = √(r² + h²). Slant height is what you need when you cut cone-shaped material from a flat sheet — for paper hats, lampshades, or sheet-metal funnels — because the lateral surface unrolls into a circular sector with radius equal to ℓ.
The volume formula V = ⅓ π r² h reflects a classic geometric fact: a cone holds exactly one third the volume of a cylinder with the same base and height. The total surface area SA = π r (r + ℓ) combines the circular base (π r²) and the lateral surface (π r ℓ). This calculator can also work in reverse — pick "Find height" or "Find radius" to solve for a missing dimension that produces a target volume.
Tips & Best Practices
Frequently Asked Questions
What is the difference between height and slant height?
The height (h) is the perpendicular distance from the centre of the base straight up to the apex — measured through the inside of the cone. The slant height (ℓ) is the distance along the outer surface from the apex down to the rim. They are related by ℓ = √(r² + h²). Use height for volume, slant height for surface area and flat-pattern cutting.
Why is the volume formula V = ⅓ π r² h?
A cone fits inside a cylinder with the same base and height, and it occupies exactly one third of that cylinder's volume. This can be proved with calculus (integrating circular cross-sections) or by physical experiment — fill a cone three times and you fill the matching cylinder. Since a cylinder's volume is πr²h, the cone's is one third of that: ⅓ π r² h.
How do I calculate just the lateral (side) surface area?
The lateral surface area — everything except the circular base — is π r ℓ, where ℓ is the slant height. This is the figure you want when cutting material for an open cone (no bottom), like a witch's hat or a paper drinking cup. The full surface area adds the base: SA = π r² + π r ℓ = π r (r + ℓ).
Can I find the radius if I know the volume and height?
Yes. Switch the calculator to Find radius mode and enter your target volume and height. The formula is r = √(3V / (π h)). For example, a target volume of 1,000 cm³ at a height of 15 cm gives a radius of √(3000 / (π × 15)) ≈ 7.98 cm.
What is a frustum, and does this calculator handle it?
A frustum is a cone with the top sliced off — like a paper cup or a traffic cone. This calculator handles full cones only. To find a frustum's volume, calculate the volume of the full cone and subtract the volume of the smaller cone you removed: V = ⅓ π h (R² + R r + r²), where R and r are the two radii.