Cone Volume and Surface Area Calculator
Find the volume, total surface area, lateral surface area, or slant height of any right circular cone. Useful for engineering, packaging design, and 3D geometry coursework.
About this calculator
A right circular cone has a circular base of radius r and an apex directly above the center at height h. The slant height l connects the apex to the edge of the base: l = √(r² + h²). The volume enclosed by the cone is one-third of the base area times the height: V = (1/3) × π × r² × h. The lateral (curved) surface area wraps around the side: Lateral Area = π × r × l. The total surface area adds the circular base: Total Area = π × r² + π × r × l = π × r × (r + l). These four quantities are fully determined once you supply the radius and height.
How to use
Suppose a cone has a base radius of 3 units and a height of 4 units. First compute the slant height: l = √(3² + 4²) = √(9 + 16) = √25 = 5 units. Volume: V = (1/3) × π × 3² × 4 = (1/3) × π × 9 × 4 ≈ 37.70 cubic units. Lateral surface area: π × 3 × 5 ≈ 47.12 square units. Total surface area: π × 3² + π × 3 × 5 = 28.27 + 47.12 ≈ 75.40 square units. Enter radius = 3 and height = 4, choose your desired output, and the calculator returns each result instantly.
Frequently asked questions
How do you calculate the volume of a cone step by step?
The volume formula is V = (1/3) × π × r² × h, where r is the base radius and h is the perpendicular height. Start by squaring the radius and multiplying by π to get the base area. Then multiply by the height and finally divide by 3, because a cone occupies exactly one-third of the volume of a cylinder with the same base and height. For a cone with r = 5 and h = 9, V = (1/3) × π × 25 × 9 ≈ 235.62 cubic units.
What is the difference between lateral surface area and total surface area of a cone?
The lateral surface area covers only the curved side of the cone, calculated as π × r × l where l is the slant height. The total surface area includes the flat circular base as well: Total = π × r × l + π × r². Use lateral surface area when you need to know how much material wraps around the outside (e.g., the label on an ice cream cone), and total surface area when the base is also enclosed (e.g., a conical tank that needs painting inside and out).
Why is the slant height of a cone different from its vertical height?
The vertical height h is the straight-line distance from the apex directly down to the center of the base, measured perpendicularly. The slant height l is the distance from the apex to any point on the circumference of the base, measured along the surface. Because the base has a nonzero radius, the slant height is always longer than the vertical height, related by the Pythagorean theorem: l = √(r² + h²). Slant height matters when computing surface area, while vertical height matters for volume.