Cone Calculator
Compute a cone's volume, lateral surface area, total surface area, and slant height from its base radius and height. Useful for geometry homework, 3D printing projects, or engineering design tasks involving conical shapes.
About this calculator
A cone is a three-dimensional solid with a circular base tapering to a single apex. Its volume is found with V = (1/3) × π × r² × h, where r is the base radius and h is the perpendicular height. The slant height l = √(r² + h²) connects the base edge to the apex along the surface. The lateral (side) surface area is A_lateral = π × r × l, while the total surface area adds the circular base: A_total = π × r × l + π × r². These formulas apply to right circular cones. Knowing just the radius and height is enough to derive every other property of the cone.
How to use
Suppose a cone has a base radius of 4 cm and a height of 9 cm. First, compute slant height: l = √(4² + 9²) = √(16 + 81) = √97 ≈ 9.849 cm. Volume: V = (1/3) × π × 4² × 9 = (1/3) × π × 144 ≈ 150.80 cm³. Lateral surface area: π × 4 × 9.849 ≈ 123.82 cm². Total surface area: 123.82 + π × 4² ≈ 123.82 + 50.27 ≈ 174.09 cm². Enter r = 4 and h = 9 to confirm all results instantly.
Frequently asked questions
How do you find the volume of a cone with radius and height?
The formula is V = (1/3) × π × r² × h, where r is the base radius and h is the vertical height. The factor of 1/3 distinguishes a cone from a cylinder with the same base and height — a cone holds exactly one-third as much. Simply square the radius, multiply by π and the height, then divide by 3. For example, r = 5 cm and h = 12 cm gives V = (1/3) × π × 25 × 12 ≈ 314.16 cm³.
What is the difference between lateral surface area and total surface area of a cone?
Lateral surface area covers only the curved sloping side of the cone: A_lateral = π × r × l, where l is the slant height. Total surface area adds the flat circular base: A_total = π × r × l + π × r². If you were painting just the outside of an ice-cream cone shell, you'd use lateral area; if you needed to wrap the entire solid including the bottom, you'd use total surface area. The distinction matters in packaging, manufacturing, and construction applications.
How is slant height different from the vertical height of a cone?
Vertical height h is the perpendicular distance from the base center straight up to the apex. Slant height l is the distance measured along the surface from the base edge to the apex. They are related by the Pythagorean theorem: l = √(r² + h²). Slant height is always longer than vertical height (unless the radius is zero). It appears directly in surface-area formulas because it represents the actual length of the cone's side.