On This Page
- What is the formula for cone volume?
- The height trap: vertical, not slant
- Four ways the numbers arrive
- Composite cone problems
- When the top is cut off
- Units and common slips
- How the calculator is verified
- FAQ
What Is the Formula for Cone Volume?
Here r is the radius of the circular base and h is the vertical height from the base center to the tip. A cone holds exactly one-third of a cylinder with the same base and height: fill a cone-shaped cup and it takes three pours to fill the matching cylinder. Why a cone is exactly one-third of a cylinder gives the proof, and the same one-third appears in the Pyramid Volume Calculator for the same reason.
By hand: square the radius, multiply by the height and π, then divide by 3. Enter the radius and height above and the calculator adds slant height, surface area, and liquid capacity in liters and gallons.
The Height Trap: Vertical, Not Slant
Cone problems often give the slant height l, the diagonal along the side, which is longer than the vertical height the formula needs. Recover the vertical height with Pythagoras:
A cone with a 6 cm radius and a 10 cm slant is √(100 − 36) = 8 cm tall, so its volume is (1/3)π(36)(8) = 96π ≈ 301.6 cm³. A leaning (oblique) cone uses the same formula with the perpendicular height. The calculator has no slant field, because it works slant height out from the radius and height you enter.
Four Ways the Numbers Arrive
- Radius and height: a traffic cone with a 4 in radius, 18 in tall: (1/3) × 3.14 × 16 × 18 = 301.4 in³.
- Diameter instead of radius: halve it first. A party hat 10 cm across and 24 cm tall is (1/3)π(5²)(24) = 200π ≈ 628.3 cm³.
- Slant height: convert it with h = √(l² − r²), as above.
- Volume known, dimension missing: h = 3V ÷ (πr²) or r = √(3V ÷ πh). A 192π in³ cone that is 16 in tall has r = √(576 ÷ 16) = 6 in.
Composite Cone Problems
For combined shapes, work out each part and add them.
Ice cream cone with a scoop: a cone of radius 3 in and height 8 in is (1/3)π(9)(8) = 24π. A hemisphere scoop of the same radius is (2/3)π(27) = 18π. The total is 42π ≈ 131.9 in³. A silo that is a cylinder topped by a cone works the same way.
Filling time: a cone of radius 60 cm and height 150 cm holds (1/3) × 3.14 × 3,600 × 150 = 565,200 cm³. A hose delivering 1,413 cm³ a minute fills it in 400 minutes.
When the Top Is Cut Off
A bucket, plant pot or paper cup is a cone with its tip sliced off parallel to the base, a frustum. It has two radii, so it needs V = (1/3)πh(R² + Rr + r²); use the Frustum Volume Calculator.
Units and Common Slips
Volume comes out in cubic units. For a funnel or container, 1,000 cm³ is 1 liter, 1 m³ is 1,000 liters, 231 in³ is 1 US gallon and 1 ft³ is about 7.48 US gallons. The usual slips are using the slant height, forgetting the one-third, putting in the diameter as the radius, mixing units, and using the full-cone formula on a frustum.
How the Calculator Is Verified
There's no slant height field on this calculator. The value comes from Math.sqrt(radius^2 + height^2), computed fresh every time you change either the radius or the vertical height, and that same slant height feeds directly into the surface area formula (PI * r * (r + slantHeight)). Because it's derived, not entered, it can't drift out of sync with the radius and height the volume figure is built from.
Frequently Asked Questions
Why does a cone hold exactly one-third of the same-sized cylinder?
A cone tapers from a full circular base to a single point, so each horizontal cross-section becomes smaller toward the apex. When you sum all those shrinking circles, the total is exactly one-third of the cylinder. You can demonstrate it physically: fill a cone-shaped cup with water and pour it into a matching cylinder, you need exactly three pours to fill it.
What common objects are cone-shaped?
Traffic cones, party hats, funnels, waffle cones, and ice cream cones are close enough to true cones for volume estimates. Sand, salt, or grain piled into a conical mound in a bin or stockpile is another practical case, volume gives you how much material is there before moving it.
References
- Cone, Wikipedia: geometric definition, volume formula derivation, oblique cone, and frustum relationship.
- Cone, Wolfram MathWorld: formal mathematical treatment including slant height, lateral surface area, and volume proof.
- Cone, Math is Fun: approachable explanation of the cone volume formula with diagrams and examples.