# Formulas and Tables

Elementary Geometry# Pyramid

Lateral edge: \(b\)

Height of a pyramid: \(h\)

Slant height: \(m\)

Number of sides in the base: \(n\)

Volume: \(V\)

Semiperimeter of the polygon in the base: \(p\)

Lateral surface area: \({S_L}\)

Area of the base: \({S_B}\)

Total surface area: \(S\)

- A pyramid whose base is a triangle is called a tetrahedron.
- The perpendicular drawn from the vertex of a pyramid to the base plane is called the height (altitude) of the pyramid. In a regular pyramid, the height is given by

\(h = {\large\frac{{\sqrt {4{b^2}{{\sin }^2}\frac{\pi }{n} – {a^2}} }}{{2\sin \frac{\pi }{n}}}\normalsize},\)

where \(b\) is the lateral edge, \(a\) is the base side, \(n\) is the number of sides of the polygon in the base. - The height of a lateral face is called the slant height. In a regular pyramid, the length of the slant height is expressed by the formula

\(m = \sqrt {{b^2} – {\large\frac{{{a^2}}}{4}\normalsize}} \) - Lateral surface area of a regular pyramid

\({S_L} = {\large\frac{1}{2}\normalsize} man =\) \({\large\frac{1}{4}\normalsize} an\sqrt {4{b^2} – {a^2}} \) \(= pm\) - Area of the base of a regular pyramid

\({S_B} = pr\),

where \(p\) is the semiperimeter of the polygon in the base, \(r\) is the inradius of the base. - Total surface area

\(S = {S_B} + {S_L}\) - Volume of a pyramid

\(V = {\large\frac{1}{3}\normalsize}{S_B}h\) - Volume of a regular pyramid

\(V = {\large\frac{1}{3}\normalsize} prh\)