Torus Calculator

Calculate the volume and surface area of a torus (donut shape) from its major and minor radii. Free online torus calculator with formulas.

Torus Properties

Volume

394.7842

cubic units

Surface Area

394.7842

square units

Major Radius R5
Minor Radius r2
R / r Ratio2.5

Radii

Formulas Used

  • Volume = 2π² × R × r² = 2π² × 5 × 2² = 394.7842
  • Surface Area = 4π² × R × r = 4π² × 5 × 2 = 394.7842
  • Equivalent: Volume = (π r²) × (2π R) — tube cross-section × centerline circumference

Quick Answer

To calculate torus properties: Volume = 2 x pi^2 x R x r^2. Surface Area = 4 x pi^2 x R x r, where R is the major radius (center to tube center) and r is the minor radius (tube radius), with R > r. For R = 5 and r = 2: Volume = 2 x 9.8696 x 5 x 4 = 394.78 cubic units, Surface Area = 4 x 9.8696 x 5 x 2 = 394.78 square units.

Key Facts

  • Formula: Volume = 2 x pi^2 x R x r^2
  • Formula: Surface Area = 4 x pi^2 x R x r
  • Torus R = 5, r = 2 has volume 394.78 and surface area 394.78
  • The major radius R must exceed the minor radius r, or the torus self-intersects
  • Volume equals tube cross-section area (pi r^2) times centerline circumference (2 pi R)
  • When R = 2r, volume and surface area have equal numerical value
  • A torus is generated by revolving a circle of radius r around an axis R away

Frequently Asked Questions

Use Volume = 2 x pi^2 x R x r^2, where R is the major radius (from the center of the hole to the center of the tube) and r is the minor radius (radius of the tube). For R = 5 and r = 2: Volume = 2 x 9.8696 x 5 x 4 = 394.78 cubic units.

Use Surface Area = 4 x pi^2 x R x r. For R = 5 and r = 2: SA = 4 x 9.8696 x 10 = 394.78 square units. This is Pappus's theorem applied to the revolved circle: area equals 2 pi R (path of the centroid) times 2 pi r (circle perimeter).

The major radius R runs from the center of the donut hole to the center of the tube. The minor radius r is the radius of the tube itself. The hole diameter is 2R - 2r; the outer diameter is 2R + 2r.

If R <= r, the tube wraps past the central axis and the surface self-intersects — it becomes a spindle torus rather than a ring. The standard volume and surface area formulas above apply only to the ring torus with R > r.

Compare the formulas: V = 2 pi^2 R r^2 and SA = 4 pi^2 R r. They are numerically equal when r = 2, since the extra factor of r in volume cancels the extra 2 in surface area. The units still differ (cubic vs square).

Donuts, bagels, O-rings, gaskets, life rings, tire inner tubes, and magnetic cores in electronics. Engineers use torus volume for material estimates on O-rings and toroidal tanks.