Cone (EntityTopic, 11)

From Hi.gher. Space

A cone is a special case of a pyramid where the base is a circle.

The cone is one of the few curved polyhedra that satisfy Euler's F + V = E + 2.

Equations

• Variables:

r ⇒ radius of base of cone
h ⇒ perpendicular height of cone

• All points (x, y, z) that lie on the surface of a cone will satisfy the following equations:

Unknown

• All points (x, y, z) that lie on the edges of a cone will satisfy the following equations:

x² + y² = r²
z = 0

• The hypervolumes of a cone are given by:

total edge length = 2πr
surface area = πr(r + √(r² + h²))
volume = ^π∕₃ · r²h

• The planar cross-sections (n) of a cone are:

[!x,!y] ⇒ isosceles triangle of base length 2r and perpendicular height h
[!z] ⇒ circle of radius (r − ^nr∕_h)

Arrinder

An arrinder is the surface of revolution of an arrow, just as a cone is the surface of revolution of a triangle. It can also be thought of as a cone with a smaller cone removed from the base. As such, this shape's volume is the difference between the volume of the two aforementioned cones.