Torus Surface Area

Q: What is the formula for torus surface area?

The surface area of a torus is calculated using A = 4pi^2Rr, where R is the major radius (distance from tube center to torus center) and r is the minor radius (tube radius). When using inner radius a and outer radius b, the formula becomes A = pi^2(b - a)(b + a), since R = (a + b)/2 and r = (b - a)/2.

Q: How do you find the inner and outer radius of a torus?

The inner radius (a) is the shortest distance from the center of the torus hole to the inner edge. The outer radius (b) is the longest distance from the center to the outer edge. They relate to the major radius R and minor radius r as follows: a = R - r and b = R + r, where R is the distance from the center of the tube to the torus center, and r is the radius of the tube cross-section.

Q: What types of tori exist?

There are three types of tori based on the relationship between major radius R and minor radius r. Ring torus (R > r): the classic doughnut shape with a visible hole, like a bagel or lifebuoy. Horn torus (R = r): the surface is tangent to itself at the center, forming a horn-like shape with no hole. Spindle torus (R = 0).

Calculate the surface area of any torus (doughnut shape) from its inner and outer radius. Free online 3D geometry calculator with formula A = pi^2(b - a)(b + a) and instant results.

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Inner Radius (a):

Outer Radius (b):

About This Calculator

This Torus Surface Area Calculator computes the surface area of a torus -- a 3D doughnut-shaped surface generated by revolving a circle around an axis in the same plane. The torus is one of the most recognizable geometric shapes, appearing everywhere from doughnuts and bagels to tires, lifebuoys, O-rings, wedding rings, and even magnetic fusion reactor chambers like tokamaks.

The calculator uses the formula A = pi^2(b - a)(b + a), where a is the inner radius (half the diameter of the hole) and b is the outer radius (half the overall diameter). This is equivalent to the more standard form A = 4pi^2Rr, where R = (a + b)/2 is the major radius (distance from the tube center to the torus center) and r = (b - a)/2 is the minor radius (radius of the tube cross-section).

Key formulas:

Surface area: A = pi^2(b - a)(b + a)
Alternative form: A = 4pi^2Rr
Major radius: R = (a + b)/2
Minor radius: r = (b - a)/2

Three types of tori exist based on the R-to-r ratio. Ring torus (R > r, a > 0): the classic doughnut with a visible hole. Horn torus (R = r, a = 0): the surface pinches at the center with no hole. Spindle torus (R < r, a < 0): self-intersecting like an apple. This calculator supports ring and horn tori. The surface area grows linearly with both the tube circumference (2pir) and the revolution circumference (2piR), giving the intuitive relationship A = (2piR)(2pir) = 4pi^2Rr.

Frequently Asked Questions

What is the formula for torus surface area?

The surface area of a torus is calculated using A = 4pi^2Rr, where R is the major radius (distance from tube center to torus center) and r is the minor radius (tube radius). When using inner radius a and outer radius b, the formula becomes A = pi^2(b - a)(b + a), since R = (a + b)/2 and r = (b - a)/2.

How do you find the inner and outer radius of a torus?

The inner radius (a) is the shortest distance from the center of the torus hole to the inner edge. The outer radius (b) is the longest distance from the center to the outer edge. They relate to the major radius R and minor radius r as follows: a = R - r and b = R + r, where R is the distance from the center of the tube to the torus center, and r is the radius of the tube cross-section.

What types of tori exist?

There are three types of tori based on the relationship between major radius R and minor radius r. Ring torus (R > r): the classic doughnut shape with a visible hole, like a bagel or lifebuoy. Horn torus (R = r): the surface is tangent to itself at the center, forming a horn-like shape with no hole. Spindle torus (R < r): the surface self-intersects, creating a spindle-like shape, similar to an apple. This calculator covers ring and horn tori (a >= 0).

What is the surface area of a torus with inner radius 1 and outer radius 3?

Using A = pi^2(b - a)(b + a): a = 1, b = 3, so A = pi^2 x (3 - 1) x (3 + 1) = pi^2 x 2 x 4 = 8pi^2 ≈ 78.96 square units. This is a ring torus where the major radius R = 2 and the minor radius r = 1.

What are real-world examples of tori?

Common real-world examples of tori include doughnuts and bagels, lifebuoys and rescue rings, vehicle tires and inner tubes, wedding rings and toroidal jewelry, tokamak fusion reactors (magnetic confinement in a toroidal shape), O-rings and gaskets in engineering, and the human intestine's toroidal-like geometry.

What units does the torus surface area calculator use?

The calculator works with any consistent unit system. If you enter inner and outer radius in meters, the result is in square meters. If you use centimeters, the result is in square centimeters. The calculator is unit-agnostic -- just ensure both inputs use the same unit. For ring-type tori, ensure the inner radius is positive and smaller than the outer radius.

Is this tool free to use?

Yes, all calculators on Calculy are completely free to use with no registration required. You can share your torus surface area calculation results via URL for easy reference.