Volume of a cone calculator
Set to a radius of 3 cm and a height of 4 cm, the volume of a cone calculator gives 12π cm³, about 37.70 cm³, a third of the 36π cylinder with the same base and height.
Volume
- Volume The exact answer is the form before the ≈ sign. Keep it that way until the last step if more arithmetic follows.
- 12π ≈ 37.69911 cm³
- Formula A third of the cylinder with the same base and height. Any cone or pyramid (any solid tapering to a point) is a third of its prism, which is what the 1/3 means.
- V = (1/3)πr²h = (1/3) × π × 3² × 4 = 12π ≈ 37.69911
- Measurements used
- r = 3 cm, h = 4 cm
- Total surface area A = πr² + πrs = π × 3 × (3 + 5).
- 24π ≈ 75.39822 cm²
- Lateral area (the sloping side) πrs, where s is the slant height √(r² + h²) = 5 cm. The height in the volume formula is the vertical one; the slant is only for area.
- 47.12389 cm²
- In liters 1 liter is 1,000 cubic centimeters, the volume of a 10 cm cube.
- 0.03769911 L
- In US gallons A US gallon is defined as exactly 231 cubic inches, 3.785411784 liters. An imperial gallon is about 20% larger.
- 0.009959052 gal
- In other cubic units Cubic conversions cube the length factor: a yard is 3 feet, so a cubic yard is 27 cubic feet.
- 0.0000376991 m³ = 0.001331332 ft³ = 0.0000493086 yd³
| Every length times | Volume | Surface area |
|---|---|---|
| 0.5 | 4.712389 cm³ (×0.125) | 18.84956 cm² (×0.25) |
| 1 | 37.69911 cm³ (×1) | 75.39822 cm² (×1) |
| 2 | 301.5929 cm³ (×8) | 301.5929 cm² (×4) |
| 3 | 1,017.876 cm³ (×27) | 678.584 cm² (×9) |
| 10 | 37,699.11 cm³ (×1,000) | 7,539.822 cm² (×100) |
- Volume scales with the cube of length and surface area with the square. Double every measurement and the volume goes up 8 times while the surface only goes up 4. That is why large animals keep heat better and why small ice cubes melt faster.
How the Volume Calculator works
The volume of a cone is V = (1/3)πr²h, where r is the radius of the circular base and h is the vertical height from the base to the tip. It is the cylinder's formula, πr²h, divided by three. For a radius of 3 cm and a height of 4 cm: (1/3) × π × 9 × 4 = 12π, about 37.699 cm³. The cylinder with the same base and height would hold 36π, three times as much.
Why a third? Slice the cone into thin disks parallel to the base. At a fraction x of the way down from the tip the disk's radius is x times r, so its area is x² times the base area. Adding up disks whose areas grow with the square of the distance gives a third of the full stack, for the same reason the area under y = x² from 0 to 1 is 1/3. It holds for any solid that tapers in straight lines from a flat base to a single point, which is why a pyramid has the same 1/3 in its formula. Fill a cone-shaped paper cup and pour it into a cylindrical glass of the same width and height, and the water comes a third of the way up.
The common mistake is the height. The volume formula wants the vertical height, measured straight down from the tip to the center of the base. The slant height, measured along the sloping side, is longer, and it belongs in the surface area formula, not the volume one. In this example the slant height is √(3² + 4²) = 5 cm, and using 5 in place of 4 gives 15π, about 47.12 cm³, which is 25% too much. If a problem gives you the slant height s, get the vertical height first: h = √(s² - r²). The second mistake is using the diameter as the radius, which makes the answer four times too big.
Volume comes out in cubic units: cubic centimeters here, and a cubic centimeter is a milliliter, so this cone holds about 37.7 mL. The surface area, in square units, is πr² for the base plus πrs for the sloping side: 9π + 15π = 24π, about 75.40 cm². An ice cream cone or a funnel has no base, so only the 15π side counts.
A cone does not have to be upright. An oblique cone, with its tip off to one side, has the same volume as the right cone with the same base and vertical height, because every slice still has the same area. A cone with its top cut off is a frustum (a bucket, a lampshade, a flower pot) and has its own shape in this calculator: (1/3)πh(R² + Rr + r²), which is the big cone minus the small one. A pile of sand or gravel settles into a rough cone too, and (1/3)πr²h is the usual way to estimate how much is in it.
Volume Calculator FAQ
What is the formula for the volume of a cone?
V = (1/3)πr²h, where r is the base radius and h the vertical height. A cone with radius 3 and height 4 has a volume of 12π, about 37.70 cubic units.
Should I use the slant height or the vertical height?
The vertical height, measured straight from the tip to the center of the base. The slant height is for the surface area. If you only have the slant height s, the vertical height is √(s² - r²).
Why is a cone one third of a cylinder?
Its cross-sections shrink in proportion to the distance from the tip, so their areas shrink with the square of it, and summing areas that grow like x² gives one third of the full stack. The same holds for every pyramid.
How do I find the volume of a cone from the diameter?
Halve the diameter first, or use V = πd²h / 12. A cone 6 cm across and 4 cm tall is 12π cubic centimeters either way.
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