What is a torus?
A torus is a 3D shape obtained by revolving a circle around an axis. This shape is commonly found in doughnuts, rings, tires, and tubes. 🍩 You have surely met it on your plate as a doughnut or a bagel, or on the roads underneath vehicles. The life-saving tube or ring — aka the rescue buoy — is also a torus. If you take a ring and circularly trace it around with a pencil, you get a torus. In modern design software, it is fairly easy to draw them by using a revolve command with a circle as a cross-section.
A torus has two radii — the first radius is the radius of the cross-section r, and the second radius R is the radius of revolution, which is the distance between the center axis and the center of the cross-section. Any point on a torus is defined using a modified coordinate system having two directions — toroidal and poloidal. Based on the combination of the two radii, we can obtain multiple types of tori:
- Ring type (R > r)
- Horn type (R = r)
- Spindle type (R < r)
In addition to these radii, the torus can also be expressed in the form of two radii — the inner radius (a) and the outer radius (b) of the torus. Mathematically, that is:
b = R + r
The surface area A of the said torus is:
The surface area can also be written in terms of the inner and outer radii, which is the formula used in this calculator:
where:
- r = (b − a) / 2 — radius of the cross-section
- R = (a + b) / 2 — radius of the torus
- π (pi) ≈ 3.14159265…
💡 Note: This calculator only applies to ring-type or horn-type tori. Furthermore, in the case of a horn-type torus, i.e., R = r, the inner radius a becomes zero.
How to use the torus surface area calculator?
Follow three simple steps to find out the surface area of a torus:
- Choose your unit system — metric (centimeters) or US/Imperial (inches).
- Enter the inner radius of the torus, a.
- Enter the outer radius of the torus, b.
The calculator will now use the above formula to return the surface area of a torus — along with the cross-section radius r, the torus radius R, the torus type, and the area in multiple units.
Example: How to calculate the surface area of a torus?
Find the surface area of a horn-type torus having a radius of cross-section r = 1 m.
Note: The torus is of horn type, i.e., r = R. Let us first convert the radii into the inner and outer radius, a and b:
b = R + r = 1 + 1 = 2 m
Step 1: Enter the inner radius of the torus, a = 0 m.
Step 2: Enter the outer radius of the torus, b = 2 m.
Step 3: Using the torus surface area formula:
A = π² × (2 − 0) × (2 + 0) = π² × 4 ≈ 39.48 m²
Using the smaller radii from the prompt (r = R = 1 m, so a = 0 m and b = 1 m) gives A = π² × 1 × 1 ≈ 9.87 m². Either way, the surface area for a horn-type torus follows the same formula. You might also be interested in determining the volume of a torus.
Unit Systems
Metric System
Enter the radii in centimeters (cm). The calculator returns the surface area in:
- cm² — square centimeters
- m² — square meters (1 m² = 10,000 cm²)
- mm² — square millimeters (1 cm² = 100 mm²)
US / Imperial System
Enter the radii in inches (in). The calculator returns the surface area in:
- in² — square inches
- ft² — square feet (1 ft² = 144 in²)
- yd² — square yards (1 yd² = 1,296 in²)
FAQs
What is a torus?
A torus is a 3D circular shape with a circle as a cross-section. The shape is commonly found in doughnuts, tires, and hoops. It is obtained when you revolve a circle along a circular path around an axis normal to the circle.
How do I calculate the surface area of a torus?
Use the formula A = π² × (b − a) × (b + a), where a is the inner radius and b is the outer radius. Equivalently, with r = (b − a) / 2 and R = (a + b) / 2, the surface area is A = 4 × π² × r × R.
What is the difference between the ring, horn, and spindle types?
The type depends on the relationship between the torus radius R and the cross-section radius r. If R > r you get a ring torus (like a doughnut), if R = r you get a horn torus (the hole shrinks to a point), and if R < r you get a spindle torus (self-intersecting). This calculator supports ring and horn types.
What happens to the inner radius for a horn torus?
For a horn-type torus, R = r, so the inner radius a = R − r becomes zero. The hole in the middle closes to a single point and the surface area equals A = π² × b², where b is the outer radius.