Torus Surface Area Calculator

What Is a Torus?

A torus is the surface you get by spinning a circle around an axis in its own plane that does not cut it. Donuts, bagels, inner tubes, O-rings and lifebuoys are all close to tori. Two sizes fix it: the center radius R (also called the major radius), from the axis to the center of the tube, and the tube radius r (the minor radius), the radius of the circle that was spun.

Torus Surface Area Formula

Multiply 4, π², the center radius and the tube radius. Other ways to write it:

• As two circles: A = (2πr)(2πR), the tube's circumference times its center circle
• With inner and outer radii: A = π²(b² − a²)
• With diameters: A = π²Dd, where D = 2R and d = 2r

The answer has π² in it, not π, because the surface is swept around two circles at once. The decimal value of π² is about 9.8696.

Why the Formula Works

Think of the torus as a tube bent into a ring. Cut it once and straighten it, and you get a cylinder whose circumference is 2πr and whose length is the center circle, 2πR. Unrolled, that side is a rectangle of area 2πr × 2πR = 4π²Rr.

A real torus can't be flattened without stretching: the outside of the ring is longer than 2πR and the inside is shorter. The extra on the outside and the shortfall on the inside cancel exactly, which is Pappus's centroid theorem: the area of a surface of revolution is the length of the spun curve times the distance its centroid travels.

From the Inner and Outer Radius

On a real ring it is easier to measure the hole and the outside than the center of the tube. With the hole radius a and the outer radius b:

• Center radius: R = (a + b) ÷ 2
• Tube radius: r = (b − a) ÷ 2
• Surface area: A = 4π²Rr = π²(b² − a²)

A donut 9 cm across with a 3 cm hole has b = 4.5 and a = 1.5, so R = 3 and r = 1.5. Its surface area is π²(4.5² − 1.5²) = 18π² ≈ 177.65 cm². Pick Inner & outer in the calculator to enter these directly.

Torus Surface Area Examples

The exact answer is 4Rr times π²; the decimal is rounded. Doubling either radius doubles the area, and doubling both makes it four times as large.

Ring, Horn and Spindle Tori

• Ring torus (R > r): the usual donut with a hole. A = 4π²Rr.
• Horn torus (R = r): the hole shrinks to a single point. With R = r = 4, A = 64π² ≈ 631.65.
• Spindle torus (R < r): the tube passes through itself, so 4π²Rr counts the inner part twice. The calculator asks for r no larger than R.

Common Mistakes

• π instead of π²: 4π × 8 × 3 = 96π ≈ 301.59 is far too small. The answer is 96π² ≈ 947.48.
• Outer radius for R: R runs to the middle of the tube, not the outside. Using 11 instead of 8 gives 132π², too big.
• Diameters as radii: 4π² × 16 × 6 = 384π² is four times the real area. Halve diameters first, or use π²Dd.
• Volume for area: 2π²Rr² is the volume, in cubic units.
• Mixing units: give both radii in the same unit before multiplying.

Key Terms

Torus

A ring shaped surface made by spinning a circle about an outside axis.

Center radius

R, the major radius, from the axis to the center of the tube.

Tube radius

r, the minor radius, of the tube's circular cross-section.

Inner radius

a = R − r, the radius of the hole.

Outer radius

b = R + r, to the outside edge.

How to Use the Torus Surface Area Calculator

1. Choose what you know. Pick Center & tube for the center radius R and tube radius r, or Inner & outer for the radius of the hole and the outer radius.
2. Enter the two radii. Type both radii, such as 8 and 3, and pick a unit if you want one (mm, cm, m, km, in, ft, yd or mi). Decimals and fractions such as 3/4 work.
3. Read and copy the result. The surface area appears as you type, with the working, the exact answer in terms of π², the other pair of radii, the tube and center circles, the volume and the area in other units. Press Copy to copy it, or Clear or Esc to start again.

Example: an O-ring has a center radius of 8 cm and a cord radius of 3 cm. Enter 8 and 3 and pick cm. The surface area is 96π² ≈ 947.48 cm², and the details show a hole radius of 5 cm, an outer radius of 11 cm and a volume of 144π² ≈ 1,421.22 cm³.

Who Can Use This Calculator?

Frequently Asked Questions (FAQs)