Right Circular Cone Calculator: Volume & Surface Area

What is a right circular cone?

A right circular cone is a solid shape with a flat circular base and one curved side that narrows to a single point, called the apex. It is "right" because the apex sits directly above the center of the base. A line from the apex straight down to the base meets the base at a right angle. This calculator finds a cone's volume, base area, lateral (side) surface area, and total surface area from its radius and height. It works out the slant height along the way but does not display it.

Common examples of the shape include traffic cones, ice cream cones, funnels, and the roofs of some towers.

Cone measurements

Two measurements describe a right circular cone:

• Radius (r): the distance from the center of the circular base to its edge.
• Height (h): the straight vertical distance from the base to the apex.

A third length, the slant height (l), is the distance along the outside of the cone from the apex down to the edge of the base. It is not measured directly. Instead it is calculated from the radius and height, because these three lengths form a right triangle.

Cone formulas

Slant height formula

The radius, height, and slant height form a right triangle, so the slant height follows from the Pythagorean theorem:

$$l = \sqrt{r^2 + h^2}$$

Base area formula

The base is a circle, so its area uses the standard circle-area formula:

$$A_b = \pi r^2$$

Lateral surface area formula

The lateral surface area covers only the curved side, not the base. If the curved side were unrolled flat, it would form a sector of a circle with area:

$$A_l = \pi r l$$

Total surface area formula

Total surface area adds the base to the curved side:

$$A = A_b + A_l = \pi r^2 + \pi r l = \pi r (r + l)$$

A cone that is open at the base, such as a paper funnel, only needs the lateral area for material calculations.

Cone volume formula

$$V = \frac{1}{3} \pi r^2 h$$

A cone's volume is exactly one third of a cylinder with the same base and height. Eudoxus proved this relationship with a method that came before calculus. It was later confirmed with integral calculus, which treats the cone as an infinite stack of shrinking circular disks from base to apex.

How to calculate cone volume: worked example

Take a cone with radius r = 5 units and height h = 12 units.

1. Find the slant height.

$$l = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \text{ units}$$

2. Find the base area.

$$A_b = \pi (5)^2 = 25\pi \approx 78.54 \text{ units}^2$$

3. Find the lateral surface area.

$$A_l = \pi (5)(13) = 65\pi \approx 204.20 \text{ units}^2$$

4. Find the total surface area.

$$A = 25\pi + 65\pi = 90\pi \approx 282.74 \text{ units}^2$$

5. Find the volume.

$$V = \frac{1}{3} \pi (5)^2 (12) = 100\pi \approx 314.16 \text{ units}^3$$

The calculator shows each result to four decimal places: 78.5398, 204.2035, 282.7433, and 314.1593.

Diagram

The dashed vertical line is the height, from the base to the apex. The dashed horizontal line is the radius, from the center of the base to its edge.

Where cones appear in real life

Cone volume and surface area calculations show up in several practical settings:

• Sheet metal fabrication: the lateral surface area gives the amount of flat material needed to roll into a conical hopper, funnel, or duct reducer, before adding a margin for seams and waste.
• Storage capacity: the volume formula gives the capacity of conical tanks and the conical bottoms of silos.
• Construction: spires, tower caps, and other conical roof elements use the lateral area to estimate roofing material.
• Everyday objects: paper cups, ice cream cones, and traffic cones are all approximately right circular cones.

This calculator assumes a perfect cone with a circular base and the apex centered over the base. A cone cut off at the top, called a frustum, needs a different set of formulas because it has two radii instead of one.

History

Ancient Greek mathematicians studied cones in detail. Euclid's Elements, written around 300 BCE, defined a cone as the solid formed by rotating a right triangle around one of its legs. The cone volume rule was first stated by Democritus. Eudoxus of Cnidus proved it with the method of exhaustion, a geometric technique that anticipated integral calculus, and Archimedes credited the proof to him. Later, Apollonius of Perga studied the curves formed by slicing a cone at different angles, now called conic sections. His book Conics introduced the terms parabola, ellipse, and hyperbola.

Frequently asked questions

What makes a cone "right circular"?

"Right" means the apex is directly above the center of the base, so the axis is perpendicular to the base. In an oblique cone, the apex is off to one side. "Circular" means the base is a circle rather than an ellipse or another shape.

What is the formula for the volume of a cone?

V = (1/3)πr²h, where r is the base radius and h is the perpendicular height. Volume does not require the slant height.

What is the formula for the surface area of a cone?

Total surface area is A = πr² + πrl, where r is the radius and l is the slant height. The first term is the base area and the second is the lateral (side) area. Leave out the first term if the cone has no base, such as an open funnel.

What is slant height, and how is it different from height?

Height is the straight vertical distance from the base to the apex, measured through the inside of the cone. Slant height is the distance along the outside surface from the apex to the edge of the base. They are related by l = √(r² + h²), with slant height always the longer of the two whenever the radius is greater than zero.

How do I find the volume of a cone from its diameter?

Divide the diameter by 2 to get the radius, then use V = (1/3)πr²h. For example, a cone with a 10 cm diameter and a 15 cm height has a 5 cm radius, so V = (1/3)π(5²)(15) ≈ 392.7 cm³.

Does this calculator work for truncated cones (frustums)?

No. A frustum is a cone with the top sliced off, so it has two radii instead of one, and it needs its own formulas for volume and surface area.

References

1. D. E. Joyce, Clark University. Euclid's Elements, Book XII, Proposition 10. Elements XII.10 — "Any cone is a third part of the cylinder with the same base and equal height."
2. D. E. Joyce, Clark University. Euclid's Elements, Book XI, Definitions 18–20. Elements XI, Definition 18 — a cone is the figure swept out when a right triangle turns about one of the sides at the right angle.
3. MacTutor History of Mathematics, University of St Andrews. Eudoxus of Cnidus. Eudoxus biography — Eudoxus proved, by the method of exhaustion, that a cone is one third of the cylinder with the same base and height; Archimedes attributed the proof to him.
4. MacTutor History of Mathematics, University of St Andrews. Apollonius of Perga. Apollonius biography — his Conics introduced the terms parabola, ellipse and hyperbola.