Cone Volume

What the Cone Volume Calculator Does

This calculator finds the volume of a right circular cone from two measurements: the radius of its circular base and its vertical height. Enter both values in the same unit (centimeters, inches, meters) and the tool returns the volume in cubic units of that measurement.

It is handy for students checking geometry homework, engineers and machinists sizing tapered parts, and anyone working with real cone shapes such as funnels, traffic cones, ice cream cones, paper cups, or piles of sand, gravel, and grain. Knowing the cone volume tells you exactly how much a container holds or how much material a pile contains.

How Cone Volume Is Calculated (The Formula)

A cone holds exactly one-third the volume of a cylinder with the same base and height. The cone volume formula is:

V = pi * r^2 * h / 3

Here r is the base radius, h is the perpendicular height from the base to the tip (apex), and pi is approximately 3.14159. The radius is half the base diameter, so if you measure across the base, divide that figure by two before entering it.

If you also need the slant height (the distance along the sloped side from the base edge to the apex), use the Pythagorean relationship: slant = sqrt(r^2 + h^2). Slant height is what you need for surface area and for cutting a flat pattern, but it is not used in the volume formula itself.

Worked Example With Real Numbers

Suppose a cone has a base radius of 3 cm and a height of 8 cm.

Square the radius: 3^2 = 9. Multiply by the height: 9 * 8 = 72. Multiply by pi: 72 * 3.14159 = 226.19. Divide by 3: 226.19 / 3 = 75.40.

The volume is about 75.40 cubic centimeters (cm^3). For the slant height: sqrt(3^2 + 8^2) = sqrt(9 + 64) = sqrt(73), which is about 8.54 cm.

Tips and Common Mistakes

The most frequent errors come from mixing up measurements. Watch for these:

• Diameter vs. radius: if you measured the full width across the base, halve it before using the formula. Using the diameter as the radius makes the result four times too large.
• Height vs. slant height: the formula needs the vertical height (straight up the center), not the sloped side length. If you only know the slant, solve h = sqrt(slant^2 - r^2) first.
• Forgetting to divide by 3: skipping that step gives the volume of a cylinder, not a cone.
• Mismatched units: keep radius and height in the same unit. Mixing inches and centimeters produces a meaningless answer.
• Cubic units: volume is always in cubic units (cm^3, in^3). To convert cm^3 to liters, divide by 1000; to convert in^3 to US gallons, divide by 231.

Factors That Affect the Result

Radius has the strongest effect because it is squared: doubling the radius multiplies the volume by four, while doubling the height only doubles the volume. So small measuring errors on the base have an outsized impact on the answer.

This calculator assumes a right circular cone, where the apex sits directly above the center of the base. An oblique cone (tilted apex) has the same volume as long as its perpendicular height and base radius are unchanged, since volume depends only on base area and height. For a truncated cone (a frustum, like a bucket with two different circular ends) you need a separate frustum formula rather than this one.