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z)(klJUvta;x2Tc?+}}!?WIIv*^CR#UPDxgFf4>^*1*qL*CeJ$VzePse;Bz$G<3n%}{`ad*!hoGO zEbw*&`@cn{GGYrud$7d!|NB27|JPan8UFw4tpDt~{~u=7U&`^PAya$%=g;B5pS-lP KRJnvv(EkH^L+3OA literal 0 HcmV?d00001 diff --git a/index.md b/index.md index 0c01b73..181eac3 100644 --- a/index.md +++ b/index.md @@ -342,3 +342,30 @@ La sezione di sfera sul piano $\alpha'$ individua un cerchio di raggio $HB$. L'area del cerchio è quindi: $A=\pi(r^2-h^2)$ Il cerchio e la corona circolare sono equivalenti quindi per il principio di Cavalieri: $\mathscr{S} \doteq \mathscr{A}$ + +### Scodella di Galileo +![Scodella di galileo](images/27.png) + +Si dice scodella di Galileo il solido ottenuto dalla differenza fra un cilindro di altezza uguale al raggio di base circoscritto ad una semisfera di centro $O$ e la sfera stessa. + +**Teorema** +![Scodella di Galileo sezioneta](images/28.png) + +La scodella di Galileo $\mathscr{S}$ è equivalente al cono $\mathscr{C}$ di vertice $O$ e di base congruente alla base della scodella. + +**Dimostrazione** +Sia $\alpha$ un qualunque piano parallelo al piano della base $\pi$ che intersechi $\mathscr{S}$ e $\mathscr{C}$ formando un cerchio $S$ e una corona circolare $S_1$. + +Siano $A$ e $B$ gli estremi del diametro della base della scodella su $\pi$. +Indicato con $\delta$ il piano per $A,B,O$, $\delta$ interseca $\alpha$ in $C$ e $N$. +I punti in comune tra $\alpha, \delta$ e la semisfera sono $D$ e $M$ e quelli tra $\alpha, \delta$ e il cilindro sono $E$ e $Z$. +Infine sia $H$ il piede della perpendicolare $\mathscr{t}$ condotta da $O$ ad $\alpha$ e K l'intersezione tra $\mathscr{t}$ e $\pi$. + +$OK \cong KB$ per definizione $\Rightarrow$ $\triangle KOB$ è rettangolo isoscele in $K$. +$\triangle OHN \sim \triangle KOB$ in quanto hanno $\widehat{O}$ in comune e $HN \parallel KB \Rightarrow \triangle OHN$ è isoscele $\Rightarrow \overline{OH}=\overline{OK}$. +Posto $h = \overline{OH}$, l'area del cerchio risulta $A_S=\pi h^2$ + +Considero $\triangle OHM$ rettangolo: per il teorema di Pitagora: $\overline{HM}=\sqrt{\overline{OM}^2-\overline{HM}^2} = \sqrt{r^2-h^2}$ (l'ipotenusa è il raggio della semisfera) +L'area della corona circolare è pertanto: $A_{S_1}=\pi\left(\overline{HZ}^2-\overline{HM}^2\right) = \pi [r^2 - (r^2 -h^2)] = \pi h^2$ + +$S \equiv S_1$ per il principio di Cavalieri: $\mathscr{S} \doteq \mathscr{C}$