Donut (Torus) Calculator — Volume & Surface Area
Find the volume and surface area of a donut-shaped torus by entering the major radius R (ring centre to tube centre) and minor radius r (tube radius).
Mértékegység
Volume enclosed by the torus (V = 2π²Rr²)
394,78
vol. (cm³)- 1
Minor radius squared
r² = 2 × 2 = 4 - 2
π² (constant)
π × π = 9,869604π² ≈ 9.8696 appears in both the torus volume and surface area formulas. - 3
Volume V = 2π²Rr²
2 × 9,869604 × 5 × 4 = 394,7842
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A torus (donut shape) has volume V = 2π²Rr² and surface area A = 4π²Rr, where R is the major radius (ring centre to tube centre) and r is the minor radius (tube thickness). Both formulas follow from Pappus's theorem: the centroid of the circle of area πr² or perimeter 2πr travels a path of length 2πR.
Képlet
How this is calculated
A torus is the mathematical name for a donut or ring shape — the solid formed by rotating a circle of radius r around an axis that is R units away from the circle's centre. Two inputs define it: R (the major radius, from the axis of revolution to the centre of the tube) and r (the minor radius, the radius of the tube itself). For a valid ring torus r must be strictly less than R so there is a hole in the centre.
The volume formula V = 2π²Rr² comes from Pappus's centroid theorem: the volume of a solid of revolution equals the cross-section area of the rotating shape (πr² for a circle) multiplied by the path length that the centroid travels (2πR), giving πr² × 2πR = 2π²Rr². The surface area A = 4π²Rr follows the same theorem applied to the perimeter of the circle (2πr) times the centroid path length (2πR).
Practical uses include calculating the capacity of toroidal tanks and O-rings, the volume of a ring-shaped garden bed, an inflatable swim ring, a bagel, or a donut-shaped architectural element. The outer diameter is 2(R+r), the inner hole diameter is 2(R−r), and the tube circumference is 2πr.
Gyakran ismételt kérdések
The major radius R is from the axis of the ring (its central axis of symmetry) to the centre of the tube. The minor radius r is the radius of the tube itself. Think of R as how large the ring is overall and r as how fat the tube is.
When r = R the inner hole vanishes and you get a horn torus that is tangent to itself. When r > R the tube intersects itself. This calculator requires r < R for a proper ring torus.
Common uses include sizing toroidal water or gas tanks, calculating the volume of circular garden borders, sizing inflatable rings, analysing O-ring seals in engineering, and computing the amount of material in any ring-shaped structure.
Más néven
TG we-Calculate Editorial Team. (2026). Donut (Torus) Calculator — Volume & Surface Area [Online calculator]. TG we-Calculate. https://we-calculate.com/hu/calculator/donut-calculator
TG we-Calculate Editorial Team. "Donut (Torus) Calculator — Volume & Surface Area." TG we-Calculate. 2026. https://we-calculate.com/hu/calculator/donut-calculator.
TG we-Calculate Editorial Team, "Donut (Torus) Calculator — Volume & Surface Area," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/hu/calculator/donut-calculator
@misc{wecalculate_donut_calculator, title = {Donut (Torus) Calculator — Volume & Surface Area}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/hu/calculator/donut-calculator}}, year = {2026}, note = {TG we-Calculate} }
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