Cone Volume Calculator
A cone holds exactly one third of the cylinder that fits around it: V = ⅓πr²h. Enter what you know — radius or diameter, height, slant height or volume — and this calculator solves the rest, including surface areas, truncated cones (frustums) and the volume of a conical pile of sand, gravel or grain.
Cone volume calculator
Cone and Solid Geometry Practice Pack
Printable cone and frustum volume and surface area worksheets with worked answer keys, a 3D solids formula card and a cone net template.
- Cone worksheet (PDF/DOCX)
- Frustum worksheet (PDF/DOCX)
- Formula card (PDF)
- Cone net (PDF)
Formats: PDF, DOCX. Instant download after payment (link valid 72 hours, up to 5 downloads). AI-assisted: the templates were drafted with AI help and reviewed and laid out by Kedop.
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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
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 — use pile mode with the base diameter and angle of repose.
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?
Yes — r = √(3V ÷ (πh)); enter the volume and height, then adjust the radius until they match, or use the formula.
Does it work for an upside-down cone?
Yes — orientation doesn’t change the volume.