Cone Calculator
Compute the volume, lateral surface area, total surface area, or slant height of a right circular cone from its base radius and height. Ideal for geometry students, engineers, and anyone working with conical shapes.
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 is found with the Pythagorean theorem: l = √(r² + h²). Volume is one-third the base area times the height: V = (1/3)πr²h. The lateral (curved) surface area is the base circumference times the slant spread: A_lateral = πrl. Adding the circular base gives the total surface area: A_total = πrl + πr². These formulas apply to any right cone and are fundamental in packaging design, architecture, and manufacturing contexts where conical geometry appears.
How to use
Suppose a cone has base radius r = 4 cm and height h = 3 cm. First compute slant height: l = √(4² + 3²) = √(16 + 9) = √25 = 5 cm. Volume: V = (1/3) × π × 4² × 3 = (1/3) × π × 48 ≈ 50.27 cm³. Lateral surface area: A = π × 4 × 5 ≈ 62.83 cm². Total surface area (include base): A = π × 4 × 5 + π × 4² = 62.83 + 50.27 ≈ 113.10 cm². Enter r = 4, h = 3, choose your desired output, and the calculator returns the result instantly.
Frequently asked questions
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 πrl where r is the base radius and l is the slant height. The total surface area adds the flat circular base (πr²) to the lateral area, giving πrl + πr². Use lateral area when the base is open or irrelevant, such as calculating the material for a funnel or party hat.
How do you find the slant height of a cone if only the radius and height are given?
The slant height l is the straight-line distance from the apex of the cone to any point on the edge of the base circle. Because the radius, height, and slant height form a right triangle, you can apply the Pythagorean theorem: l = √(r² + h²). For example, r = 6 and h = 8 gives l = √(36 + 64) = √100 = 10. This value is required for all surface area calculations.
Why is the volume of a cone one-third of a cylinder with the same base and height?
A classic result in geometry shows that exactly three congruent cones fill one cylinder of the same base radius and height, which is why V_cone = (1/3)πr²h. This can be proven rigorously using Cavalieri's principle or integral calculus by summing infinitely thin circular cross-sections from apex to base. Intuitively, the cone tapers uniformly, so it occupies a third of the space that a full cylinder would.