Cone Calculator

What is slant height and how is it different from height?

Height h is the perpendicular distance from the base to the apex — measured straight up. Slant height l is the distance along the slanted surface from the base edge to the apex. They are the two legs and hypotenuse of a right triangle: l = √(r² + h²).

How do you "unroll" a cone's lateral surface?

When you cut along a slant line and flatten the lateral surface, it becomes a sector of a circle with radius equal to the slant height l and arc length equal to the base circumference 2πr. The sector angle is 2πr/l radians. The sector's area is (1/2) × l × 2πr = πrl — the lateral surface area formula.

How much bigger is a cylinder than a cone with the same base and height?

Exactly three times bigger. V_cylinder = πr²h, V_cone = (1/3)πr²h. This relationship was proved by Archimedes and can be verified physically: filling a cone three times and pouring into a cylinder of the same dimensions will fill it exactly.

Where do cones appear in real life?

Traffic cones, ice cream cones, funnels, party hats, rocket nose cones, megaphones, volcanic mountains (perfect cones), drill bits, and stalactites. In engineering, conical transitions smoothly redirect fluid flow between pipes of different diameters.