# Definition:Right Circular Cone

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## 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.

### Generating Angle

Let $K$ be a right circular cone.

Let point $A$ be the apex of $K$.

Let $O$ be the center of the base of $K$.

Let $B$ be the endpoint of a diameter of the base of $K$.

Then the angle $\angle OAB$ is the **generating angle** of $K$.

## 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 $24$)

## Also see

## Sources

- 1933: D.M.Y. Sommerville:
*Analytical Conics*(3rd ed.) ... (previous) ... (next): Chapter $\text {IV}$. The Ellipse: $1$. - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**cone** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**right-circular**