Torus Volume

A torus is a doughnut: take a circle of radius $r$ and sweep it around an axis at distance $R$ from its centre. Pappus of Alexandria worked out around 340 AD that the volume of any such solid of revolution is simply the area of the moving shape times the distance its centroid travels. Here that is $\pi r^2$ times $2\pi R$, giving $V = 2\pi^2 R r^2$ with no integration at all.

The result is used far outside the bakery. O-ring manufacturers compute rubber volume per part this way, tyre engineers estimate inner-tube capacity, and fusion physicists quote the plasma volume of a tokamak, which is a torus with $R$ of a few metres, straight from this expression. Vacuum-chamber and pipe-bend volumes come from the same relation, since a 360-degree bend is a torus and a 90-degree bend is a quarter of one.

Almost every error with a torus is a measuring error, not an arithmetic one. $R$ runs from the centre of the hole to the centreline of the tube, not to its inner or outer face. If you have the outer radius $R_o$ and the tube radius, then $R = R_o - r$. Get that wrong on a doughnut and nobody dies; get it wrong on a seal groove and the O-ring will not fit.