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Cone Volume Calculator

Calculate the volume of a cone from its radius and height using V = (1/3)πr²h.

Reviewed by Christopher FloiedPublished Updated

This free online cone volume calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Minimum: 0

Minimum: 0

Perpendicular height from base to apex

Results

Volume

261.7994 cubic units

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Cone Volume Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Cone Volume Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Cone Volume Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

Related Calculators

Cylinder Volume Calculator

Calculate the volume of a cylinder from its radius and height using V = πr²h.

Frustum Volume Calculator

Calculate the volume of a frustum (truncated cone) using V = (PI*h/3)*(R^2 + R*r + r^2), where R and r are the base radii and h is the height. Used in architecture, earthwork calculations, container design, and manufacturing of conical vessels.

Surface Area of a Cone Calculator

Calculate the total surface area of a cone from its radius and slant height.

Circular Segment Area Calculator

Calculate the area of a circular segment (the region between a chord and its arc) given the radius and central angle. Uses A = (r^2/2)(theta - sin(theta)), essential for engineering, architecture, tank volume calculations, and curved surface analysis.

Pyramid Volume Calculator

Calculate the volume of a pyramid from its base area (or base length and width) and height.

Rectangular Prism Calculator

Calculate the volume and surface area of a rectangular prism (box) from its length, width, and height.

About Cone Volume Calculator

A cone is a three-dimensional shape that tapers smoothly from a flat circular base to a point called the apex. The volume of a cone is exactly one-third of the volume of a cylinder with the same base and height: V = (1/3)πr²h. Cones appear in many contexts — from ice cream cones and traffic cones to volcanic shapes and satellite dish reflectors. The factor of 1/3 arises because the cone narrows linearly from base to apex, reducing its cross-sectional area quadratically with height. This relationship was first proved rigorously by Eudoxus of Cnidus using the method of exhaustion, a precursor to integral calculus. This calculator computes the cone's volume given the base radius and perpendicular height. Understanding cone volume is important in manufacturing, civil engineering, and physics.

The Math Behind It

The cone volume formula V = (1/3)πr²h can be derived by integration. At height y from the apex, the cone's cross-section is a circle with radius (r/h)y. The volume is V = ∫₀ʰ π((r/h)y)² dy = πr²/h² × ∫₀ʰ y² dy = πr²/h² × h³/3 = (1/3)πr²h. The 1/3 factor is general: any pyramid or cone (regardless of base shape) has volume (1/3) × base area × height. This holds for oblique cones as well, by Cavalieri's principle. The slant height l of a right cone is l = √(r² + h²), and the lateral surface area is πrl. The total surface area is πrl + πr² = πr(l + r). A frustum (a cone with the top cut off) has volume V = (πh/3)(R² + Rr + r²), where R and r are the radii of the two circular faces. Conical shapes minimize material use for a given volume in some engineering applications, which is why many hoppers and funnels are conical.

Formula Reference

Cone Volume

V = (1/3) π r² h

Variables: r = base radius, h = height

Worked Examples

Example 1: Sand pile

A conical pile of sand has base radius 3 m and height 2 m.

Step 1:V = (1/3) × π × 3² × 2
Step 2:V = (1/3) × π × 9 × 2
Step 3:V = (1/3) × 18π

V ≈ 18.85 m³

Common Mistakes & Tips

  • !Forgetting the 1/3 factor, which gives the cylinder volume instead.
  • !Using the slant height instead of the perpendicular height.
  • !Confusing the radius of the base with the diameter.

Related Concepts

Cylinder Volume

A cone is 1/3 the volume of a cylinder with the same base and height.

Surface Area of a Cone

The total outer area of a cone.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Cylinder Volume Calculator

Pyramid Volume Calculator

Surface Area of a Cone Calculator

Frequently Asked Questions

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

Because the cross-sectional area of a cone decreases as the square of the distance from the base. Integrating this quadratic decrease yields the factor of 1/3.

Does the formula work for oblique cones?

Yes. By Cavalieri's principle, an oblique cone has the same volume as a right cone with the same base and perpendicular height.

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