A Torus is the mathematical name for the shape of a donut, a bagel, or an inner tube. It is a 3D surface generated by revolving a circle (the “tube”) around an axis coplanar with the circle.
While “Donut” is the tasty common name, knowing the geometry of a Torus is crucial for everything from designing magnetic fusion reactors (Tokamaks) to modeling 3D video game assets.
Calculator Features
1. Interactive 3D Model
Because a Torus is a 3D object, a 2D drawing doesn’t do it justice. Our calculator generates a live 3D wireframe. Click and drag to rotate the view and verify the proportions of your “Major” and “Minor” radii.
2. Volume & Area Calculation
Instantly compute the total Surface Area (for coating/painting) and the Volume (for material quantity). The tool assumes a perfect “Ring Torus” where the hole is open ($R > r$).
3. Smart Validation
If you make the tube too fat ($r > R$), the shape self-intersects and becomes a “Spindle Torus” (like a pumpkin). The calculator alerts you to prevent physically impossible designs.
The Theorems of Pappus
The volume and area of a rotated solid are best found using the Theorems of Pappus-Guldinus.
Let $R$ be the Major Radius (distance from center of hole to center of tube).
Let $r$ be the Minor Radius (radius of the tube itself).
Surface Area
Area = (Circumference of Tube) $\times$ (Distance traveled by its center).
Surface Area $A = (2\pi r) \times (2\pi R) = 4\pi^2 R r$
Volume
Volume = (Area of Tube Circle) $\times$ (Distance traveled by its centroid).
Volume $V = (\pi r^2) \times (2\pi R) = 2\pi^2 R r^2$
Uses in Science & Engineering
Nuclear Fusion (Tokamaks)
To hold plasma at millions of degrees, scientists use magnetic fields shaped like a Torus. The “Tokamak” reactor design relies heavily on toroidal geometry calculations.
O-Rings and Seals
Rubber O-rings are tori. Engineers calculate their volume to ensure they will squish correctly into a groove to form a watertight seal.
Topology
A coffee mug is topologically equivalent to a donut because they both have exactly one hole. This shape is fundamental in the study of topology and genus-1 surfaces.
Tips for Input
Major Radius vs Outer Radius
Be careful! The formulas use the **Major Radius** ($R$), which goes to the *center of the tube*. If you measured the full *Outer Radius*, you need to subtract $r$ first ($R = R_{outer} – r$).
Units
Volume results are cubic units (e.g., $cm^3$), while Area is in square units (e.g., $cm^2$). Keep your input units consistent.
Frequently Asked Questions (FAQs)
1. What if $R = r$?
This is called a “Horn Torus.” The hole in the middle closes completely to a single point. The surface area formula still works, but the hole diameter is zero.
2. How do I convert from Diameter?
If you have Tube Diameter ($d$), then $r = d/2$. If you have the Ring Diameter (center-to-center $D$), then $R = D/2$.
3. Why is there a $\pi^2$ in the formula?
Because you are multiplying two circular components: the circle of the tube ($2\pi r$) and the circular path it travels ($2\pi R$). $\pi \times \pi = \pi^2$.
Final Words
The Donut Area Calculator proves that math can be delicious. Whether you are dealing with subatomic plasma or just trying to tile a texture onto a 3D donut model, these formulas are the key to unlocking the geometry of the torus.
