Definition:Right Circular Cone

From ProofWiki
Jump to navigation Jump to search


This page has been identified as a candidate for refactoring of medium complexity.
In particular: Establish the equivalence between a) the definition as a solid of revolution of a right-angled triangle and b) the definition as a cone whose base is a circle and whose apex is on a perpendicular to the center of the base. As it stands, the two definitions are munged together.
Until this has been finished, please leave {{Refactor}} in the code.

New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only.

Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Refactor}} from the code.


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.
RightCircularCone.png


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 $24$)


Also see


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Right_Circular_Cone&oldid=570740"