Pyramid

Definition

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. It is a conic solid with polygonal base. A pyramid with an n-sided base has n + 1 vertices, n + 1 faces, and 2n edges.

Properties

• The base of a pyramid is a polygon with at least three edges.
• The number of edges on the base determines how many side surfaces the pyramid has.
• The sides of a pyramid are triangular. They run inwards from the base surfaces and meet at the tip.
• The center of gravity of the pyramid divides the distance between the center of the base and the top in a ratio of 1:3.

Formulas

The following formulas refer to the calculation of a straight square pyramid

Side length of the base

\(\displaystyle a=\sqrt{\frac{P}{4}}\)

Radius to the straight lines

\(\displaystyle r_s=\sqrt{\frac{A}{2}}\)

Radius to a corner

\(\displaystyle r_v=\sqrt{(a/2)^2+{r_s}^2}\)

Perimeter of the base

\(\displaystyle P=4·a\)

Base area

\(\displaystyle A=a^2\)

Height

\(\displaystyle h=\frac{3·V}{A} \) \(\displaystyle \ \ \ =\sqrt{m^2-{r_s}^2}\)

Lateral height

\(\displaystyle m=\sqrt{h^2+{r_s}^2}\)

Edge length

\(\displaystyle k=\sqrt{m^2+(a^2/4)}\)

Area of one side

\(\displaystyle M_1=\frac{m · a}{2}\)

Lateral surface without base

\(\displaystyle M=\frac{m · P}{2}\)

Volume

\(\displaystyle V=\frac{A · h}{3}\)

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