Cone Volume and Surface Area

Written byOleh Soloninupdated 27 September 2026

What to find

Base radius

Height

Result

Volume37.699112
Volume in terms of πV = 12π ≈ 37.7
Slant height5
Base area28.274334
Lateral surface47.12389
Total surface75.398224
Apex angle, °73.7398
WorkingV = π · 3² · 4 ÷ 3 = 12π ≈ 37.7; l = √(3² + 4²) = 5

A third of the cylinder with the same height

The volume of a cone, V = πr²h/3, and of a frustum, with slant height, base, lateral and total surface area, apex angle, and the height or radius from a known volume, answered in terms of π with the working.

A cone's volume is a third of base area times height: V = πr²h ÷ 3. For radius 3 and height 4, V = π · 9 · 4 ÷ 3 = 12π ≈ 37.70, exactly a third of the cylinder with the same base and height.

The slant height is the distance from the apex to the rim of the base: l = √(r² + h²), exactly 5 for r = 3 and h = 4. It gives the lateral surface S = πrl, and the total surface is πrl + πr².

How to use

Choose what to find: everything from radius and height, height or radius from volume, or a frustum.
Enter the known dimensions in one unit; areas come out in square units and volume in cubic units.
Read the volume, also in terms of π, the slant height, areas and the working.

Cone and frustum formulas

Cone: V = πr²h ÷ 3; l = √(r² + h²); lateral area πrl; total area πr(l + r). Backwards: height from volume h = 3V ÷ (πr²), radius r = √(3V ÷ (πh)).

Frustum with base radii R and r and height h: V = πh(R² + Rr + r²) ÷ 3; slant height l = √(h² + (R − r)²); lateral area π(R + r)l. For R = 3, r = 1, h = 4: V = π · 4 · (9 + 3 + 1) ÷ 3 = 52π/3 ≈ 54.45.

Worked out with this calculator.
Radius	Height	Volume	Slant height	Total surface
3	4	12π ≈ 37.70	5	24π ≈ 75.40
1	3	π ≈ 3.14	3.1623	≈ 13.08
5	12	100π ≈ 314.16	13	90π ≈ 282.74
6	8	96π ≈ 301.59	10	96π ≈ 301.59

What this calculator shows

The cone's volume as a number and in terms of π
Slant height, base area, lateral and total surface
Height or radius from a known volume
Volume and areas of a frustum
Apex angle and the working

What to keep in mind

The cone is a right circular cone: the apex sits directly above the centre of the base.
All dimensions use one unit; areas come out squared and volume cubed.

FAQs

How do I find the volume of a cone?

Multiply the base area πr² by the height and divide by 3: V = πr²h ÷ 3. For r = 3 and h = 4 that is 12π ≈ 37.70.

How do I find a cone's height from its volume?

h = 3V ÷ (πr²). With a volume of 12π and radius 3, the height is 3 · 12π ÷ 9π = 4. Choose "Height from volume and radius".

How do I find the slant height?

With Pythagoras: l = √(r² + h²). Radius 5 and height 12 give a slant height of 13.

How do I find the volume of a frustum?

V = πh(R² + Rr + r²) ÷ 3, with R and r the radii of the two bases. For R = 3, r = 1, h = 4 it is 52π/3 ≈ 54.45. The calculator has a frustum mode.

What is the volume of a cone of radius 3 and height 4?

37.70, exactly a third of the cylinder that encloses it. The slant height is 5, because 3, 4 and 5 form a right triangle.

Where does the one third come from?

A cone fills exactly a third of the cylinder around it. It falls out of integrating the cross-section over the height, and it is easy to verify: three full cones empty into one cylinder of the same radius and height.

Worked example

A 3 by 4 cone

Input: Radius 3, height 4

Output: Volume 12π ≈ 37.7, slant height 5

Note: The slant height is a whole number because 3, 4 and 5 form a Pythagorean triple. A cylinder of the same size would hold 36π ≈ 113.1, exactly three times as much.

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Cone Volume and Surface Area

Result

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