Right Circular Cone Calculator - Volume and Surface Area

A right circular cone calculator is a geometry tool that returns every standard measurement of a right circular cone in one pass. Enter the radius (or diameter) and the perpendicular height, and the page returns the volume, slant height, base area, lateral and total surface area, and apex angle in the unit of your choice.

• • Geometry homework and textbook problems: Solve a chapter on solids of revolution: compute the volume, lateral area, and apex angle of a right circular cone given a base radius and a perpendicular height, including the classic 3-4-5 and 5-12-13 slant-height examples.
• • Cone-shaped objects and 3D models: Estimate the capacity of a traffic cone, a paper cup, a funnel, or a grain hopper, or confirm the apex angle of a CAD model where a tangent cone meets a sphere or a cylinder.

A right circular cone has its axis perpendicular to the base. An oblique cone leans to one side; V = (1/3) · π · r² · h still holds (with h the perpendicular height), but its lateral surface area needs 3D coordinates, so this page focuses on right circular cones.

For a tool that focuses on the cone volume formula alone with metric and imperial unit selection, the Cone Volume Calculator page covers the (1/3) · π · r² · h relationship in a stripped-down form.

The right circular cone calculator applies the standard formulas in one pass. It takes the base dimension and the perpendicular height, converts the diameter to a radius when needed, and uses the radius and the height to compute the slant height, volume, base area, lateral and total surface areas, and the apex angle.

• r: Radius of the circular base, in the chosen length unit.
• h: Perpendicular (vertical) height from the base to the apex.
• ℓ: Slant height, equal to sqrt(r² + h²).
• A_lateral: Curved side area, equal to π r ℓ.
• A_total: Lateral area plus the circular base, equal to π r (r + ℓ).
• 2α: Full apex (tip) angle in degrees, equal to 2 arctan(r / h).

All six outputs come from the same two inputs, so the slant height is always consistent with the radius and the height you typed.

According to Wolfram MathWorld, the volume of a right circular cone of radius r and height h is V = π r^2 h / 3, the slant height is l = sqrt(r^2 + h^2), the lateral surface area is π r l, and the apex half-angle satisfies tan(α) = r / h

Because a right circular cone is exactly one third of the cylinder that surrounds it, the Cylinder Volume Calculator page is a useful companion: run the cylinder formula and divide by 3 as a manual check on the cone volume.

Four ideas explain why these formulas look the way they do and what each result really means geometrically.

If you need the top cut off (a frustum), the radius changes with height and a separate formula applies.

If you only need the curved side of a right circular cone, the Lateral Area of Cone Calculator page applies π · r · ℓ directly and skips the volume and the apex angle.

Five short steps cover every common case, from a textbook example to a flat zero-height cone.

1. 1 Pick the base input mode: Choose 'Radius' or 'Diameter' depending on whether you know the distance from the center of the base to the rim or the full width across the base.
2. 2 Enter the base length: Type the radius or the diameter in the chosen length unit. A typical textbook problem uses 3 cm or 5 cm; a real funnel might use 8 cm; a paper cup might use 2 in.
3. 3 Enter the perpendicular height: Type the vertical distance from the base plane to the apex. Use the perpendicular height (not the slant height) so the volume and apex angle come out correctly.
4. 4 Pick the length unit: Choose centimeters, meters, inches, or feet. Volume, area, and slant-height outputs use the matching cubic, squared, and linear unit label.
5. 5 Read the results and reset: Read the volume, slant height, base area, lateral and total surface area, and apex angle. Click Reset to restore the default example and start a new problem.

Try radius = 5 cm, height = 12 cm (the 5-12-13 right triangle). The page shows ℓ = 13 cm, V ≈ 314.16 cm³, base area ≈ 78.54 cm², lateral area ≈ 204.20 cm², total area ≈ 282.74 cm², and apex angle 2α ≈ 45.24°.

A common follow-up is to compare the cone volume with a sphere of the same radius, and the Sphere Volume Calculator page returns the (4/3) · π · r³ result with the same unit selection.

These benefits matter most when you are working a problem by hand and need a quick check, or when you are designing a real conical object.

• • Six cone measurements in one pass: Get the volume, slant height, base area, lateral and total surface area, and apex angle from the same two inputs.
• • Skips the arithmetic mistakes: Manual cone problems are easy to slip up on the squaring step, the slant-height square root, or the (1/3) factor. The page applies the formulas exactly.
• • Works for radius or diameter: Switch between radius and diameter input with a single selector, so a problem that gives you the diameter and a problem that gives you the radius both work without manual conversion.
• • Metric and imperial in the same tool: Choose centimeters, meters, inches, or feet, and the volume, area, and slant-height labels update to the matching cubic, squared, and linear unit.
• • Includes the apex angle: The apex (tip) angle is the most often-missed cone measurement. The page returns 2 arctan(r / h) in degrees.
• • Connects to the rest of the math toolkit: The page links to peer calculators for cone volume, cylinder volume, sphere volume, lateral cone area, and surface area.

Use it to confirm a homework answer, sanity-check a fabrication estimate, or pre-validate a cone before passing it to a 3D modeling script.

If your real problem is a closed container that is not a cone, the Surface Area Calculator page covers other common 3D shapes with the same metric and imperial unit selection.

The formulas are the same in every case, but a few factors change how the result should be read and how it is applied in a real problem.

• • This page is for right circular cones only. An oblique cone shares the same volume formula V = (1/3) · π · r² · h, but its lateral and total surface areas need 3D coordinates instead of the π · r · ℓ and π · r · (r + ℓ) forms used here.
• • The page assumes a single solid cone with one circular base. For a frustum or a truncated cone (top cut off) the radius changes with height and a separate formula is needed.
• • The page does not include an inscribed cylinder, circumscribed cylinder, or tangent sphere, which are common extensions in calculus and engineering problems.

The volume formula V = (1/3)π r² h has a clean geometric check: a cone is exactly one third of the cylinder that surrounds it. Run a cylinder calculator and divide by 3 to verify the cone volume.

According to Wikipedia (Cone article), a right circular cone is a cone whose axis is perpendicular to the base, and the slant height is the distance from the apex to any point on the base circle, equal to sqrt(r^2 + h^2)

If the top of the cone has been cut off, the radius is no longer constant and the Truncated Cone Volume Calculator page handles a frustum with its own formulas for volume and surface area.