Pyramid
Properties and formulas for calculating a 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 nsided 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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