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Math • Torus (Donut) Volume Calculator

Torus (Donut) Volume Calculator

Calculate the volume and surface area of a torus (donut shape) from its major radius and minor (tube) radius.

Torus Volume (V = 2π²Rr²)
1776.529
Surface Area (A = 4π²Rr)
1184.353
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Calculation Guide & Reference

Torus (Donut) Volume & Surface Area Mathematics

Torus calculator computes the volume and surface area of a donut-shaped solid from its major radius and minor (tube) radius.

Standardized Mathematical Formula
V = 2π²Rr² | Surface Area = 4π²Rr

A torus is generated by revolving a circle of radius r (the tube) around an axis at distance R (the major radius) from the circle's center. Its volume and surface area formulas come from Pappus's centroid theorems for solids of revolution.

Variables:
R:Major radius — distance from the torus's center to the middle of the tube
r:Minor radius — radius of the tube itself
How It Works (Step-by-Step)
  • 1Enter the major radius (R, center to tube middle) and minor radius (r, tube thickness).
  • 2View the torus's volume and surface area.
Real-World Numerical Example
A Torus with Major Radius 10, Minor Radius 3

A donut-shaped ring has a major radius of 10 units and a tube (minor) radius of 3 units.

•Volume: 2 × π² × 10 × 3² = 2 × π² × 10 × 9 = 180π² ≈ 1,776.53 cubic units.
•Surface area: 4 × π² × 10 × 3 = 120π² ≈ 1,184.35 square units.
Result: The torus has a volume of approximately 1,776.53 cubic units and a surface area of about 1,184.35 square units.
Calculation Best Practices & Tips
Examine the discriminant (b² - 4ac) before solving quadratics to instantly know if roots are real, distinct, repeated, or complex.
In statistical analysis, always check both the mean and median to identify if data distribution is skewed by outliers.
When calculating percentage changes, ensure you divide by the original starting baseline value: ((New - Old) / Old) * 100.

Frequently Asked Questions (FAQ)

The major radius (R) is the distance from the center of the whole torus to the center of the tube; the minor radius (r) is the radius of the tube itself — think of R as the donut's overall size and r as its thickness.

Donuts, inner tubes, O-ring seals, and bagels are all approximately torus-shaped — the formulas here apply anywhere a tube is bent into a complete loop.

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