térfogat
$$ V = \frac{1}{3} T_a \cdot m $$
felszín
$$ F = T_a + T_{p} $$
alapterület
$$
\begin{aligned}
&T_a = \frac{1}{4} n a^2 \cot\frac{180^\circ}{n} \\ \\
& n = 3 \ \Rightarrow \ T_a = \frac{\sqrt{3}}{4} a^2 \\ \\
& n = 4 \ \Rightarrow \ T_a = a^2
\end{aligned}
$$
palást terület
$$ T_{p} = \frac{n a m_a}{2} $$
oldalél
$$
\begin{aligned}
s &= \frac{m}{\sin\alpha_1} \\ \\
s &= \sqrt{m^2 + R^2} \\ \\
s &= \sqrt{m_a^2 + \left(\frac{a}{2}\right)^2}
\end{aligned}
$$
magasság az a oldalra
$$
\begin{aligned}
m_a &= \frac{m}{\sin\alpha_2} \\ \\
m_a &= \sqrt{m^2 + r^2} \\ \\
m_a &= \sqrt{s^2 - \left(\frac{a}{2}\right)^2}
\end{aligned}
$$
köréírt kör (sugár)
$$
\begin{aligned}
&R = \frac{a}{2\cdot\sin\frac{180^\circ}{n}} \\ \\
& n = 3 \ \Rightarrow \ R = \frac{a}{\sqrt{3}} \\ \\
& n = 4 \ \Rightarrow \ R = \frac{a}{\sqrt{2}}
\end{aligned}
$$
beírt kör (sugár)
$$
\begin{aligned}
&r = \frac{a}{2\cdot\tan\frac{180^\circ}{n}} \\ \\
& n = 3 \ \Rightarrow \ r = \frac{\sqrt{3}}{6}a \\ \\
& n = 4 \ \Rightarrow \ r = \frac{a}{2}
\end{aligned}
$$
