Definition:Right Circular Cone
Definition
A right circular cone is a cone:
- whose base is a circle
- in which there is a line perpendicular to the base through its center which passes through the apex of the cone:
- which is made by having a right-angled triangle turning along one of the sides that form the right angle.
In the words of Euclid:
- When, one side of those about the right angle in a right-angled triangle remaining fixed, the triangle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cone.
And, if the straight line which remains fixed be equal to the remaining side about the right angle which is carried round, the cone will be right-angled; if less, obtuse-angled; and if greater, acute-angled.
(The Elements: Book $\text{XI}$: Definition $18$)
Parts of Right Circular Cone
Axis
Let $K$ be a right circular cone.
Let point $A$ be the apex of $K$.
Let point $O$ be the center of the base of $K$.
Then the line $AO$ is the axis of $K$.
In the words of Euclid:
- The axis of the cone is the straight line which remains fixed and about which the triangle is turned.
(The Elements: Book $\text{XI}$: Definition $19$)
Base
Let $\triangle AOB$ be a right-angled triangle such that $\angle AOB$ is the right angle.
Let $K$ be the right circular cone formed by the rotation of $\triangle AOB$ around $OB$.
Let $BC$ be the circle described by $B$.
The base of $K$ is the plane surface enclosed by the circle $BC$.
In the words of Euclid:
- And the base is the circle described by the straight line which is carried round.
(The Elements: Book $\text{XI}$: Definition $20$)
Directrix
Let $K$ be a right circular cone.
Let $B$ be the base of $K$.
The circumference of $B$ is the directrix of $K$.
Generatrix
Let $K$ be a right circular cone.
Let $A$ be the apex of $K$.
Let $B$ be the base of $K$.
Then a line joining the apex of $K$ to its directrix is a generatrix of $K$.
Opening Angle
Let $K$ be a right circular cone.
Let point $A$ be the apex of $K$.
Let $B$ and $C$ be the endpoints of a diameter of the base of $K$.
Then the angle $\angle BAC$ is the opening angle of $K$.
In the above diagram, $\phi$ is the opening angle of the right circular cone depicted.
Types of Right Circular Cone
Acute-Angled
Let $K$ be a right circular cone.
Then $K$ is acute-angled if and only if the opening angle of $K$ is an acute angle.
Right-Angled
Let $K$ be a right circular cone.
Then $K$ is right-angled if and only if the opening angle of $K$ is a right angle.
Obtuse-Angled
Let $K$ be a right circular cone.
Then $K$ is obtuse-angled if and only if the opening angle of $K$ is an obtuse angle.
Similar Cones
Let $h_1$ and $h_2$ be the lengths of the axes of two right circular cones.
Let $d_1$ and $d_2$ be the lengths of the diameters of the bases of the two right circular cones.
Then the two right circular cones are similar if and only if:
- $\dfrac {h_1} {h_2} = \dfrac {d_1} {d_2}$
In the words of Euclid:
- Similar cones and cylinders are those in which the axes and the diameters of the bases are proportional.
(The Elements: Book $\text{XI}$: Definition $24$)
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: cone
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: right-circular