據(jù)

祖暅原理

設(shè)

棱臺(tái)

與

等上下底面面積與高

圓臺(tái)

上下底面面積與高

分別為

S1，S2，h

有

V棱臺(tái)

=

V圓臺(tái)

=

lim(n→+∞)

Σ(i=1，n)

π(r1+(r2-r1)i/n)2h/n

=

lim(n→+∞)

Σ(i=1，n)

π(r12+2(r2-r1)r1i/n+(r2-r1)2i2/n2)h/n

=

lim(n→+∞)

hπ

(r12

+(r2-r1)r1(1+n)/n+

(r2-r1)2(n+1)(2n+1)/(6n2))

=

hπ(r12+(r2-r1)r1+1/3(r2-r1)2)

=

hπ(r12+r1r2+r22)/3

=

(S1+√（S1S2)+S2)h/3

得證

ps.

微積分證法

詳見

CV18157464