Geometry
Cone Volume Calculator
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How to use it
Using the cone volume calculator
- 01
Enter radius and vertical height
Height must be perpendicular, not measured along the slope.
- 02
Read volume and slant
Slant height feeds construction: pattern layout, fabric panels, roof framing.
- 03
Estimate materials
Lateral area approximates the outer skin; total surface adds the base disc.
Good to know
The one-third rule
Cones taper linearly from full base to a point, averaging one third of the constant cross-section a cylinder maintains. Ice cream scoopers, sand piles and funnel design all live with this economy; sharp tips cost little material but hold little too.
Why slant height matters practically
Nobody builds along the invisible vertical axis: fabricators cut patterns along the surface. The 3-4-5 triple hiding in our example (legs r = 3, h = 4 → l = 5) shows how often slant heights come out clean.
How it's calculated
The math behind this calculator
V = πr²h⁄3 l = √(r² + h²) Lateral area = πrlA cone holds exactly one third of the cylinder sharing its base and height; demonstrable by filling three cone-cupfuls into one cylinder. The slant height l comes from the Pythagorean theorem applied to the right triangle formed by radius, height and slope.
Lateral (side) surface area unrolls into a circular sector of radius l and arc length 2πr, giving πrl. Add the base circle πr² for the total enclosing area, shown as an extra row.
Assumptions & limitations
- Right circular cone with apex directly above the center.
- Perpendicular height, not slant length, goes in the height field.
- Positive lengths in one consistent unit.
Worked example
A cone of radius 3 and height 4 has slant height exactly 5, volume ≈ 37.70 cubic units, and lateral area ≈ 47.12 square units.
FAQ
Frequently asked questions
- I measured the slant, not the height; now what?
- Square the slant, subtract the radius squared, take the root: that converts slant back to vertical height before entering it.
- Does lateral area include the base?
- No; it is just the sloped side. The total surface row adds the πr² base disc.
- What about truncated cones (buckets)?
- Subtract a smaller cone from a larger one: V = πh(R² + Rr + r²)/3 handles the frustum case.
- Why is my pile’s volume off in practice?
- Loose material reposes at its angle of repose and piles are rarely perfect cones; treat results as upper-bound estimates.
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