What is the Torus Surface Area Calculator?
A torus is the donut-shaped surface generated by revolving a circle of radius r around an axis at distance R from the circle's center. This calculator returns the total surface area of that shape from just two measurements: the major radius R and the minor radius r. It works in any consistent unit — millimeters, inches, meters — and the result is simply in those units squared.
How to use it
Enter the major radius R (the distance from the very center of the torus to the center of the tube) and the minor radius r (the radius of the tube cross-section). Click calculate and the surface area appears immediately. Make sure both radii use the same unit so the squared output is meaningful.
The formula explained
The surface area of a torus is given by:
$$A = 4\pi^2 R r$$This comes from Pappus's theorem: the surface area of a surface of revolution equals the length of the generating curve (the tube's circumference, \(2\pi r\)) multiplied by the distance traveled by its centroid (\(2\pi R\)). Multiplying these gives \(2\pi r \times 2\pi R = 4\pi^2 R r\).
Worked example
Suppose a torus has a major radius \(R = 5\) and a minor radius \(r = 2\). Then $$A = 4 \times \pi^2 \times 5 \times 2 = 40\pi^2 \approx 394.78$$ square units. Doubling the tube radius to \(r = 4\) would double the area to \(80\pi^2 \approx 789.57\).
FAQ
What's the difference between R and r? R (major) spans from the torus center to the tube center; r (minor) is the thickness/radius of the tube itself.
Does R need to be larger than r? For a standard ring torus, R > r. If R = r the hole closes (horn torus); if R < r it self-intersects (spindle torus), but the formula still computes a value.
What units does the answer use? Whatever unit you input, squared. If R and r are in cm, the area is in cm².
