Cone calculator
Volume, slant height and surface area of a cone from radius and height.
Fill in the fields and the result will appear here automatically.
Works a cone — the shape of a sand pile, a funnel, a hopper or a tent roof. Besides the volume it gives the slant height: the distance along the slope from the apex to the rim, which is what you need for cutting cladding, whereas the height is the vertical from apex to centre. Those two are the ones people mix up, and the gap is real: at radius 3 and height 4 the slant is 5.
How it works
Formula and logic
V = π · r² · h ÷ 3, the slant height is l = √(r² + h²), the lateral surface is πrl and the total is πr(r + l). Unrolling the lateral surface gives a sector of radius l with arc length 2πr, so its area is (2πr)l/2 = πrl.
Example
A cone of radius 3 m and height 4 m has a slant height of 5 m and a volume of 37.699 m³.
Fields and units
- Length unit — list option
- Base radius — cm
- Height — cm
How to use
- — Choose the length unit.
- — Enter the base radius and the vertical height.
- — Read the volume, slant height and surfaces.
- — Use positive radius and height, with the apex above the centre of the circular base. Switching cm/m changes input interpretation and does not convert the numbers.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax: solid volumes and surface areas OpenStax: sine and tangent in a right triangle
- Limitation
- A right circular cone with positive r and h. Slant height follows the surface; height is perpendicular to the base. Volume is geometric; cladding thickness and cutting allowance are excluded.
FAQ
How does the slant height differ from the height?
The height is the vertical from apex to the centre of the base; the slant is the sloping line from apex to the rim. The slant is always longer, and it is the one you need for cutting cladding.
Why is a cone a third of a cylinder?
Because a cone with the same base and height occupies exactly one third of that cylinder — a classic result of solid geometry.
How do I measure a sand pile?
Measure the circular base radius and vertical height. An ideal cone approximates a pile: uneven shape, base slope and compaction can shift the result in either direction. Geometric volume does not determine mass without density.
What does the total surface include?
The lateral surface plus the circular base. For an open funnel only the lateral part applies.