In geometria solida il tronco di cono è un cono al quale è stata tagliata la punta con un piano parallelo alla base. Qualora il piano non sia parallelo alla base, la sezione ottenuta è un'ellisse anziché un cerchio.
Sia T {\displaystyle T} un tronco di cono d'altezza h {\displaystyle h} e le cui basi hanno raggi R {\displaystyle R} e r {\displaystyle r} . Il volume del tronco è pari a
La superficie laterale S l {\displaystyle S_{l}} del tronco di cono è data dalla formula
dove a {\displaystyle a} è l'apotema, la lunghezza del lato obliquo del tronco di cono, pari a
La superficie totale del cono è data dalla formula:
oppure
È dato un tronco di cono T in cui R sia il raggio della base maggiore, r quello della minore e h l'altezza.
Si prolunghi la superficie laterale dalla parte di r fino ad ottenere il cono V1 di base in R e altezza pari a h + h2, in cui h2 è l'altezza del cono V2 con base in r. Il volume del tronco è quindi:
V T = V 1 − − --> V 2 {\displaystyle V_{T}=V_{1}-V_{2}}
I triangoli di lati r e h2 e di lati h e R-r sono simili, poiché hanno tutti gli angoli uguali. Pertanto possiamo scrivere:
h : ( R − − --> r ) = h 2 : r {\displaystyle h:(R-r)=h_{2}:r}
Per cui: h 2 = h r R − − --> r {\displaystyle h_{2}={\frac {hr}{R-r}}}
Partendo dalla formula del volume del cono:
V 1 = π π --> R 2 ( h + h 2 ) 3 {\displaystyle V_{1}={\frac {\pi R^{2}(h+h_{2})}{3}}}
V 2 = π π --> r 2 h 2 3 {\displaystyle V_{2}={\frac {\pi r^{2}h_{2}}{3}}}
Sostituendo in h2:
V 1 = π π --> R 2 h 3 + π π --> R 2 h r 3 ( R − − --> r ) {\displaystyle V_{1}={\frac {\pi R^{2}h}{3}}+{\frac {\pi R^{2}hr}{3(R-r)}}}
V 2 = π π --> r 2 h r 3 ( R − − --> r ) {\displaystyle V_{2}={\frac {\pi r^{2}hr}{3(R-r)}}}
Tornando alla formula iniziale:
V T = π π --> R 2 h 3 + π π --> R 2 h r 3 ( R − − --> r ) − − --> π π --> r 2 h r 3 ( R − − --> r ) {\displaystyle V_{T}={\frac {\pi R^{2}h}{3}}+{\frac {\pi R^{2}hr}{3(R-r)}}-{\frac {\pi r^{2}hr}{3(R-r)}}}
V T = π π --> h 3 ( R 2 + R 2 r R − − --> r − − --> r 3 R − − --> r ) {\displaystyle V_{T}={\frac {\pi h}{3}}(R^{2}+{\frac {R^{2}r}{R-r}}-{\frac {r^{3}}{R-r}})}
V T = π π --> h 3 R 3 − − --> R 2 r + R 2 r − − --> r 3 R − − --> r {\displaystyle V_{T}={\frac {\pi h}{3}}{\frac {R^{3}-R^{2}r+R^{2}r-r^{3}}{R-r}}}
V T = π π --> h 3 R 3 − − --> r 3 R − − --> r {\displaystyle V_{T}={\frac {\pi h}{3}}{\frac {R^{3}-r^{3}}{R-r}}}
V T = π π --> h 3 ( R − − --> r ) ( R 2 + r 2 + R r ) R − − --> r {\displaystyle V_{T}={\frac {\pi h}{3}}{\frac {(R-r)(R^{2}+r^{2}+Rr)}{R-r}}}
V T = π π --> h 3 ( R 2 + r 2 + R r ) {\displaystyle V_{T}={\frac {\pi h}{3}}(R^{2}+r^{2}+Rr)}
La formula per calcolare il volume di un tronco di cono ellittico è la seguente:
V = π π --> 3 { r 3 tan --> α α --> − − --> 1 2 b [ 4 a 2 − − --> ( H − − --> h ) 2 ( r tan --> α α --> − − --> h ) − − --> ( H − − --> h ) ( r − − --> h cot --> α α --> ) ] } {\displaystyle V={\frac {\pi }{3}}\left\{r^{3}\tan {\alpha }-{\frac {1}{2}}b\left[{\sqrt {4a^{2}-(H-h)^{2}}}(r\tan {\alpha }-h)-(H-h)(r-h\cot {\alpha })\right]\right\}}
dove V è il volume del tronco di cono, r è il raggio, α è l'inclinazione dell'apotema del cono sezionato, a e b sono i semiassi dell'ellisse ottenuta dal sezionamento del cono e H e h sono rispettivamente l'altezza massima e minima del tronco di cono.
Un cilindro può essere pensato come un tronco di cono con basi di uguali dimensioni. Partendo quindi dalla formula del volume di un tronco di cono C per il quale il raggio R risulta anche uguale a r, si ha:
V C = π π --> h 3 ( R 2 + R 2 + R R ) {\displaystyle V_{C}={\frac {\pi h}{3}}(R^{2}+R^{2}+RR)}
V C = π π --> h 3 ( 3 R 2 ) {\displaystyle V_{C}={\frac {\pi h}{3}}(3R^{2})}
V C = π π --> h R 2 {\displaystyle V_{C}=\pi hR^{2}}
che è la formula del volume di un cilindro.
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