Torus Volume Calculator
Calculate volume and surface area of torus (doughnut shape) using radii
Calculate Torus Volume
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: Cross-section radius r = 40 mm, Revolution radius R = 100 mm
Convert to inner/outer: a = R-r = 60 mm, b = R+r = 140 mm
Volume: V = 2×π²×r²×R = 2×π²×40²×100 ≈ 3,158,273 mm³
Type: Ring type (R > r, since 100 > 40)
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
Volume: V = 2×π²×r²×R = 2×π²×1²×1 ≈ 19.74 m³
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
Volume
V = 2×π²×r²×R
Alternative Volume
V = 0.25×π²×(b-a)²×(b+a)
Surface Area
A = 4×π²×r×R
Radius Relations
r = (b-a)/2
R = (a+b)/2
Cross-section Area
A_cross = π×r²
r = minor radius, R = major radius
a = inner radius, b = outer radius
Understanding Torus Volume
What is a Torus?
A torus is a 3D shape formed by revolving a circle around an axis in 3D space. The resulting doughnut-like shape is commonly found in everyday objects such as bagels, tires, inner tubes, and various engineering components like O-rings and bearings.
Volume Calculation Methods
- •Inner & Outer Radii: Using distances from center to edges
- •Major & Minor Radii: Using revolution and cross-section radii
- •Alternative Formula: V = 0.25×π²×(b-a)²×(b+a)
Mathematical Derivation
Cross-section Method
V = A_cross × circumference
V = (π×r²) × (2π×R) = 2π²r²R
Pappus's Theorem
Volume = area × distance traveled by centroid
Centroid travels 2πR distance
Integral Calculus
Triple integration in cylindrical coordinates
∫∫∫ ρ dρ dφ dθ over torus region
Applications: Fluid dynamics (flow through pipes), mechanical engineering (O-rings, gaskets), architecture (decorative elements), and plasma physics (magnetic confinement).