Cone Calculator

What is a Cone?

A cone is a three-dimensional geometric shape that tapers smoothly from a flat circular base to a point called the apex or vertex. More specifically, this calculator deals with a right circular cone, where the apex is positioned directly above the center of the circular base, making the axis perpendicular to the base.

Cones are fundamental shapes found throughout nature and human-made structures. They appear in ice cream cones, traffic cones, volcanoes, party hats, speaker cones, funnels, and many architectural elements. Understanding cone properties is essential for construction, manufacturing, and various engineering applications. For related 3D shapes, try the Cylinder Calculator and Conical Frustum Calculator.

Formulas and Calculations

The key formulas for a right circular cone are:

Volume (V): V = (1/3)πr²h

The volume of a cone is exactly one-third of the volume of a cylinder with the same base radius and height.

Slant Height (s): s = √(r² + h²)

The slant height is the distance from the apex to any point on the circumference of the base, calculated using the Pythagorean theorem.

Lateral Surface Area (L): L = πrs

This is the area of the cone's curved surface, excluding the base.

Base Surface Area (B): B = πr²

This is the area of the circular base.

Total Surface Area (A): A = L + B = πr(s + r)

The total surface area combines the lateral surface area and the base surface area.

Angles

The half-angle (θ) is the angle between the center axis and any side of the cone, calculated as θ = arctan(r/h). The aperture angle (φ or 2θ) is the full angle at the apex between opposite sides. The base angle (β) is the angle at the base where the side meets the base plane.

How to Use the Cone Calculator

Using this calculator is simple. Enter the radius (r) of the circular base and the height (h) of the cone. The tool will instantly compute the slant height, volume, lateral surface area, base surface area, total surface area, and all cone angles. All calculations are performed in real-time in your browser.

Both values must be positive numbers. You can enter whole numbers or decimals for precise calculations.