Matemaatika:Gümnaasium/Kordamine eksamiks/Geomeetria ruumis

Valemid

Risttahukas

Risttahukas

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

$\begin{array}{c}V=abc\end{array}$

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

Kuup

Kuup

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

$\begin{array}{c}V=a^3\end{array}$

Püst- ja kaldprisma

Prisma

Silinder

$\begin{array}{c}S=S_k+2Sp=2\pi r(h+r)\end{array}$

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

Koonus

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

$\begin{array}{c}S=S_k+S_p=\pi r(m+r)\end{array}$

$\begin{array}{c}V=\frac{1}{3}{S}_{p}h=\frac{1}{3}\pi r^{2}h\end{array}$

Sfäär ja kera

Kera

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

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

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.