Torus Surface Area Calculator
Calculate surface area and volume of torus (doughnut shape) using radii
Calculate Torus Surface Area
Distance from center to inner edge of torus
Distance from center to outer edge of torus
Results
Calculated Parameters
Minor radius (r): 0.000000 m
Major radius (R): 0.000000 m
Inner radius (a): 0 m
Outer radius (b): 0 m
Example Calculations
Ring Type Torus Example
Given: Inner radius a = 2 m, Outer radius b = 6 m
Calculate: r = (6-2)/2 = 2 m, R = (2+6)/2 = 4 m
Surface Area: A = 4×π²×r×R = 4×π²×2×4 ≈ 315.83 m²
Type: Ring type (R > r, since 4 > 2)
Horn Type Torus Example
Given: Inner radius a = 0 m, Outer radius b = 2 m
Calculate: r = (2-0)/2 = 1 m, R = (0+2)/2 = 1 m
Surface Area: A = 4×π²×r×R = 4×π²×1×1 ≈ 39.48 m²
Type: Horn type (R = r, since 1 = 1)
Torus Types
Ring Type
R > r (Most common)
Like a doughnut or tire
Horn Type
R = r (Special case)
Inner radius becomes zero
Spindle Type
R < r (Not supported)
Self-intersecting surface
Key Formulas
Surface Area
A = 4×π²×r×R
Volume
V = 2×π²×r²×R
Radius Relations
r = (b-a)/2
R = (a+b)/2
Alternative Form
A = π²×(b-a)×(b+a)
r = minor radius, R = major radius
a = inner radius, b = outer radius
Understanding Torus Geometry
What is a Torus?
A torus is a 3D shape obtained by revolving a circle around an axis in 3D space. This creates a doughnut-like shape commonly seen in everyday objects like bagels, tires, life rings, and various engineering components.
Radius Definitions
- •Minor radius (r): Radius of the circular cross-section
- •Major radius (R): Distance from center axis to tube center
- •Inner radius (a): Distance from center to inner edge
- •Outer radius (b): Distance from center to outer edge
Mathematical Properties
Surface Area Formula
A = 4×π²×r×R
Product of both circumferences
Volume Formula
V = 2×π²×r²×R
Cross-section area × revolution path
Coordinate System
Toroidal and poloidal directions
Two-parameter surface mapping
Applications: Engineering (pipes, bearings), architecture, computer graphics, topology, and plasma physics (tokamak reactors).