Math
Torus Calculator
The torus calculator finds the volume and surface area of a donut-shaped solid from its two radii: the major radius R from the center of the hole to the center of the tube, and the minor radius r of the tube itself.
Results stay exact in terms of π² (R = 6, r = 2 gives 48π²) and the calculator tells you whether the torus is a ring, horn, or self-intersecting spindle.
Distance from the center of the hole to the center of the tube.
Radius of the tube (half its thickness).
Pick the unit your measurements use. Area is reported in square units and volume in cubic units.
Try an example
Result
Volume
—
- Volume (exact)
- —
- Surface area
- —
- Outer / inner radius
- —
- Torus type
- —
- Capacity
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the torus calculator solves
The torus calculator finds the volume and surface area of a donut-shaped solid from its two radii: the major radius R from the center of the hole to the center of the tube, and the minor radius r of the tube itself. Results stay exact in terms of π² (R = 6, r = 2 gives 48π²) and the calculator tells you whether the torus is a ring, horn, or self-intersecting spindle.
Formula
How to use it
- Measure the major radius R (center of the hole to the middle of the tube) and the minor radius r (half the tube's thickness). If you have the outer and inner radii instead, R = (outer + inner)/2 and r = (outer − inner)/2.
- Enter both in one unit and choose the unit.
- Read the volume and surface area, exact in terms of π² and as decimals.
How to read the answer
By Pappus's theorem the torus is a circle of area πr² swept around a path of length 2πR, so V = πr² × 2πR = 2π²Rr². Likewise the surface is the circle's circumference 2πr swept the same distance: 4π²Rr. When r is less than R you get a ring donut; equal gives a horn torus with no hole; larger gives a spindle that overlaps itself.
Common mistakes and edge cases
- Entering the outer radius as the major radius. R is measured to the center of the tube, not its outer edge.
- Entering the tube diameter instead of its radius.
- Expecting a multiple of π: torus formulas involve π², so 48π² is the The answer kept as a fraction or radical instead of a rounded decimal..
- Zero or negative radii cannot form a torus.
Worked examples
R = 6, r = 2 (48π²)
Volume
473.741
R = 10, r = 3 (180π²)
Volume
1,776.5288
Horn torus R = r = 2
Volume
157.9137
Decimal R = 4.5, r = 1.25
Volume
138.7913
Spindle torus R = 1.5, r = 2.5
Volume
185.0551
Zero minor radius (impossible)
Volume
Error
Frequently asked questions
What is the volume of a torus?+
V = 2π²Rr², where R is the major radius and r the minor (tube) radius. R = 6 and r = 2 give 48π² ≈ 473.74.
What is the surface area of a torus?+
SA = 4π²Rr. For R = 6, r = 2 that is 48π² ≈ 473.74 square units (numerically equal to the volume only because r = 2 here).
How do I find R and r from the outer and inner radius?+
R = (outer + inner)/2 and r = (outer − inner)/2. A donut 16 across with a 8-wide hole has outer radius 8, inner radius 4, so R = 6 and r = 2.
Why does the formula have π squared?+
Two circles are involved: the tube's cross-section (πr²) and the circular path it sweeps around (2πR). Multiplying them gives π².
What is the difference between a ring, horn, and spindle torus?+
Ring: r < R, the familiar donut with a hole. Horn: r = R, the hole shrinks to a point. Spindle: r > R, the tube passes through itself.
Does Pappus's theorem apply to a torus?+
Yes — it is the cleanest way to derive both formulas: volume = (area of the tube circle) × (distance its centroid travels), and surface area = (circumference) × (same distance).
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
Related calculators
Sphere Calculator
Enter one sphere measurement — radius, diameter, circumference, surface area, or volume — and get all five with exact π forms (36π), the great-circle area, steps, and liters or gallons of capacity.
Cylinder Calculator
Cylinder calculator for volume, lateral area, total surface area, and diagonal from radius and height — or solve the height or radius from a known volume or lateral area. Exact π answers plus liters and gallons.
Volume Calculator
Find the volume of 11 solids — cube, box, cylinder, cone, sphere, hemisphere, pyramid, triangular prism, ellipsoid, capsule, and frustum — with exact π forms, cubic-unit labels, surface area, and liters/gallons capacity.
Circle Calculator
Enter any one circle measurement (radius, diameter, circumference, or area) to get all four, with exact π forms, unit-aware labels, arc length, sector area, and chord length for any central angle.
Surface Area Calculator
Find the total surface area of a cube, box, cylinder, cone, sphere, rectangular pyramid, or triangular prism with the formula for each solid, the lateral and base breakdown, exact π forms, and square-unit labels.
Area Calculator
Find the area of 11 shapes — rectangle, square, triangle (base-height, three sides, or SAS), circle, trapezoid, parallelogram, rhombus, ellipse, regular polygon, sector, and kite — with the formula, substitution, square-unit labels, and a diagram.
Last updated: September 4, 2026