Cone Volume Calculator
A cone holds exactly one third of the cylinder that fits around it: V = ⅓πr²h. Enter radius or diameter and vertical height. A filled diameter takes priority over radius; a known volume takes priority over height and solves for height. Slant height is an output, not an input. Frustum mode needs both radii; pile mode uses the angle only when height and known volume are blank.
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Quick answer
Cone volume formula: V = πr²h/3, using radius r and vertical height h. For r = 3 cm and h = 8 cm, V = 24π ≈ 75.40 cm³. Enter a radius or diameter and height below. Diameter overrides radius; known volume overrides height and solves for height.
Cone volume calculator
Cone and Solid Geometry Practice Pack
12 cone-volume questions and 8 frustum-volume questions with answer keys, a solids formula card and a cone net. All 4 PDFs are free individually; the $3 ZIP adds editable Word versions of the two worksheets.
- Cone volume: 12 questions and answers (PDF/DOCX)
- Frustum volume: 8 questions and answers (PDF/DOCX)
- Volume and surface area formula card (PDF)
- Cone net cut-out template (PDF)
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Printable Cone Net TemplateCone calculator formulas
Use the same length unit for all lengths; when volume is known, use the matching cubic unit, such as cm with cm³ or ft with ft³. Radius from volume and height from slant are manual formulas, not input modes. Slant height must be converted to vertical height before entry.
| Goal | Inputs | Formula | Result unit |
|---|---|---|---|
| Cone volume | r, h | V = ⅓πr²h | cubic unit |
| Using diameter | d, h | r = d/2; V = ⅓πr²h | cubic unit |
| Height from volume | r, V | h = 3V ÷ (πr²) | length unit |
| Radius from volume (manual) | h, V | r = √(3V ÷ (πh)) | length unit |
| Slant height | r, h | s = √(r² + h²) | length unit |
| Height from slant (manual) | r, s | h = √(s² − r²) | length unit |
| Lateral area | r, s | A = πrs | square unit |
| Total surface area | r, s | A = πr(s + r) | square unit |
| Frustum volume | R, r, h | V = ⅓πh(R² + Rr + r²) | cubic unit |
| Pile volume | base diameter D, angle θ from horizontal | r = D/2; h = r tan θ; V = ⅓πr²h | cubic unit |
Cone formulas
| Quantity | Formula |
|---|---|
| Volume | V = ⅓πr²h |
| Slant height | s = √(r² + h²) |
| Lateral (curved) surface area | πrs |
| Total surface area | πrs + πr² = πr(s + r) |
| Height from volume | h = 3V ÷ (πr²) |
| Radius from volume | r = √(3V ÷ (πh)) |
| Frustum volume | V = ⅓πh(R² + Rr + r²) |
| Frustum lateral area | π(R + r)·s, where s = √(h² + (R − r)²) |
A cone’s volume is exactly one third of a cylinder with the same base and height — Archimedes’ result, which you can check by filling a cone with water and pouring it into the matching cylinder three times.
Worked example
A cone has radius 3 cm and height 8 cm. Volume = ⅓ × π × 3² × 8 = 24π ≈ 75.40 cm³, about 0.075 litres. The slant height is √(9 + 64) = √73 ≈ 8.544 cm, so the curved surface is π × 3 × 8.544 ≈ 80.53 cm² and the total surface, including the base, ≈ 108.80 cm². These are the calculator’s default values.
Piles of sand, gravel and grain
Loose material poured onto the ground forms a cone whose side slope is its angle of repose — often around 30–40° for dry sand and gravel. If you know the pile’s base diameter and the angle, the height is radius × tan(angle), and the volume follows. For example, a gravel pile 3 m across at 34° is about 1.5 × tan 34° ≈ 1.01 m high, with a volume of about 2.4 m³. Use the “Pile” mode, or measure the height directly if you can.
Everyday cones
| Object | Typical shape | Tip |
|---|---|---|
| Ice-cream cone | Cone | Measure inside radius and depth |
| Funnel | Frustum + tube | Use frustum mode for the cone part |
| Paper cup | Frustum | Top and bottom radius plus height |
| Traffic cone | Frustum | Often has a flat top |
| Hopper or silo bottom | Inverted frustum | Add a cylinder for the upper part |
| Lampshade | Frustum (surface area) | Use lateral area for fabric |
Unit conversions
- 1 litre = 1,000 cm³ = 1 dm³.
- 1 US gallon ≈ 3.785 litres ≈ 231 in³.
- 1 ft³ ≈ 28.317 litres ≈ 7.48 US gallons.
- 1 cubic yard = 27 ft³ ≈ 0.765 m³.
- Always use the same unit for radius and height before calculating.
Deriving the formula
Slice a cone into thin horizontal discs. At a distance y below the apex, the disc’s radius grows in proportion: r·y/h, so its area is π(r·y/h)². Adding up all the discs from the apex to the base — integrating from 0 to h — gives π r²/h² × h³/3 = ⅓πr²h. The same idea explains why any pyramid, whatever the shape of its base, has volume ⅓ × base area × height: the cross-sections all shrink in the same proportion towards the apex.
Cones in the kitchen and garden
| Question | How to use the calculator |
|---|---|
| How much batter fills a cone-shaped mould? | Measure the inside radius and depth; the result in cm³ equals millilitres |
| How much soil in a conical planter? | Use frustum mode with the top and bottom radii |
| How big is my compost pile? | Use pile mode with the base diameter and height |
| How much popcorn fits in a paper cone? | Cone mode with the top radius and depth |
| How much fabric for a lampshade? | Use the lateral surface area of the frustum, plus seam allowance |
Cones and angles
The apex half-angle — the angle between the axis and the side — is tan⁻¹(r ÷ h). Cones used as funnels often have a 60° apex angle (30° half-angle); a traffic cone is much narrower. The calculator reports the half-angle and the side slope measured from the base, which is the same angle as a pile’s angle of repose.
Similar cones
If you scale a cone by a factor k — doubling every length, for example — its surface area scales by k² and its volume by k³. A cone twice as tall and twice as wide holds eight times as much. That’s why a small increase in the height of a gravel or grain pile adds a surprising amount of material, and why half-filling a conical glass by height fills it only one eighth by volume.
Common mistakes
- Using the diameter instead of the radius — divide the diameter by 2.
- Using the slant height as the vertical height.
- Forgetting the ⅓ — that gives a cylinder’s volume.
- Mixing units, such as a radius in inches and a height in feet.
Privacy
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Frequently asked questions
How do I calculate volume of a cone with slant height?
Use the slant height only after converting it to vertical height: h = √(s² − r²). Then calculate V = ⅓πr²h. This works when s is the side length from rim to apex.
Can a cone volume calculator use diameter instead of radius?
Yes. Enter diameter in its own field; it takes priority over radius when both are filled. The calculator divides it by 2 before applying V = πr²h/3.
What units should I use for a cone volume calculator?
Use one length unit for radius, diameter, height, and slant height. The volume will be in the matching cubic unit, such as in³, ft³, cm³, or m³.
What is the formula for the volume of a cone?
V = ⅓πr²h.
How do I find the slant height?
s = √(r² + h²).
What is a frustum?
A cone with its top cut off parallel to the base.
How do I find the height from the volume?
h = 3V ÷ (πr²).
Why is a cone one third of a cylinder?
It’s a result from geometry (and calculus); you can demonstrate it by pouring water.
Can I calculate a sand pile?
Yes. Choose pile mode, enter a base radius or diameter, and leave height and known volume blank to use the angle of repose (greater than 0° and less than 90°).
How full is a cone filled halfway up?
Only one eighth of its volume, because volume scales with the cube of the height.
How many litres in a cubic foot?
About 28.3 litres.
What is the lateral area of a cone?
πrs — the curved surface only, without the base.
Can I solve for the radius?
The formula is r = √(3V ÷ (πh)), but this tool has no radius-solving input mode. Calculate radius separately, then enter it here. The known-volume field solves for height.
Does it work for an upside-down cone?
Yes — orientation doesn’t change the volume.