What is the volume of a cone?

A cone is a 3D shape with a circular base that tapers smoothly to a single point (the apex). The volume of a cone tells you how much space the cone holds inside — think “how much sand fits in a party-hat shape,” “how much concrete fills a conical footing,” or “how much soft-serve is in a cone if it were solid.”

This calculator gives you the cone’s volume instantly from the two measurements that matter most: the radius (or diameter) of the base circle and the height (the perpendicular distance from base to apex). It also shows the steps and a quick reference slant height (useful for surface-area problems).

The cone volume formula (the one you actually need)

The standard formula is:

V = (1/3) · π · r² · h

• V = volume
• π (pi) ≈ 3.14159
• r = radius of the base circle
• h = height (straight up and down, perpendicular to the base)

If you have the diameter instead of the radius, remember: r = d/2. That’s why this calculator lets you choose radius or diameter — it keeps you from doing that extra conversion every time.

Why is there a “1/3” in the formula?

This is the key intuition that makes the formula feel obvious instead of “just memorize it”: a cone is exactly one-third the volume of a cylinder with the same base radius and the same height.

Imagine a cylinder and a cone that share the same base circle (same r) and the same height (h). The cylinder volume is:

V_cylinder = π · r² · h

The cone has the same base area (πr²), but it narrows to a point, so it “contains less” as you move upward. The beautiful result from geometry (and can be demonstrated with physical water/sand experiments) is:

V_cone = (1/3) · V_cylinder

That’s the origin of the 1/3. Once you remember “cone = one-third of a matching cylinder,” you’ll never forget the formula again.

Units: why your answer becomes “cubic”

Volume is always length × length × length, so your units become cubic:

• If r and h are in centimeters → volume is in cm³
• If they’re in meters → m³
• If they’re in inches → in³
• If they’re in feet → ft³

Big real-world tip: if you’re working with “container” problems, you may want liters (L) or gallons. A quick conversion reminder:

• 1,000 cm³ = 1 L
• 1 m³ = 1,000 L

If you need that kind of conversion often, you can use the Volume Converter below in the math interlinks.

Step-by-step example (radius + height)

Let’s say you have a cone with:

• radius r = 4 cm
• height h = 9 cm

Plug into the formula:

• r² = 4² = 16
• π · r² = π · 16 ≈ 50.265
• π · r² · h ≈ 50.265 · 9 ≈ 452.389
• Divide by 3 → V ≈ 150.796 cm³

So the cone’s volume is about 150.8 cm³.

Step-by-step example (diameter + height)

Now suppose you only know the diameter:

• diameter d = 10 in
• height h = 12 in

First convert diameter to radius:

• r = d/2 = 10/2 = 5 in

Then compute:

• V = (1/3) · π · 5² · 12
• 5² = 25
• π · 25 · 12 = 300π
• (1/3) · 300π = 100π ≈ 314.16 in³

That’s the volume. If you were comparing cones of different sizes, this diameter-to-radius step is where most mistakes happen — which is why having a radius/diameter toggle is such a time saver.

How this calculator works (under the hood)

This page follows the exact cone formula: V = (π · r² · h) / 3. Here’s what happens when you tap “Calculate Volume”:

1. Read inputs (radius or diameter, plus height, plus units).
2. Validate that all numbers are positive.
3. Convert diameter → radius if needed (r = d/2).
4. Compute volume using π from the browser’s math engine.
5. Show results in cubic units (cm³, m³, in³, ft³).
6. Optionally compute slant height s = √(r² + h²) to help with related geometry tasks.

Everything runs locally in your browser. If you save results, they are stored in your browser’s local storage — so you can compare multiple cones without signing in.

Real-world uses (where cone volume shows up)

• Construction: concrete for conical footings, tapered posts, or pile caps.
• Manufacturing: tapered parts, funnels, hoppers, and nozzles.
• Food: cone-shaped servings, piping, or packaging estimates.
• Science & labs: approximate volume of conical flasks (when modeled as cones).
• 3D printing: estimating material use for conical solids.

Common mistakes (and how to avoid them)

• Using slant height instead of height: the formula needs the perpendicular height, not the side length.
• Forgetting the 1/3: cones are not cylinders — always divide by 3.
• Mixing units: if r is in cm and h is in inches, the result is meaningless unless you convert first.
• Using diameter as radius: if you input diameter, make sure it’s halved (or just use the diameter toggle).

FAQ

• Do I need the slant height to find volume?

  No. Volume only needs radius (or diameter) and height. Slant height is mainly used for surface area and for right-triangle relationships.

• What if I know the base area instead of the radius?

  If the base area is A, then A = πr². Solve r = √(A/π), then use V = (1/3)Ah. If you’re doing that often, try the Circle Calculator and Surface Area Calculator from the math links.

• Is the cone volume formula different for “oblique” cones?

  Surprisingly, the same formula still works as long as h is the perpendicular height. The base area is the same, and the volume depends on base area × perpendicular height × 1/3.

• How accurate is this calculator?

  The math is exact; the only approximation is π (pi) which your browser represents with very high precision. Your final accuracy mostly depends on the accuracy of your measurements.

• How do I convert cm³ to liters?

  Divide by 1,000. For example, 2,500 cm³ = 2.5 L. For other conversions (like to gallons), use a dedicated volume converter.

• Can I use this for a cone frustum (a cut-off cone)?

  A frustum uses a different formula because it has two radii (top and bottom). Use the Volume Calculator (general) or a frustum-specific tool if you add one later.

Educational note: This calculator is for learning and estimation. For engineering or safety-critical work, verify assumptions and measurement units.