A torus is the doughnut shape made by sweeping a circle of radius r around an axis at distance R from its centre. Inner tubes, O-rings, bagels, lifebuoys and tokamak fusion reactors are tori.
How to use the Torus Surface Area Calculator
- Dimensions given as
- Major radius R (centre to tube centre)
- Minor radius r (tube radius)
- Inner radius a
- Outer radius b
Results update as you type, and everything is calculated in your browser.
How it works
Pappus’s theorems make the formulas simple: the volume is the tube’s cross-section area times the distance its centre travels (2πR), and the surface area is the tube’s circumference times that same distance. If you measured the hole and the overall size instead, choose inner and outer radii.
Inputs explained
| Field | What to enter |
|---|---|
| Dimensions given as | Choose one: Major radius R and minor radius r, Inner radius a and outer radius b |
| Major radius R (centre to tube centre) | Units: cm, m, mm, in, ft, yd … |
| Minor radius r (tube radius) | Units: cm, m, mm, in, ft, yd … |
| Inner radius a | Units: cm, m, mm, in, ft, yd … |
| Outer radius b | Units: cm, m, mm, in, ft, yd … |
What each result means
- Volume = 2×pi^2×[Major radius R]×[Minor radius r]^2
- Surface area = 4×pi^2×[Major radius R]×[Minor radius r]
- Inner radius (hole) = [Major radius R] – [Minor radius r]
- Outer radius = [Major radius R] + [Minor radius r]
Example values
How the main result changes when Major radius R changes and everything else stays at the example values above:
| Major radius R (centre to tube centre) (cm) | Surface area (cm²) |
|---|---|
| 5 | 592.1763 |
| 7.5 | 888.2644 |
| 10 | 1,184.3525 |
| 12.5 | 1,480.4407 |
| 15 | 1,776.5288 |
| 20 | 2,368.7051 |
Formula
V = 2π²·R·r² = (π r²)(2π R)
A = 4π²·R·r = (2π r)(2π R)
R = (a + b)/2, r = (b − a)/2
Worked example
Inputs — Dimensions given as: Major radius R and minor radius r; Major radius R (centre to tube centre): 10 cm; Minor radius r (tube radius): 3 cm.
Results — Major radius R = 10 cm; Minor radius r = 3 cm; Volume = 1,776.528792 cm³; Surface area = 1,184.352528 cm²; Inner radius (hole) = 7 cm; Outer radius = 13 cm.
Key terms
- Major radius
- Distance from the centre of the torus to the centre of the tube.
- Minor radius
- Radius of the tube.
Frequently asked questions
How much glaze covers a doughnut?
Its surface area 4π²Rr.
Why is the surface area 4π²Rr?
Tube circumference 2πr times the path length 2πR.
Is the inside of the ring included?
Yes – the whole closed surface.
What are the units?
Square units of whatever length you entered.