Cone: properties & formulas
A right circular cone narrows from a circular base to a single apex directly above the center. Its volume is one third of a cylinder with the same base and height.
- 1 circular face
- 1 curved surface
- 1 curved edge
- 1 vertex (apex)
Properties of a cone
| Flat faces | 1 circular base |
|---|---|
| Curved surfaces | 1 |
| Edges | 1 circular edge where the base meets the curved surface |
| Vertices | 1 apex |
| Slant height | ℓ = √(r² + h²) |
| Cross-sections | Parallel to the base: circles; through the apex and axis: an isosceles triangle |
| Net | A circle and a circular sector |
Cone formulas
| Volume | V = πr²h / 3 |
|---|---|
| Total surface area | S = πr(r + ℓ) = πr² + πrℓ |
| Lateral (curved) area | L = πrℓ |
| Slant height | ℓ = √(r² + h²) |
| Height from slant height | h = √(ℓ² − r²) |
Cone volume & surface area calculator
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Use the same unit for every dimension. Changing the unit relabels the measurements; it does not convert them.
From dimensions to an answer
A worked example
A cone with radius 3 and vertical height 4 has slant height 5, volume 12π ≈ 37.699 cubic units, and total surface area 24π ≈ 75.398 square units.
Cone: common questions
Wondering about the why? Start here.
How many faces, edges, and vertices does a cone have?
A cone has 1 flat circular face, 1 curved surface, 1 curved edge around the base, and 1 vertex at the apex. Some curricula list the edge count as zero because the edge is curved rather than straight.
Is a cone a 2D or 3D shape?
A cone is a three-dimensional solid. Its side view is a triangle and its base is a circle, which are two-dimensional. Unrolled, its net is a circle and a circular sector.
See printable netsWhat is the cross-sectional area of a cone?
A cut parallel to the base is a circle. At distance y from the apex, its radius is r × y / h, so its area is π(ry / h)². A vertical cut through the apex gives an isosceles triangle with area r × h.
Why is cone volume one third of cylinder volume?
A cone has one third the volume of a cylinder with the same base area and perpendicular height. Their formulas are πr²h/3 and πr²h respectively. The comparison only applies when both radius and height match.
Compare with a cylinderHow do perpendicular height and slant height differ?
Perpendicular height runs from the tip straight down to the base plane. Slant height runs along the cone’s side. For a right circular cone, slant height is √(r² + h²). This calculator accepts perpendicular height and derives slant height for surface area.
See the Pythagorean connection