volum
$$ V = \frac{1}{3} G \cdot h $$
overflateareal
$$ A = G + S $$
grunnflateareal
$$
\begin{aligned}
&G = \frac{1}{4} n a^2 \cot\frac{180^\circ}{n} \\ \\
& n = 3 \ \Rightarrow \ G = \frac{\sqrt{3}}{4} a^2 \\ \\
& n = 4 \ \Rightarrow \ G = a^2
\end{aligned}
$$
sideflateareal
$$ S = \frac{n a h_a}{2} $$
sidekant
$$
\begin{aligned}
s &= \frac{h}{\sin\alpha_1} \\ \\
s &= \sqrt{h^2 + R^2} \\ \\
s &= \sqrt{h_a^2 + \left(\frac{a}{2}\right)^2}
\end{aligned}
$$
høyde
$$
\begin{aligned}
h_a &= \frac{h}{\sin\alpha_2} \\ \\
h_a &= \sqrt{h^2 + r^2} \\ \\
h_a &= \sqrt{s^2 - \left(\frac{a}{2}\right)^2}
\end{aligned}
$$
omsirkel (radius)
$$
\begin{aligned}
&R = \frac{a}{2\cdot\sin\frac{180^\circ}{n}} \\ \\
& n = 3 \ \Rightarrow \ R = \frac{a}{\sqrt{3}} \\ \\
& n = 4 \ \Rightarrow \ R = \frac{a}{\sqrt{2}}
\end{aligned}
$$
innsirkel (radius)
$$
\begin{aligned}
&r = \frac{a}{2\cdot\tan\frac{180^\circ}{n}} \\ \\
& n = 3 \ \Rightarrow \ r = \frac{\sqrt{3}}{6}a \\ \\
& n = 4 \ \Rightarrow \ r = \frac{a}{2}
\end{aligned}
$$
