# Formulas

## Elementary Geometry # Cone

Radius of the base of a circular cone: $$R$$
Generatrix of a cone: $$m$$
Height of cone: $$H$$
Volume: $$V$$
Area of the base: $${S_B}$$
Lateral surface area: $${S_L}$$
Total surface area: $$S$$
1. A cone or a conical surface is a three-dimensional shape formed by the movement of a straight line (called the generatrix) that passes through a fixed point (the vertex of the cone) and crosses a given curve called the directrix. The cone is often defined as a three-dimensional shape bounded by the interior of a plane crossing the conical surface and the portion of the conical surface between the vertex and the boundary of crossing. The portion of the plane lying inside the conical surface is called the base of the cone and the portion of the conical surface is called the lateral surface.
2. A cone is called a circular cone if its base is a circle.
3. A cone is called a right circular cone if the line from the vertex of the cone to the centre of its base is perpendicular to the base.
4. A right circular cone is formed by rotating a right triangle about its leg. A right circular cone is determined by the radius of the base $$R$$ and height $$H$$ (or similarly, by the radius of the base $$R$$ and generatrix $$m$$).
5. Relationship between the height, radius of the base and generatrix (slant height) of a right circular cone
$$H = \sqrt {{m^2} – {R^2}}$$
6. Lateral surface area of a right circular cone
$${S_L} = \pi Rm$$
7. Base area of a right circular cone
$${S_B} = \pi {R^2}$$
8. Total surface area of a right circular cone
$$S = {S_L} + {S_B}$$ $$= \pi R\left( {m + R} \right)$$
9. Volume of a right circular cone
$$V = {\large\frac{{{S_B}H}}{3}\normalsize}$$ $$= {\large\frac{{\pi {R^2}H}}{3}\normalsize}$$