Torus Calculator (Volume & Surface Area)
Torus Calculator (Volume & Surface Area)
A torus is a donut-shaped solid, formed by sweeping a circle (the tube, radius r) around an axis at a distance R from the tube's own center - inner tubes, bagels, and O-rings are everyday examples. Enter the tube radius and the center radius (in whatever unit you like), and this calculator finds the volume and surface area, along with a diagram and the full step-by-step working.
- V = 2π²Rr², so a torus with tube radius 1.5 and center radius 6 has a volume of 2 × π² × 6 × 1.5² ≈ 266.48 cubic units.
- SA = 4π²Rr, for the same torus that's 4 × π² × 6 × 1.5 ≈ 355.31 square units - both formulas come from treating the torus as a circle of area/circumference πr²/2πr swept around a circle of radius R (Pappus's centroid theorems).
- This calculator assumes a standard (ring) torus where r < R - the tube radius must be smaller than the center radius, otherwise the tube would overlap itself in the middle and the shape wouldn't have a hole.
How do I find the volume of a torus?
Use V = 2π²Rr², where R is the center radius (hole center to tube center) and r is the tube radius. For r = 1.5, R = 6: V = 2 × π² × 6 × 1.5² ≈ 266.48.
What is the difference between R and r for a torus?
R (center radius) is measured from the very center of the donut's hole out to the center of the tube; r (tube radius) is the radius of the tube itself, measured from the tube's centerline to its outer surface.
Torus Calculator (Volume & Surface Area)
A torus is a donut-shaped solid, formed by sweeping a circle (the tube, radius r) around an axis at a distance R from the tube's own center - inner tubes, bagels, and O-rings are everyday examples. Enter the tube radius and the center radius (in whatever unit you like), and this calculator finds the volume and surface area, along with a diagram and the full step-by-step working.
- V = 2π²Rr², so a torus with tube radius 1.5 and center radius 6 has a volume of 2 × π² × 6 × 1.5² ≈ 266.48 cubic units.
- SA = 4π²Rr, for the same torus that's 4 × π² × 6 × 1.5 ≈ 355.31 square units - both formulas come from treating the torus as a circle of area/circumference πr²/2πr swept around a circle of radius R (Pappus's centroid theorems).
- This calculator assumes a standard (ring) torus where r < R - the tube radius must be smaller than the center radius, otherwise the tube would overlap itself in the middle and the shape wouldn't have a hole.
How do I find the volume of a torus?
Use V = 2π²Rr², where R is the center radius (hole center to tube center) and r is the tube radius. For r = 1.5, R = 6: V = 2 × π² × 6 × 1.5² ≈ 266.48.
What is the difference between R and r for a torus?
R (center radius) is measured from the very center of the donut's hole out to the center of the tube; r (tube radius) is the radius of the tube itself, measured from the tube's centerline to its outer surface.