What is the formula for the volume of a cone?

V = (1/3) times pi times r squared times h, where r is the base radius and h is the perpendicular height from base to apex. The factor of one-third arises because a cone occupies exactly one-third of the volume of the cylinder with the same base and height.

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

Three identical cones with base radius r and height h fit exactly inside a cylinder of radius r and height h. This can be proved with Cavalieri's principle by comparing horizontal cross-sections: at any height y the cone's circular cross-section has area pi times (r times (h minus y) / h) squared, and integrating from 0 to h gives exactly (1/3) times pi times r squared times h.

How do I find the radius of a cone if I know the volume and height?

Rearrange the formula: r = sqrt(3V / (pi times h)). Enter the volume and height in the "Solve for Base radius" mode and the calculator applies this rearrangement automatically, showing the full working.

How do I find the height of a cone from its volume and radius?

Rearrange to h = 3V / (pi times r squared). Switch the "Solve for" dropdown to Height, enter the known volume and base radius, and the result appears instantly with step-by-step working.

Does this calculator work for oblique cones?

Yes — the volume formula V = (1/3) times pi times r squared times h holds for any cone (right or oblique) as long as h is the true perpendicular height between the base plane and the apex, not the slant height. The diagram shows a right cone for clarity, but the arithmetic is the same.

What units can I use?

You can enter values in millimetres, centimetres, metres, inches, or feet. The result is automatically expressed in the matching cubic unit (mm³, cm³, m³, in³, or ft³). To convert between volume units after calculating, use a separate unit-conversion tool.

Cone Volume Calculator

A cone volume calculator that works in three directions: give it the base radius and height to get the volume, or supply the volume and one dimension to recover the missing one. It covers the most common practical need — filling a funnel, hopper, party cone, or measuring flask — and the worked-step display makes it useful for checking geometry homework or engineering notes.

The formula

For a right circular cone the volume is:

V = (1/3) · π · r² · h

where r is the base radius and h is the perpendicular height from the base plane to the apex. Rearranging for the two reverse cases:

Goal	Rearrangement
Volume from r and h	V = (1/3) · π · r² · h
Radius from V and h	r = sqrt(3V / (π · h))
Height from V and r	h = 3V / (π · r²)

All three share the same single formula — the calculator simply applies algebra before reaching for the square root. The SVG diagram highlights the variable you are solving for in amber so the geometry stays clear.

Why one-third?

The factor of 1/3 is one of the most elegant results in elementary solid geometry. At height y above the base (measuring downward from the apex), the cone’s horizontal cross-section is a circle whose radius scales linearly with y. Integrating the area from apex to base gives:

V = integral from 0 to h of pi * (r * y / h)^2 dy
  = (pi * r^2 / h^2) * [y^3 / 3] from 0 to h
  = (1/3) * pi * r^2 * h

The exponent 3 in y^3 is why the denominator is 3 — and why a cone is exactly one-third of its enclosing cylinder regardless of base size or height.

Worked example

A traffic cone has a base radius of 15 cm and a height of 45 cm. What is its volume?

Substitute into the formula: V = (1/3) · π · 15² · 45
Square the radius: 15² = 225
Multiply: (1/3) · π · 225 · 45 = (1/3) · π · 10 125
Divide by 3: π · 3 375
Result: approximately 10 602.9 cm³

Now suppose you know a conical flask holds 500 cm³ and the cone section is 12 cm tall. What is the base radius?

r = sqrt(3 × 500 / (π × 12)) = sqrt(1500 / 37.699) = sqrt(39.789) ≈ 6.31 cm

Both calculations run instantly in the tool above, with the intermediate steps shown on screen.

Practical uses

Funnels and hoppers: estimating how much liquid or grain a conical section holds before overflow.
Ice-cream cones and party hats: checking whether a given cone shape fits a target volume.
Engineering and manufacturing: sizing tapered nozzles, silos, and drill-bit voids where the cone geometry is specified by one dimension and a capacity constraint.
3-D printing and CNC: verifying that a conical pocket or boss has the intended volume for material or weight calculations.
Education: geometry assignments that require showing step-by-step working for V, r, or h.

Keep all inputs in the same unit and the result will be in the matching cubic unit automatically.