A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex.
More precisely, it is the solid figure bounded by a base in a plane and by a surface (called the lateral surface) formed by the locus of all straight line segments joining the apex to the perimeter of the base, such that there is a circular cross section. The term "cone" sometimes refers just to the surface of this solid figure, or just to the lateral surface.
The axis of a cone is the straight line (if any), passing through the apex, about which the base has a rotational symmetry.
In common usage in elementary geometry, cones are assumed to be right circular, where right means that the axis passes through the centre of the base (suitably defined) at right angles to its plane, and circular means that the base is a circle. Contrasted with right cones are oblique cones, in which the axis does not pass perpendicularly through the centre of the base. In general, however, the base may be any shape that permits a circular cross section of the cone, and the apex may lie anywhere (though it is usually assumed that the base is bounded and therefore has finite area, and that the apex lies outside the plane of the base).
The more general conic solid also has an an apex and lines connecting the apex to all the points on a planar base, which can be of any shape. If the result has a circular cross section, it is a cone; if it has a polygonal base, it is a pyramid.
Surface areaThe lateral surface area of a right circular cone is where is the radius of the circle at the bottom of the cone and is the lateral height of the cone (given by the Pythagorean theorem where is the height of the cone). The surface area of the bottom circle of a cone is the same as for any circle, . Thus the total surface area of a right circular cone is:
VolumeThe volume of any conic solid is one third of the product of the area of the base and the height (the perpendicular distance from the base to the apex).
Center of massThe center of mass of a conic solid of uniform density lies one-quarter of the way from the center of the base to the vertex, on the straight line joining the two.
Right circular coneFor a circular cone with radius R and height H, the formula for volume becomes
- where is the slant height.
A right circular cone with height and aperture , whose axis is the coordinate axis and whose apex is the origin, is described parametrically as
In implicit form, the same solid is defined by the inequalities