volume
$$ V = \frac{1}{3} A_b \cdot h $$
surface area
$$ A = A_b + A_{l} $$
base area
$$
\begin{aligned}
&A_b = \frac{1}{4} n a^2 \cot\frac{180^\circ}{n} \\ \\
& n = 3 \ \Rightarrow \ A_b = \frac{\sqrt{3}}{4} a^2 \\ \\
& n = 4 \ \Rightarrow \ A_b = a^2
\end{aligned}
$$
lateral area
$$ A_{l} = \frac{n a s}{2} $$
lateral edge
$$
\begin{aligned}
s &= \frac{h}{\sin\alpha_1} \\ \\
s &= \sqrt{h^2 + r_o^2} \\ \\
s &= \sqrt{s^2 + \left(\frac{a}{2}\right)^2}
\end{aligned}
$$
slant height
$$
\begin{aligned}
s &= \frac{h}{\sin\alpha_2} \\ \\
s &= \sqrt{h^2 + r_v^2} \\ \\
s &= \sqrt{s^2 - \left(\frac{a}{2}\right)^2}
\end{aligned}
$$
circumcircle (radius)
$$
\begin{aligned}
&r_o = \frac{a}{2\cdot\sin\frac{180^\circ}{n}} \\ \\
& n = 3 \ \Rightarrow \ r_o = \frac{a}{\sqrt{3}} \\ \\
& n = 4 \ \Rightarrow \ r_o = \frac{a}{\sqrt{2}}
\end{aligned}
$$
incircle (radius)
$$
\begin{aligned}
&r_v = \frac{a}{2\cdot\tan\frac{180^\circ}{n}} \\ \\
& n = 3 \ \Rightarrow \ r_v = \frac{\sqrt{3}}{6}a \\ \\
& n = 4 \ \Rightarrow \ r_v = \frac{a}{2}
\end{aligned}
$$
