This cone calculator finds a right cone’s volume, total surface area, lateral (side) area and slant height from its radius and height — with the πr² vs πrℓ distinction that decides whether you have enough material to actually build one.

Volume–
Slant Height ℓ–
Lateral Area (side)–
Total Surface Area–

The lateral area is the material of the cone’s wall — cut and unrolled it’s a circular sector of radius ℓ. (A paper party hat is this sector with a slice removed; the sector angle = 360° × r ÷ ℓ.)

Three Formulas, One Right Triangle

Slant height ℓ = √(r² + h²) (Pythagoras on the cone’s axial slice) unlocks the rest: volume V = ⅓πr²h (exactly one third of the enclosing cylinder — a fact worth a classroom demo with sand), lateral area = πrℓ, total area = πr(r + ℓ) adding the base disc. More solids live in our Volume Calculator. Part of our Math Calculators collection.

Worked Example (Verified)

  • r = 3, h = 4 (defaults): ℓ = 5 (the 3-4-5 triangle standing upright) · V = 12π ≈ 37.70 · lateral = 15π ≈ 47.12 · total = 24π ≈ 75.40
  • Ice-cream-cone reality: a wafer cone r ≈ 2.5 cm, h ≈ 11 cm holds ~72 cm³ of cone before the scoop — the hemispherical scoop above the rim is dessert’s geometric bonus, not part of this volume
  • Frustum preview: slice the top off and you get a frustum (bucket shape) — its volume formula differs; a traffic cone is a frustum on a slab, which is why the “cone” of everyday speech is often secretly this calculator’s cousin
  • Material check: the lateral area is the sheet you must cut — for the default cone, 47.12 square units of it, arranged as a 5-radius sector spanning 216°

Common Confusions, Settled

  • Height ≠ slant height: formulas are allergic to swapping them; ℓ is always the longer of the two.
  • Volume’s ⅓ is not an approximation — it’s exact, provable by calculus or by pouring rice three times.
  • Diameter inputs: halve first. Radius is the formula’s native tongue everywhere in solid geometry.
  • Oblique cones (apex off-center) keep the same volume but lose the clean lateral formula — this calculator is the right cone, like nearly all real ones.

Related Tools

Frequently Asked Questions

How do you find the volume of a cone?

V = ⅓ × π × r² × h. A cone holds exactly one third of the cylinder with the same base and height: for r = 3, h = 4 that’s 12π ≈ 37.70 cubic units.

What is slant height and how is it used?

The distance from apex to base edge along the surface: ℓ = √(r² + h²). It feeds the lateral area (πrℓ) and any pattern-making (the unrolled cone is a sector of radius ℓ). Vertical height h is for volume; slant ℓ is for surface — swapping them is the classic cone error.

How do you find a cone’s total surface area?

Add the base disc to the lateral wall: πr² + πrℓ = πr(r + ℓ). The r = 3, h = 4 cone totals 24π ≈ 75.40. For an open cone (a funnel, a lampshade), quote the lateral figure only.

Why is cone volume one third of a cylinder’s?

Integration (or Cavalieri’s principle): at every height, the cone’s cross-section is the cylinder’s scaled by the square of the remaining fraction, and ∫₀¹ x² dx = 1/3. The same ⅓ governs pyramids of any base shape — a hint that the law is about tapering, not circles.

Can this calculator do a frustum (truncated cone)?

No — a frustum needs two radii and its own formula V = ⅓πh(R² + Rr + r²). This page is the complete cone; the frustum is a separate tool in spirit (and a common follow-up request).

Is this cone calculator free?

Yes — free, instant, no sign-up.