339.292
cu units
113.0973
sq units
1,017.876
cu units
0.333333
339.292
cu units
113.0973
sq units
1,017.876
cu units
0.333333
The Cone Volume Calculator computes the volume of a cone given its base radius and height using the classical formula:
$$V = \frac{1}{3}\pi r^2 h$$
where $$r$$ is the radius of the circular base and $$h$$ is the perpendicular height from the base to the apex. This formula applies to both right and oblique circular cones, provided $$h$$ is measured perpendicular to the base.
The factor of $$\frac{1}{3}$$ is what distinguishes cone volume from cylinder volume. A cone holds exactly one-third the volume of a cylinder with the same base radius and height. This relationship was known to ancient Greek mathematicians and can be demonstrated by a classic water-pouring experiment: fill a cone with water and pour it into a matching cylinder -- you need exactly three cones to fill the cylinder.
The mathematical proof uses integral calculus. Consider the cone with its apex at the origin and base at height $$h$$. At height $$y$$, the cross-sectional circle has radius $$\frac{ry}{h}$$, so the area is $$\pi\left(\frac{ry}{h}\right)^2$$. Integrating from 0 to $$h$$:
$$V = \int_0^h \pi \frac{r^2 y^2}{h^2}\,dy = \frac{\pi r^2}{h^2} \cdot \frac{h^3}{3} = \frac{1}{3}\pi r^2 h$$
This derivation elegantly produces the $$\frac{1}{3}$$ factor from the integration of $$y^2$$.
Cone volume calculations are essential in industrial applications. Conical tanks and hoppers store granular materials, liquids, and powders. Engineers calculate their volume to determine storage capacity and estimate fill levels. The conical shape promotes gravity-assisted flow, which is why silos and bins often have conical bottoms.
In geology, volcanic cinder cones and alluvial fans approximate conical shapes. Estimating the volume of erupted material requires cone volume calculations, often using aerial measurements of base diameter and height. Similarly, stockpiles of sand, gravel, and ore at mining sites form conical heaps whose volumes are estimated with this formula.
The food and beverage industry uses cone volume for ice cream cones (determining how much ice cream fits inside the cone portion), conical wine glasses, and funnel-shaped containers. In traffic engineering, the volume of a standard traffic cone helps determine material costs and shipping logistics.
This calculator also shows the equivalent cylinder volume (the cone volume multiplied by 3) and the base area for additional context. These companion values help visualize the 1/3 relationship and provide useful information for related calculations.
Enter the base radius and perpendicular height. The calculator computes $$V = \frac{1}{3}\pi r^2 h$$ using JavaScript's Math.PI and Math.pow functions. It also calculates the equivalent cylinder volume $$\pi r^2 h$$ (showing the 3:1 ratio) and the base area $$\pi r^2$$.
The Volume is the space enclosed by the cone, in cubic units matching your input units. The Equivalent Cylinder Volume shows the volume of a cylinder with the same base and height -- always exactly three times the cone volume. The Base Area is the circular bottom. Ensure height is the perpendicular distance from base to apex, not the slant height.
Inputs
Results
V = (1/3)*pi*36*9 = 108*pi ~ 339.29. Cylinder volume = 324*pi ~ 1017.88. The cone is exactly 1/3 of the cylinder, as expected.
Inputs
Results
V = (1/3)*pi*9*2 = 6*pi ~ 18.85 m3. At typical sand density of 1600 kg/m3, this pile weighs about 30,159 kg (30 tonnes).
If you know the base diameter $$d$$, the radius is $$r = d/2$$. Substituting: $$V = \frac{1}{3}\pi \left(\frac{d}{2}\right)^2 h = \frac{\pi d^2 h}{12}$$. So divide diameter squared times height times pi by 12.
Use the Pythagorean theorem to find the perpendicular height: $$h = \sqrt{l^2 - r^2}$$, where $$l$$ is the slant height and $$r$$ is the base radius. Then use $$V = \frac{1}{3}\pi r^2 h$$ with this computed height.
Yes. The volume formula $$V = \frac{1}{3}\pi r^2 h$$ applies to any cone (right or oblique) as long as $$h$$ is the perpendicular distance from the base plane to the apex. This follows from Cavalieri's principle: any two solids with equal cross-sectional areas at every height have equal volumes.
Rearrange the formula: $$h = \frac{3V}{\pi r^2}$$. For example, to hold 500 cubic units in a cone with radius 8: $$h = \frac{3 \times 500}{\pi \times 64} = \frac{1500}{64\pi} \approx 7.46$$ units.
Both use the same principle: volume equals one-third of the base area times height. For a cone, $$V = \frac{1}{3}(\pi r^2)h$$. For a pyramid, $$V = \frac{1}{3}Bh$$ where $$B$$ is the base area (square, triangular, etc.). The 1/3 factor applies universally to all pointed solids with a flat base.
Measure the inner radius at the top ($$r$$) and the depth ($$h$$) of the cup. Apply $$V = \frac{1}{3}\pi r^2 h$$. For a typical paper cone cup with $$r = 4$$ cm and $$h = 10$$ cm: $$V = \frac{1}{3}\pi(16)(10) \approx 167.6$$ cm3, or about 168 ml -- roughly 5.7 fluid ounces.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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