terminology
$s$- slant height (lateral side length) - Mantellinie$A_{\text{G}}$- base area - Grundfläche$h$- height - Höhe (the height measured perpendicular to the base area)$A_{M}$- lateral (surface) area - Mantelfläche$A_{O}$- total surface area - Oberfläche- apex (tip) - Spitze
calculating the length of the slant height (Mantellinie)
$$s^2 = h^2 + r^2$$
$$s = \sqrt{h^2 + r^2}$$
base area - Grundfläche
$$A_{\text{base}} = A_G = \pi \cdot r^2$$
lateral surface - Mantelfläche
To find the surface area we need to first look at the arc length of the sector of a circle with radius equal to the slant height s.
The ratio of the sector area to the entire area of the circle (radius s) is proportional to the arc length l and the entire circumference of the circle (radius s)
$$\frac{A_{\text{sector}}}{A_{\text{circle}}} = \frac{l}{2\pi s}$$
We also know that the arc length l is equal to the circumference of the base (radius r).
So,
$$ \frac{2\pi r}{2\pi s} = \frac{r}{s}$$
The area of the sector is the fraction $\frac{r}{s}$ of the whole circle with area $2\pi s^2$.
Therefore,
$$A_{\text{lateral}} = A_M = \frac{r}{s} \cdot \pi \cdot s^2 = \pi \cdot r \cdot s$$
total surface area - Oberfläche
$$A_{\text{total}} = A_O = \pi \cdot r^2 + \pi \cdot r \cdot s = \pi \cdot r (r + s)$$
volume - Volumen
$$V_{\text{cone}} = \frac{1}{3} \cdot A_{\text{G}} \cdot h = \frac{1}{3} \cdot \pi \cdot r^2 \cdot h$$