Torus Volume
Volume of a donut-shaped solid.
Formula
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Example
Doughnut geometry
How it works
This formula gives the volume enclosed by a torus, which is formed by revolving a circle of radius r around an axis that lies R units from the circle’s center. The result comes from the fact that the torus volume equals the area of the circular cross-section, πr², multiplied by the distance that cross-section travels around the central circle, 2πR. So the volume is the product of two circular measurements, which explains the 2π²Rr² structure.
When to use it
Use this when you need the capacity or material volume of a ring-shaped object such as a pipe bend, toroidal tank, or idealized torus in geometry. It is also useful in calculus and engineering problems involving solids of revolution.
Worked example, step by step
A torus has center radius R = 12 cm and tube radius r = 3 cm. Step 1: Write the formula: V = 2π²Rr² Step 2: Substitute the values: V = 2π²(12)(3²) Step 3: Square the tube radius: 3² = 9 Step 4: Multiply: V = 2π²(12)(9) = 216π² Step 5: Approximate numerically: V ≈ 216 × 9.8696 ≈ 2131.7 cm³ Final answer: V = 216π² cm³ ≈ 2131.7 cm³
Common mistakes to avoid
- using the diameter instead of the tube radius r
- confusing the center radius R with the tube radius r
- forgetting to square r before multiplying
Did You Know?
A torus's volume can be found by Pappus's theorem: just spin its cross-section around and multiply by the distance travelled.
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