Volume of Torus/Formulation 2

Theorem

Let $\TT$ be a torus.

Let $a$ and $b$ be the inner radius and outer radius respectively of $\TT$.


Then the volume $\VV$ enclosed by $\TT$ is given by:

$\VV = \dfrac {\pi^2 \paren {a + b} \paren {b - a}^2} 4$


Proof


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Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 4$: Geometric Formulas: Torus of Inner Radius $a$ and Outer Radius $b$: $4.45$
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 7$: Geometric Formulas: Torus of Inner Radius $a$ and Outer Radius $b$: $7.45.$
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