Torus Surface Area Calculator — Volume & Geometry
Find the surface area and volume of a torus (donut shape) from its two radii. The major radius R is the distance from the torus centre to the tube centre; the minor radius r is the tube's own radius.
Total outer surface area of the torus: A = 4π²Rr
A = 394.78
V = 394.78- 1
π²
π × π = 9.869604 - 2
4π² × R
4 × 9.869604 × 5 = 197.392088 - 3
Surface area A = 4π²Rr
197.392088 × 2 = 394.7842By Pappus’s theorem: tube circumference 2πr times path 2πR.
How does this calculator work?
A torus (donut) with major radius R and tube radius r has surface area A = 4π²Rr and volume V = 2π²Rr². The outer radius is R + r and the hole radius is R − r. Enter R and r (with r < R) to get area, volume, and key dimensions instantly.
Formula
How this is calculated
A torus is a surface of revolution generated by revolving a circle of radius r about an axis in the same plane at a distance R from its centre. The surface area and volume follow elegantly from Pappus's centroid theorem: the measure of a surface or solid of revolution equals the measure of the generating curve (or region) multiplied by the distance travelled by its centroid. For a circle the centroid is its centre, which travels a distance of 2πR, so: surface area A = (2πr) × (2πR) = 4π²Rr, and volume V = (πr²) × (2πR) = 2π²Rr².
The torus is classified by how r compares to R. When r < R (ring torus, the common donut shape) there is a hole of radius R − r. When r = R the inner hole closes to a point (horn torus). When r > R the torus self-intersects (spindle torus) — this calculator excludes that case. The outer radius is R + r; the inner hole radius is R − r.
All values use the same unit as the radii you enter (e.g. if R and r are in centimetres, the area is in cm² and the volume in cm³).
Frequently asked questions
The major radius R is measured from the central axis of the torus to the centre of the circular tube. The minor radius r is the radius of the tube itself. Think of R as 'how big the donut is overall' and r as 'how thick the dough is'.
If r ≥ R the inner hole vanishes (r = R, horn torus) or the surface folds through itself (r > R, spindle torus). These are mathematically valid surfaces but their area and volume formulas are the same — this calculator validates r < R to avoid confusing self-intersecting geometry.
Pappus's centroid theorem states that the volume of a solid of revolution equals the area of the generating shape times the distance its centroid travels (2πR for a circle at radius R). For a circle of radius r: area = πr², centroid path = 2πR, so volume = 2π²Rr². Similarly the surface area = circumference × path = 2πr × 2πR = 4π²Rr.
Also known as
TG we-Calculate Editorial Team. (2026). Torus Surface Area Calculator — Volume & Geometry [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/torus-surface-area-calculator
TG we-Calculate Editorial Team. "Torus Surface Area Calculator — Volume & Geometry." TG we-Calculate. 2026. https://we-calculate.com/calculator/torus-surface-area-calculator.
TG we-Calculate Editorial Team, "Torus Surface Area Calculator — Volume & Geometry," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/torus-surface-area-calculator
@misc{wecalculate_torus_surface_area_calculator, title = {Torus Surface Area Calculator — Volume & Geometry}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/torus-surface-area-calculator}}, year = {2026}, note = {TG we-Calculate} }
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