# Formulas

## Elementary Geometry # Pyramid

Side of the base: $$a$$
Lateral edge: $$b$$
Height of a pyramid: $$h$$
Slant height: $$m$$
Number of sides in the base: $$n$$
Volume: $$V$$
Radius of the inscribed circle in the base: $$r$$
Semiperimeter of the polygon in the base: $$p$$
Lateral surface area: $${S_L}$$
Area of the base: $${S_B}$$
Total surface area: $$S$$
1. A pyramid is a polyhedron whose base is a polygon and other faces are triangles with a common vertex (also called the apex).
2. A pyramid whose base is a triangle is called a tetrahedron.
3. 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.
4. 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
$$m = \sqrt {{b^2} – {\large\frac{{{a^2}}}{4}\normalsize}}$$
5. Lateral surface area of a regular pyramid
$${S_L} = {\large\frac{1}{2}\normalsize} man =$$ $${\large\frac{1}{4}\normalsize} an\sqrt {4{b^2} – {a^2}}$$ $$= pm$$
6. 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.
7. Total surface area
$$S = {S_B} + {S_L}$$
8. Volume of a pyramid
$$V = {\large\frac{1}{3}\normalsize}{S_B}h$$
9. Volume of a regular pyramid
$$V = {\large\frac{1}{3}\normalsize} prh$$