Right Cone Calculator

A right circular cone has a circular base and a pointed apex directly above the center. The radius r is the base circle radius, and h is the perpendicular height from base to apex.

The slant height l is the straight-line distance from any point on the base edge up to the apex. By the Pythagorean theorem:

Slant height l = √(r² + h²)

Volume is one-third of the cylinder that would enclose the same base and height. That one-third holds for every cone and every pyramid, whatever the base shape:

Volume = (1/3) × π × r² × h

The lateral surface area is the area of the curved side only. If you unroll the cone into a flat shape, you get a sector of a circle with radius l and arc length 2πr:

Lateral SA = π × r × l Total SA = π × r × (r + l)

That unrolled sector is the pattern you cut if you are making a cone out of sheet metal, card or paper. Its radius is the slant height, and the angle you need is:

Sector angle = 360° × r / l

A quick sanity check: lateral SA is always larger than the base area (π × r²), because l is always longer than r for any cone with height above zero. If your lateral area comes out smaller, you have entered the slant height where the calculator wanted the vertical height.

Real-world cones you might want to measure: ice cream cones (radius about 2.5 cm, height about 12 cm), traffic cones (radius about 15 cm, height 70 cm), party hats, funnels, and conical flasks. Run the traffic cone through and you get 16,493 cm³, about 16.5 litres, which is the volume it would displace if it were solid rather than the hollow shell it really is.

The formulas assume a perfect right cone. If the apex is off-center (oblique cone), you need calculus to get exact values.