Everyday utility

Cone Calculator

Enter the base radius and perpendicular height to get cone volume, slant height, lateral surface area, and total surface area. The height is the straight distance from base center to apex, not the slant.

Last reviewed May 19, 2026 by ToolSpilo Editorial Team.

Review method: Reviewed against standard cone geometry formulas and cylinder volume comparison.

Calculator tool

How this calculator works

Use the explanation to understand the formula, assumptions, and practical limits behind the calculator result.

What a Cone Is

A cone is a 3D solid with a circular base and a curved surface that narrows to a single point called the apex. Ice cream cones, traffic cones, and party hats are everyday examples.

Formulas

Let rr be the base radius, hh the perpendicular height, and ll the slant height.

MeasurementFormula
Slant heightl=r2+h2l = \sqrt{r^2 + h^2}
VolumeV=13πr2hV = \dfrac{1}{3}\pi r^2 h
Lateral surface areaAL=πrlA_L = \pi r l
Total surface areaA=πr(r+l)A = \pi r(r + l)

Worked Example

For a cone with radius 5 cm and height 12 cm:

l=25+144=169=13 cml = \sqrt{25 + 144} = \sqrt{169} = 13\text{ cm}
V=13π×25×12=100π≈314.16 cm3V = \frac{1}{3}\pi \times 25 \times 12 = 100\pi \approx 314.16\text{ cm}^3
A=π×5×(5+13)=90π≈282.74 cm2A = \pi \times 5 \times (5 + 13) = 90\pi \approx 282.74\text{ cm}^2

Cone vs Cylinder

A cone with the same base radius and height as a cylinder holds exactly one-third of the cylinder's volume. This is a well-known result: Vcone=13VcylinderV_{cone} = \frac{1}{3}V_{cylinder}.

The Slant Height

Slant height ll is the distance from the base edge to the apex along the surface. It appears in the surface area formulas because it measures how much material covers the curved side. The perpendicular height hh and slant height ll form a right triangle with the radius rr at the base.

Frequently asked questions

What is the slant height and how is it different from the perpendicular height?

The perpendicular height (hh) is the straight vertical distance from the center of the base to the apex. The slant height (ll) is the distance along the curved surface from the base edge to the apex. They form a right triangle: l=r2+h2l = \sqrt{r^2 + h^2}. For a cone with r=5r = 5 and h=12h = 12, the slant height is 25+144=13\sqrt{25 + 144} = 13.

Why is the cone's volume one-third of a cylinder's volume?

A cone with radius rr and height hh fits exactly three times inside a cylinder with the same rr and hh. This can be demonstrated by filling a cone with water and pouring it into the matching cylinder — it takes exactly three cones to fill the cylinder. The formula V=13πr2hV = \frac{1}{3}\pi r^2 h reflects this relationship.

What is the lateral surface area used for?

Lateral surface area (πrl\pi r l) measures only the curved side — the material needed for the cone surface without the base. Use it to calculate fabric for a party hat or material for a traffic cone body. Total surface area adds the circular base: A=πr(r+l)A = \pi r(r + l), used when the base is also covered.

How do I use the cone formula if I know the slant height but not the perpendicular height?

Rearrange l=r2+h2l = \sqrt{r^2 + h^2} to find h=l2−r2h = \sqrt{l^2 - r^2}. For a cone with r=5r = 5 and l=13l = 13, the height is 169−25=144=12 cm\sqrt{169 - 25} = \sqrt{144} = 12\text{ cm}. Then use h=12h = 12 in the volume formula.