🍩 Torus Volume Calculator
Calculate volume, surface area, and dimensions of donut-shaped objects
Enter Torus Dimensions
R = major, r = minor
(tube)
📊 Torus Results
Major Radius (R)
10 cm
Minor Radius (r)
3 cm
Outer Diameter
26 cm
Inner Diameter
14 cm
Volume
1,776.5 cm³
Surface Area
1,184.4 cm²
Band Width
6 cm
Tube Circumference
18.85 cm
📝 Step-by-Step Solution
Given: R = 10 cm (major radius), r = 3 cm (minor radius)
Torus volume formula: V = 2π²Rr²
V = 2 × π² × 10 × 3² = 2 × 9.8696 × 10 × 9
V = 1776.53 cm³ = 1.777 liters
Surface area: SA = 4π²Rr = 4 × π² × 10 × 3 = 1184.35 cm²
📐 Torus Formulas
Volume: V = 2π²Rr² = (π²/4)(D² - d²) × (D - d)/2
Surface Area: SA = 4π²Rr = π²(D² - d²)
Outer Diameter: D = 2(R + r)
Inner Diameter (hole): d = 2(R - r)
Band Width: w = 2r = (D - d)/2
Understanding Torus Geometry
🍩 What is a Torus?
Donut-shaped 3D surface. Created by rotating a circle around an external axis. Common in tires, O-rings, and donuts.
📏 Two Radii
R = major radius (center to tube center). r = minor radius (tube thickness). Both needed for volume.
⭕ Band Width
w = 2r. The width of the ring material. Also equals (D - d)/2, outer minus inner diameter divided by 2.
🔄 Pappus Connection
V = 2πR × πr². Circle area (πr²) times path traveled (2πR). Elegant application of Pappus theorem.
Frequently Asked Questions
What is the volume formula for a torus?
V = 2π²Rr². Where R is major radius (center to tube center), r is
minor radius (tube radius).
What is major vs minor radius?
Major (R) = center of torus to center of tube. Minor (r) = radius
of the tube itself. Think of R as the "ring" and r as the "thickness."
How do I find surface area?
SA = 4π²Rr. Four pi-squared times both radii. Alternative: SA =
π²(D² - d²) using diameters.
What is torus band width?
w = 2r = (D - d)/2. The thickness of the ring material, measured
as the difference of outer and inner radii.
How do I use outer and inner diameter?
R = (D + d)/4, r = (D - d)/4. Convert diameters to radii, then use
standard formulas.
What is a ring torus?
Standard torus where R > r. Has a hole in the center. If R = r,
it's a horn torus. If R < r, it's a spindle torus.
How is volume related to Pappus theorem?
V = Area × Path = πr² × 2πR. Circle area times circumference
traveled. Elegant proof of torus volume.
What are real-world applications?
Tires, O-rings, donuts, bagels, lifebuoys. Any ring-shaped object
with circular cross-section.
How do I calculate tire volume?
Find R and r from tire dimensions. R = wheel radius + tire
height/2. r = tire width/2 (approximately).
What if the hole disappears?
Not a standard torus. When R ≤ r, geometry changes. Formula still
works mathematically but shape is different.
How do I find major radius from volume?
R = V / (2π²r²). Rearrange the volume formula. Need volume and
minor radius.
How do I find minor radius from volume?
r = √(V / (2π²R)). Square root of volume divided by (2π²R). Need
volume and major radius.
What is the hole diameter?
d = 2(R - r). Inner diameter = 2 times (major radius minus minor
radius). Zero if R = r.
How do I convert to liters?
1 L = 1000 cm³. Our calculator can output in liters, gallons, or
other units automatically.
What is tube circumference?
C = 2πr. Circumference of the tube cross-section. Different from
the ring circumference (2πR).
Is a bagel a perfect torus?
Approximately, not exactly. Real bagels have irregular shapes.
Torus formula gives close approximation.
What is aspect ratio of a torus?
R/r ratio. Large ratio = thin ring. Small ratio = thick ring. R/r
= 1 is horn torus.