Quick answer: the volume of a cone is V = (1/3) π r² h — exactly one-third of the cylinder with the same base and height. Use the vertical height, not the slanted side; if you only have the slant length s, height = √(s² − r²).
The cone volume formula, step by step
where π ≈ 3.14159 · r = base radius · h = vertical height (base to tip)
Worked example: a cone with a 3 cm radius and 5 cm height:
- Square the radius: 3² = 9
- Multiply by π and the height: 9 × 3.14159 × 5 = 141.37
- Divide by 3: 141.37 ÷ 3 = 47.12 cm³
That ÷3 is not an approximation — it's exact, and it applies to every shape that tapers linearly to a point, pyramids included. The intuition: as the cone narrows, its cross-sections shrink with the square of the distance from the tip, and those squares average out to a third of the full base. Formally a cone is a solid of revolution — spin the line y = rx/h about the x-axis and the disk method returns this same ⅓πr²h.
Solve a cone from any two measurements
Textbook and workshop problems rarely hand you the radius and the height together. Choose the two values you have, and the solver works out the missing ones, the slant height and the surface area.
Cone solver: any two measurements
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Pick the pair you know. The solver returns the radius, height, slant height, volume, surface areas and apex angle.
Find the radius from the volume and height
Rearrange the volume formula: r = √(3V ÷ (πh)). A cone holds 220 cm³ and stands 10 cm tall. Then 3 × 220 = 660, π × 10 = 31.416, and √(660 ÷ 31.416) = √21.01 = 4.58 cm.
Find the height from the volume and radius
Use h = 3V ÷ (πr²). A conical container holds 300 in³ of sand and has a 5 in radius: 900 ÷ (π × 25) = 900 ÷ 78.54 = 11.46 in.
Use the slant height
The radius, height and slant height form a right triangle, so h = √(s² − r²). A cone with a 13 cm slant and a 5 cm radius stands √(169 − 25) = 12 cm tall and holds (1/3) × π × 25 × 12 = 314.16 cm³.
Cone surface area
The curved side covers πrs and the base covers πr², so the total surface area is πr(r + s). The 5 cm by 13 cm cone above has a side area of 204.2 cm² and a total of 282.7 cm². An open party hat or paper cup uses the side area alone.
Conical stockpile volumes — quick reference
Dumped gravel, sand and soil settle into cones. At a typical 32° angle of repose, the pile height is roughly 0.31 × its diameter — which lets you estimate a pile from one tape measurement across its base:
| Pile diameter | Typical height | Volume | Cubic yards |
|---|---|---|---|
| 6 ft | 1.9 ft | 17.9 ft³ | 0.66 |
| 8 ft | 2.5 ft | 41.9 ft³ | 1.55 |
| 10 ft | 3.1 ft | 81.2 ft³ | 3.0 |
| 12 ft | 3.7 ft | 139 ft³ | 5.2 |
| 15 ft | 4.7 ft | 277 ft³ | 10.3 |
| 20 ft | 6.2 ft | 649 ft³ | 24.0 |
For material ordering — converting these cubic yards into tons of gravel or sand — the gravel and sand volume calculators add density tables on top of the same math.
Where cone volume shows up in real life
Ice cream, honestly measured
A standard sugar cone about 2 in across and 4.5 in deep holds (1/12)π × 4 × 4.5 ≈ 4.7 in³ — barely a quarter cup before the scoop overhangs. Waffle cones at 3 in × 6 in hold (1/12)π × 9 × 6 ≈ 14.1 in³, three times as much. Shape sells.
Funnels and hoppers
Industrial hoppers drain through a cone. A hopper cone 4 ft across at the top, tapering over 3 ft, holds (1/3)π × 4 × 3 ≈ 12.6 ft³ ≈ 94 gallons of material in the cone section alone. If the taper stops at a chute instead of a point, that's a frustum — use the truncated cone calculator.
Roof turrets and teepees
A conical turret roof 10 ft across and 8 ft tall encloses (1/3)π × 25 × 8 ≈ 209 ft³ of attic space — the number an HVAC installer needs before quoting ventilation.
Common mistakes & pro tips
- Slant height used as height. The sloped side is always longer than the vertical height, so this inflates the result. Convert first: h = √(s² − r²).
- Diameter entered as radius. A 4× error, same as any circular shape. Pile and funnel widths measured across the top are diameters — use Diameter mode.
- Forgetting the ÷3. If your answer exactly triples the calculator's, you computed the cylinder instead.
- Pro tip: measuring a pile's height. Don't climb it. Stand a straight pole vertically beside the pile, sight across the peak, and mark the pole. Or use the 0.31 × diameter estimate from the table above.
- Pro tip: cone + cylinder combos. Silo bottoms and funnels sit under cylindrical sections. Compute each part with its own calculator and add — see the cylinder calculator for the straight section.
How to use this calculator
- Measure the base. Measure the radius or diameter of the circular base. Pick the matching mode in the calculator.
- Measure the vertical height. Use the perpendicular height from base to tip — not the slanted side. If you only have the slant length s, the height is √(s² − r²).
- Let it apply V = (1/3)πr²h. Square the radius, multiply by π and the height, then divide by 3. The calculator shows every substitution.
- Convert the result. Switch the output between cubic units, liters and gallons as needed.
Frequently asked questions
Why is a cone's volume one-third of a cylinder's?
A cone tapers linearly from a full circle to a point, and integrating that shrinking cross-section gives exactly one-third of the cylinder with the same base and height. You can prove it in the kitchen: a conical cup fills its matching cylindrical container in exactly three pours.
How do I find cone volume with the diameter?
Halve the diameter first, or use V = (π/12)d²h. A cone 6 inches across and 8 inches tall: V = (π/12) × 36 × 8 = 75.4 in³. The calculator's Diameter mode does this automatically.
What's the difference between height and slant height?
Height is the straight vertical drop from tip to base center; slant height runs along the sloped surface. Only the vertical height belongs in the volume formula. Convert with h = √(s² − r²): a cone with a 5 cm slant and 3 cm radius has h = √(25 − 9) = 4 cm.
How much gravel is in a cone-shaped pile?
Dumped material settles into a cone at its angle of repose — about 30–37° for gravel and sand. Measure the pile's diameter and height, then apply V = (1/3)πr²h. A pile 12 ft across and 3.5 ft high holds (1/3)π × 36 × 3.5 ≈ 132 ft³ ≈ 4.9 cubic yards.
What is the volume of a cone with radius 3 and height 5?
V = (1/3) × π × 3² × 5 = (1/3) × π × 45 = 47.12 cubic units — exactly one-third of the 141.37 the matching cylinder holds.
How do I find the volume of a conical glass or container?
Measure the inside diameter across the rim and the vertical depth from the rim to the point, then use V = ⅓πr²h with r as half the diameter. A cone-shaped paper water cup 7 cm across and 9 cm deep holds ⅓ × π × 3.5² × 9 ≈ 115 mL, and a martini-style glass bowl 10 cm across and 7 cm deep holds about 183 mL. If the sides curve instead of running straight, the glass is not a true cone and the formula gives an estimate.
What measurements do you need to find the volume of a cone?
Two: the radius of the circular base and the vertical height. A diameter works too, because the radius is half of it. If you have the slant height instead of the height, find the height first with h = √(s² − r²). With the volume and one dimension, the solver on this page works backwards to the missing one.
How do I find the radius of a cone from its volume?
Rearrange V = (1/3)πr²h into r = √(3V ÷ (πh)). A cone of 220 cm³ with a height of 10 cm has a radius of √(660 ÷ 31.416) = 4.58 cm. To find the height instead, use h = 3V ÷ (πr²).