গোলক

৩ মাত্রার একটি গোলক ভিতরে আয়তন (অর্থাৎ একটি বল-এর আয়তন)-এর সূত্র দেওয়া হলো

${\displaystyle \!V={\frac {4}{3}}\pi r^{3}}$

যেখানে R হল গোলকের ব্যাসার্ধ এবং π হল ধ্রুবক পাই. This formula was first derived by Archimedes, who showed that the volume of a sphere is 2/3 that of a circumscribed cylinder. (This assertion follows from Cavalieri's principle.) In modern mathematics, this formula can be derived using integral calculus, e.g. disk integration to sum the volumes of an infinite number of circular disks of infinitesimal thickness stacked centered side by side along the x axis from x = 0 where the disk has radius r (i.e. y = r) to x = r where the disk has radius 0 (i.e. y = 0).

At any given x, the incremental volume (δV) is given by the product of the cross-sectional area of the disk at x and its thickness (δx):

${\displaystyle \!\delta V\approx \pi y^{2}\cdot \delta x.}$

The total volume is the summation of all incremental volumes:

${\displaystyle \!V\approx \sum \pi y^{2}\cdot \delta x.}$

In the limit as δx approaches zero[1] this becomes:

${\displaystyle \!V=\int _{-r}^{r}\pi y^{2}dx.}$

At any given x, a right-angled triangle connects x, y and r to the origin, hence it follows from Pythagorean theorem that:

${\displaystyle \!r^{2}=x^{2}+y^{2}.}$

Thus, substituting y with a function of x gives:

${\displaystyle \!V=\int _{-r}^{r}\pi (r^{2}-x^{2})dx.}$

This can now be evaluated:

${\displaystyle \!V=\pi \left[r^{2}x-{\frac {x^{3}}{3}}\right]_{x=-r}^{x=r}=\pi \left(r^{3}-{\frac {r^{3}}{3}}\right)-\pi \left(-r^{3}+{\frac {r^{3}}{3}}\right)={\frac {4}{3}}\pi r^{3}.}$

সুতরাং গোলকের আয়তন হল:

${\displaystyle \!V={\frac {4}{3}}\pi r^{3}.}$