objem
$$ V = \frac{1}{3} S_p \cdot v $$
povrch
$$ S = S_p + S_{pl} $$
obsah podstavy
$$
\begin{aligned}
&S_p = \frac{1}{4} n a^2 \cot\frac{180^\circ}{n} \\ \\
& n = 3 \ \Rightarrow \ S_p = \frac{\sqrt{3}}{4} a^2 \\ \\
& n = 4 \ \Rightarrow \ S_p = a^2
\end{aligned}
$$
obsah pláště
$$ S_{pl} = \frac{na v_a}{2} $$
délka boční hrany
$$
\begin{aligned}
s &= \frac{v}{\sin\alpha_1} \\ \\
s &= \sqrt{v^2 + r_o^2} \\ \\
s &= \sqrt{v_a^2 + \left(\frac{a}{2}\right)^2}
\end{aligned}
$$
výška
$$
\begin{aligned}
v_a &= \frac{v}{\sin\alpha_2} \\ \\
v_a &= \sqrt{v^2 + r_v^2} \\ \\
v_a &= \sqrt{s^2 - \left(\frac{a}{2}\right)^2}
\end{aligned}
$$
kružnice opsaná (poloměr)
$$
\begin{aligned}
&r_o = \frac{a}{2\cdot\sin\frac{180^\circ}{n}} \\ \\
& n = 3 \ \Rightarrow \ r_o = \frac{a}{\sqrt{3}} \\ \\
& n = 4 \ \Rightarrow \ r_o = \frac{a}{\sqrt{2}}
\end{aligned}
$$
kružnice vepsaná (poloměr)
$$
\begin{aligned}
&r_v = \frac{a}{2\cdot\tan\frac{180^\circ}{n}} \\ \\
& n = 3 \ \Rightarrow \ r_v = \frac{\sqrt{3}}{6}a \\ \\
& n = 4 \ \Rightarrow \ r_v = \frac{a}{2}
\end{aligned}
$$
