# Cylinder

Radius of the base of a cylinder: $$R$$
Generatrix of a cylinder: $$L$$
Height of a cylinder: $$H$$
Heights of a truncated cylinder: $${h_1},$$ $${h_2}$$
Area of the base: $${S_B}$$
Lateral surface area: $${S_L}$$
Total surface area: $$S$$
Volume: $$V$$
1. A cylinder is a geometric solid bounded by a cylindrical surface and two parallel planes crossing it. The cylindrical surface is formed by a straight line (called the generatrix) moving parallel to itself, so that any fixed point of the line moves along a plane curve called the directrix.
2. A cylinder is called a circular cylinder if its directrix is a circle.
3. A cylinder is called a right cylinder if it generatrix is perpendicular to the bases.
4. A right circular cylinder is determined by the radius of the base $$R$$ and the generatrix $$L,$$ which is equal to the height $$H$$ of the cylinder.
5. Lateral surface area of a right circular cylinder
$${S_B} = 2\pi RH$$
6. Total surface area of a right circular cylinder
$$S = {S_L} + 2{S_B}$$ $$= 2\pi R\left( {H + R} \right)$$
7. Volume of a right circular cylinder
$$V = {S_B}H$$ $$= \pi {R^2}H$$
8. A truncated right circular cylinder or briefly a truncated cylinder is determined by the radius of the base $$R,$$ the shortest height $${h_1}$$ and the greatest height $${h_2}$$.
9. Lateral surface area of a truncated cylinder
$${S_L} = \pi R\left( {{h_1} + {h_2}} \right)$$
10. Area of the bases of a truncated cylinder
$${S_B} = \pi {R^2}$$ $$+\;\pi R\sqrt {{R^2} + {{\left( {{\large\frac{{{h_1} – {h_2}}}{2}}\normalsize} \right)}^2}}$$
11. Total surface area of a truncated cylinder
$$S = {S_L} + {S_B} =$$ $$\pi R\Big[ {{h_1} + {h_2} + R }$$ $$+\;{ \sqrt {{R^2} + {{\left( {{\large\frac{{{h_1} – {h_2}}}{2}}\normalsize} \right)}^2}} } \Big]$$
12. Volume of a truncated cylinder
$$V = {\large\frac{{\pi {R^2}\left( {{h_1} + {h_2}} \right)}}{2}\normalsize}$$