🍩 Torus Calculator 2026
Calculate surface area, volume, and dimensions of a torus (donut shape)
Torus has two radii: R = major radius (center to tube
center), r = minor radius (tube radius)
Enter Torus Dimensions
R = major radius, r =
minor (tube) radius
📊 Results
Major Radius (R)
8 cm
Minor Radius (r)
3 cm
Outer Diameter
22 cm
Inner Diameter
10 cm
Surface Area
947.5 cm²
Volume
1,421 cm³
📝 Step-by-Step Solution
Given: R = 8 cm (major), r = 3 cm (minor)
Outer Diameter = 2(R + r) = 2(8 + 3) = 22 cm
Inner Diameter = 2(R - r) = 2(8 - 3) = 10 cm
Surface Area = 4π²Rr = 4π² × 8 × 3 = 947.48 cm²
Volume = 2π²Rr² = 2π² × 8 × 9 = 1,421.22 cm³
📐 Torus Formulas
Surface Area: A = 4π²Rr = (2πr)(2πR)
Volume: V = 2π²Rr²
Outer Diameter: Dout = 2(R + r)
Inner Diameter: Din = 2(R - r)
Valid torus: R > r (otherwise self-intersecting)
Understanding Torus Geometry
🍩 What is a Torus?
Circle revolved around an axis. Creates donut shape. Two radii: major (R, center distance) and minor (r, tube radius). Common in math, physics, and design.
📏 Major Radius (R)
Distance from torus center to tube center. Determines overall size. Must be greater than minor radius for standard (ring) torus.
⭕ Minor Radius (r)
Radius of the tube itself. Cross-section is a circle with this radius. Determines tube thickness.
📐 Surface & Volume
Surface = 4π²Rr. Product of two circumferences. Volume = 2π²Rr². Cross-section area × path length.
Frequently Asked Questions
What is the surface area formula for a torus?
A = 4π²Rr. Also written as (2πr)(2πR) — circumference of tube
times circumference of path. R = major radius, r = minor radius.
What is the volume formula for a torus?
V = 2π²Rr². Cross-sectional area (πr²) times path length (2πR).
Example: R=8, r=3 → V = 2π²×8×9 = 1421 cm³.
What is the major radius?
Distance from torus center to tube center. The "big" radius.
Determines how far the tube is from center. Usually denoted R or a.
What is the minor radius?
Radius of the tube cross-section. The "small" radius. Determines
tube thickness. Usually denoted r or b.
What happens if R = r?
Horn torus. The tube just touches center. If R < r, it's a
spindle torus (self-intersecting). Standard torus has R > r.
How do I find outer diameter?
Dout = 2(R + r). Outer edge to outer edge through
center. Add both radii, then double.
How do I find inner diameter (hole size)?
Din = 2(R - r). Diameter of the hole. Subtract minor
from major, then double. Zero if R = r.
What are real-world examples of tori?
Donuts, inner tubes, bagels, O-rings, lifebuoys, some furniture.
Tokamak fusion reactors use toroidal magnetic fields.
Is a torus the same as a donut?
Mathematically similar. Torus is the precise geometric term. Donut
is informal. Both describe the ring/donut shape.
How do I calculate material for a torus?
Use surface area: 4π²Rr. This gives exterior area. For hollow
torus (tube), also consider inner surface if applicable.
What is the cross-section of a torus?
A circle with radius r. Cut through tube perpendicular to path.
Area = πr². This rotates around center to form torus.
How do I find volume of a partial torus?
V = (θ/360°) × 2π²Rr². For angle θ in degrees. E.g., half torus
(180°) has half the volume.
What is a torus in topology?
Surface with genus 1 (one hole). Different from sphere (genus 0).
Coffee cup and donut are topologically equivalent (can be continuously deformed).
How does doubling radii affect volume?
Doubling both: volume × 8. V ∝ Rr². Doubling R only: ×2. Doubling
r only: ×4.
What is the aspect ratio of a torus?
R/r ratio. Thin torus has high ratio. Small inner hole: ratio
close to 1. Standard ring shape: ratio typically 2-5.
How do I measure a torus in real life?
Measure outer diameter and tube diameter. Then: r = tube
diameter/2, R = (outer diameter/2) - r.
How accurate is this calculator?
Uses precise mathematical formulas with π² to full precision.
Assumes perfect circular cross-section.