# Cone

A cone (or circular cone) is a three-dimensional solid. It consists of a circular base, a point (called the vertex), and all the points that lie on line segments connecting the vertex to the base. Thus, the cone is the special case of the pyramid in which the base is circular. $[asy] import three; currentprojection = perspective(0,-3,1); defaultpen(linewidth(0.7));triple vertex = (0,0,1.5); path3 rightanglemark(triple A, triple B, triple C, real s=8) { triple P,Q,R; P=s*markscalefactor*unit(A-B)+B; R=s*markscalefactor*unit(C-B)+B; Q=P+R-B; return P--Q--R; } path3 unitc=(1,0,0)..(0,1,0)..(-1,0,0)..(0,-1,0)..cycle; draw(unitc);dot(vertex);draw((1,0,0)--vertex--(-1,0,0));draw(vertex--(vertex.x,vertex.y,0)--(1,0,0));draw(rightanglemark(vertex,(vertex.x,vertex.y,0),(1,0,0),2)); label("h",(vertex.x,vertex.y,vertex.z/2),W);label("r",((1+vertex.x)/2,(vertex.y)/2,0),S); label("s",((1+vertex.x)/2,(1+vertex.y)/2,vertex.z/2),NE); [/asy]$ $[asy] import three; currentprojection = perspective(0,-3,1); defaultpen(linewidth(0.7));triple vertex = (0.4,0,1.5); path3 rightanglemark(triple A, triple B, triple C, real s=8) { triple P,Q,R; P=s*markscalefactor*unit(A-B)+B; R=s*markscalefactor*unit(C-B)+B; Q=P+R-B; return P--Q--R; } path3 unitc=(1,0,0)..(0,1,0)..(-1,0,0)..(0,-1,0)..cycle; draw(unitc);dot(vertex);draw((1,0,0)--vertex--(-1,0,0));draw(vertex--(vertex.x,vertex.y,0)--(1,0,0));draw(rightanglemark(vertex,(vertex.x,vertex.y,0),(1,0,0),2)); label("h",(vertex.x,vertex.y,vertex.z/2),W);label("r",((1+vertex.x)/2,(vertex.y)/2,0),S); label("s",((1+vertex.x)/2,(1+vertex.y)/2,vertex.z/2),ENE); [/asy]$

## Important terminology

The distance from the vertex to the plane containing the base is the height of the cone, and is frequently denoted $h$. The radius of the base is called the radius of the cone and is frequently denoted $r$. If the vertex lies directly above the center of the base, we call the cone a right circular cone (or right cone for short). In this case, the vertex is the same distance from every point on the boundary of the base; this distance is called the slant height of the cone, and is sometimes denoted $s$ or $\ell$. If a cone is not a right cone (that is, if the vertex is not directly above the center of the base), we call it an oblique cone.

## Properties

• A cone with radius $r$ and height $h$ has volume $V = \frac{1}{3} \cdot \pi r^2 \cdot h$. This is a special case of the general formula for the volume of a pyramid, $V = \frac{1}{3} \cdot B \cdot h$, where $V$ is the volume, $B$ is the area of the base and $h$ is the height.
• A right cone of radius $r$ and slant-height $s$ has surface area $\pi r^2 + \pi r s$ (the lateral area is $\pi rs$, and the area of the base is $\pi r^2$).