Matemaatika:Gümnaasium/Kordamine eksamiks/Geomeetria ruumis

Valemid

Risttahukas

Risttahukas

${\begin{matrix}S=2(ab+ac+bc)\end{matrix}}$

${\begin{matrix}V=abc\end{matrix}}$

${\begin{matrix}d={\sqrt {a^{2}+b^{2}+c^{2}}}\end{matrix}}$

Kuup

Kuup

${\begin{matrix}S=6a^{2}\end{matrix}}$

${\begin{matrix}V=a^{3}\end{matrix}}$

Püst- ja kaldprisma

Prisma

Silinder

${\begin{matrix}S=S_{k}+2Sp=2\pi r(h+r)\end{matrix}}$

${\begin{matrix}V=S_{p}h=\pi r^{2}h\end{matrix}}$

Koonus

${\begin{matrix}m^{2}=r^{2}+h^{2}\end{matrix}}$

${\begin{matrix}S=S_{k}+S_{p}=\pi r(m+r)\end{matrix}}$

${\begin{matrix}V={\frac {1}{3}}S_{p}h={\frac {1}{3}}\pi r^{2}h\end{matrix}}$

Sfäär ja kera

Kera

${\begin{matrix}S=4\pi R^{2}\end{matrix}}$

${\begin{matrix}V={\frac {4}{3}}\pi R^{3}\end{matrix}}$

Näited

• Sfääri sisse on moodustatud maksimaalne kuup. Leia kera ruumala, kui kuubi serv a=2 cm.

$R={\frac {a}{2}}{\sqrt {3}}$

$V={\frac {4}{3}}\pi R^{3}\Rightarrow V={\frac {4}{3}}\pi {{\frac {a}{2}}{\sqrt {3}}}^{3}\Rightarrow V=2{\sqrt {3}}\pi a$

$V=2{\sqrt {3}}\pi 2\Rightarrow V=4{\sqrt {3}}\pi$

Vastus: Ruumala pindala on $4{\sqrt {3}}\pi$ cm.