What is a cone calculator?
A cone calculator computes the volume, surface area, and slant height of a right circular cone from the radius and height. A cone has a circular base and a single vertex (apex); the height is the perpendicular distance from the apex to the base. You enter the radius of the base and the height; the calculator gives you volume = (1/3)πr²h, total surface area = πr(r + slant height), and slant height = √(r² + h²). No sign-up or download is required; the tool runs in your browser and is useful for geometry, engineering, and any task involving conical shapes.
Cones appear in funnels, traffic cones, ice cream cones, and many containers. Knowing the volume helps with capacity; surface area helps with material or coating. This free cone calculator gives you volume in cubic units and surface area in square units—meters, feet, inches—as long as you use the same unit for radius and height. Slant height is the straight-line distance from the apex to a point on the base edge and is used in the surface area formula.
How to use this cone calculator
Enter the radius of the circular base and the height of the cone. Both must be positive numbers. The height is the perpendicular distance from the apex to the center of the base, not the slant length. As soon as both values are valid, the results appear: volume in cubic units, surface area in square units, and slant height in the same linear unit as radius and height. The formulas are shown under the inputs. If you enter zero or a negative value, the calculator will ask you to enter positive dimensions. Use the same unit for radius and height; volume will be in cubic units and surface area in square units of that measure.
Cone volume formula
The volume of a cone is V = (1/3)πr²h, where r is the radius of the base and h is the height. So the cone’s volume is one-third the volume of a cylinder with the same base and height. For example, if the radius is 4 units and the height is 6 units, volume = (1/3) × π × 16 × 6 ≈ 100.53 cubic units. The formula comes from calculus (integrating circular cross-sections) or from comparing the cone to a cylinder. The cone calculator applies this formula so you get an accurate volume without doing the arithmetic by hand.
Surface area of a cone
The total surface area of a cone is the area of the circular base plus the lateral (curved) surface. The lateral surface unwraps into a sector of a circle with radius equal to the slant height s. So total surface area = πr² + πrs = πr(r + s), where s = √(r² + h²). For radius 4 and height 6, slant = √(16 + 36) = √52 ≈ 7.21, so surface area = π × 4 × (4 + 7.21) ≈ 141.37 square units. Use surface area when you need to cover the cone (paint, wrap) or when comparing heat transfer or exposure. This calculator computes both the lateral and total surface area via the slant height.
Slant height
The slant height of a cone is the distance from the apex to any point on the edge of the base, measured along the surface. It is the hypotenuse of a right triangle with legs r (radius) and h (height), so slant height s = √(r² + h²). It is always greater than the vertical height. The slant height is used in the lateral surface area formula (πrs) and in the total surface area πr(r + s). This cone calculator displays the slant height so you can use it for construction or design. All three outputs—volume, surface area, and slant height—are derived from the same radius and height.
Cone vs cylinder
A cylinder with the same radius and height as a cone has volume πr²h; the cone has one-third of that, (1/3)πr²h. So if you know the volume of a cone, the volume of a “matching” cylinder is three times larger. The cone’s lateral surface is a single curved sheet; the cylinder’s lateral surface is a rectangle wrapped around. This cone calculator focuses on the cone; enter radius and height to get volume, surface area, and slant height. For a cylinder, you would use a different formula (volume = πr²h, no 1/3 factor).
Applications of cone calculations
Cone volume and surface area are used in geometry, engineering, and packaging. In education, the cone is a standard solid when studying volume and surface area. In engineering, funnels, hoppers, and some tanks are conical. In manufacturing, cone dimensions affect capacity and material use. This cone calculator gives you quick, accurate results from radius and height. Always measure height as the perpendicular from apex to base, and keep units consistent: same unit for radius and height; volume in cubic units; surface area in square units.
Summary
This cone calculator finds the volume, surface area, and slant height of a right circular cone from the radius and height. Enter radius and height (positive numbers) and get volume = (1/3)πr²h, surface area = πr(r + slant), and slant height = √(r² + h²) instantly. The tool is free, runs in the browser, and requires no account. Use it for homework, design, or any conical shape. Remember: height is the perpendicular from apex to base. Keep units consistent for accurate results.