Cone Formulas (Volume, Surface Area, Slant Height)

A cone has a circular base and a single apex. Its volume is V = ⅓πr²h, one third of the enclosing cylinder. Its slant height is ℓ = √(r² + h²), its lateral surface is πrℓ, and its total surface area is πr(r + ℓ).

Cone formulas

Slant height: $\ell = \sqrt{r^2 + h^2}$
Volume: $V = \tfrac{1}{3}\pi r^2 h$
Lateral surface: $LSA = \pi r \ell$
Surface area: $SA = \pi r(r + \ell)$

where r = base radius, h = vertical height, ℓ = slant height, V = volume

Slant height

\ell = \sqrt{r^2 + h^2}

Worked example

Its slant height is 5 (a 3-4-5 right triangle), so SA = πr(r + ℓ) = 24π.

\ell = \sqrt{r^2 + h^2} = \sqrt{9 + 16} = 5

Volume

V = \tfrac{1}{3}\pi r^2 h

Worked example

A cone with r = 3 and h = 4 has volume 12π ≈ 37.70 cubic units.

V = \tfrac13\pi r^2 h = \tfrac13\pi\cdot 9\cdot 4 = 12\pi \approx 37.70

Lateral surface

LSA = \pi r \ell

Surface area

SA = \pi r(r + \ell)

Interactive cone explorer

Drag a slider (or type a value, or use the arrow keys) and watch the diagram, the results, and the full step-by-step substitution update live.

Radius r

Vertical height h

Slant height ℓ 5units

Volume V 37.7units³

Lateral surface LSA 47.12units²

Surface area SA 75.4units²

Step-by-step substitution

slant ℓ = √(r² + h²) = √(9 + 16) = 5
V = ⅓πr²h = ⅓ · π · 3² · 4 = 12π ≈ 37.7
LSA = πrℓ = π · 3 · 5 = 15π ≈ 47.12
SA = πr(r + ℓ) = π · 3(3 + 5) = 24π ≈ 75.4

Classroom use: cone formulas

Who it's for: Grades 8–12 · Grade 8 and high-school geometry students working on cone volume, surface area, and slant height.

Learning objectives

After using this page, students will be able to:

Compute cone volume using V = ⅓πr²h.
Find slant height from radius and height via the Pythagorean theorem.
Distinguish height from slant height when computing surface area.

Suggested classroom activity

Fill a cone with water and empty it into a cylinder of equal radius and height — it takes three cones. Then use the explorer to confirm the ⅓ numerically. Students routinely substitute h where ℓ belongs, so make them state which is which before each calculation.

Standards alignment

Aligned to the Common Core State Standards for Mathematics. Codes are given exactly as published so they can be matched against state crosswalks.
Code	Standard
`8.G.C.9`	Know and use the formulas for the volumes of cones, cylinders, and spheres.
`HSG.GMD.A.1`	Give an informal argument for circumference, area, and volume formulas.
`HSG.GMD.A.3`	Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
`8.G.B.7`	Apply the Pythagorean theorem to find unknown side lengths in right triangles.

Free to use and adapt in class under the terms on our license page — no account, no sign-up, no ads. Content last reviewed August 2026.

Common mistakes

Using the vertical height h in a surface-area formula that needs the slant height ℓ.
Forgetting the ⅓ factor — a cone is one third of its cylinder, not equal to it.

Related & next steps

Add cone to your formula sheet →

Frequently asked questions

What is the volume of a cone?

A cone’s volume is V = ⅓πr²h — exactly one third of a cylinder with the same base and height. A cone with radius 3 and height 4 has volume 12π ≈ 37.70.

What is the difference between slant height and height in a cone?

The height h is the vertical distance from base to apex. The slant height ℓ is the distance along the surface, ℓ = √(r² + h²). Surface-area formulas use ℓ, volume uses h.

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