Enter radius and height to calculate volume, surface area, and slant height.
A cone is a three-dimensional solid with a circular base that tapers smoothly to a point called the apex. Common examples include ice cream cones, traffic cones, and party hats. A right circular cone has its apex directly above the center of the base.
The slant height is the distance from the apex to any point on the edge of the base. Using the Pythagorean theorem:
l = sqrt(r^2 + h^2)
The volume of a cone is exactly 1/3 of a cylinder with the same base and height:
V = (1/3) * Pi * r^2 * h
The area of the cone's slanted surface:
A_lateral = Pi * r * l
Including the circular base:
A_total = Pi * r * l + Pi * r^2
= Pi * r * (l + r)
A_base = Pi * r^2
The (1/3) factor comes from calculus. When you integrate the areas of circular cross-sections from base to apex, the result is exactly 1/3 of a cylinder's volume. This is true for any cone, regardless of whether the apex is directly above the center.
The slant height (l) forms the hypotenuse of a right triangle with the radius (r) and height (h) as the two legs. This relationship is important for calculating surface area.
Given radius = 3 units and height = 4 units:
Slant Height = sqrt(3^2 + 4^2)
= sqrt(9 + 16)
= sqrt(25)
= 5 units
Volume = (1/3) * Pi * 3^2 * 4
= (1/3) * Pi * 9 * 4
= 12 * Pi
= 37.7 cubic units
Lateral Area = Pi * 3 * 5
= 15 * Pi
= 47.12 square units
Total Surface Area = Pi * 3 * (5 + 3)
= 24 * Pi
= 75.4 square units
A party hat has radius 5 cm and height 15 cm:
Slant Height = sqrt(5^2 + 15^2)
= sqrt(25 + 225)
= sqrt(250)
= 15.81 cm
Material needed (lateral area) = Pi * 5 * 15.81
= 248.5 square cm
Designing funnels, conical containers, and mechanical parts.
Roof designs, traffic cones, and architectural elements.
Ice cream cones, waffle cones, and conical packaging.