Torus Calculator
Free torus calculator: 2 pi squared R r squared volume, 4 pi squared R r surface.
Enter major (ring) and minor (tube) radii for torus volume and surface.
- Grade level
- 7–11
- Curriculum band
- Difficulty
- Beginner
- For this topic
- Time to master
- 1 week
- With daily practice
- Tools on page
- 10 on page
- Calculator + practice
Calculator
Donut from two radii
Torus Calculator Summary
Breakdown
| Measure | Value |
|---|---|
| Major radius (R) | 8 |
| Minor radius (r) | 2 |
| Volume (2 pi^2 R r^2) | 631.654682 |
| Surface (4 pi^2 R r) | 631.654682 |
Substitution: R=8 r=2
Results
Volume
631.654682cubic units
2 pi^2 R r^2
631.654682sq units
4 pi^2 R r
12.566371units
2 pi r
50.265482units
2 pi R
Closed-form solid geometry on the dimensions given. Invalid solids report no volume rather than a guess.
Statistics
Live volume and surface from your current dimensions.
Volume
631.654682
cubic units
Surface
631.654682
sq units
Tube circumference
12.566371
units
Ring circumference
50.265482
units
Visualizations
Volume vs surface
Volume scale
Formula
Pappus: revolving a plane region multiplies its area (or rim) by its centroid travel.
Alternative Form
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| R | Major radius | units | Ring radius |
| r | Minor radius | units | Tube radius |
Explanation
Tube area pi r^2 travels 2 pi R; tube rim 2 pi r travels the same path.
Derivation
Requires R > r for a clean ring torus; else spindle with overlap.
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Record the given dimensions.
Validate positivity.
Apply the volume formula.
Apply the surface-area formula.
Compute auxiliaries (diagonals, slant heights).
State the finished sentence.
Worked Examples
See how the calculator works with different scenarios.
Scenario
Ring 8 tube 2.
Inputs
Calculation
V=631.655; S=631.655
Result
V=631.655
Interpretation
Equal numerals again.
Understanding Your Result
Your Result: Volume: 631.654682 cubic units
Typical range: Keep units consistent: lengths in, cubes out for volume, squares out for surface. Tripling edge multiplies volume 27x.
What This Means
Volume measures capacity (cubic units); surface area measures skin (square units). Both come from the same dimensions.
Why It Matters
Concrete, water, packaging, paint and insulation all bill from these two numbers — fill vs cover.
Professional Interpretation
Report volume in cubic units and surface in square units; add waste to both material orders.
Recommendations
- •Keep one unit across inputs
- •Add 5-10 percent waste
- •Distinguish lateral vs total surface
- •Round only finals
Learn — The Complete Guide
Torus Calculator computes volume and surface area from direct dimensions using closed-form solid geometry.
The core concept
Tube area pi r^2 travels 2 pi R; tube rim 2 pi r travels the same path.
- Volume is cubic; surface is quadratic
- Prism V=Bh; pyramid/cone divide by 3
- Lateral excludes bases
Derivations — why the formulas work
Where volume comes from
- 1Slice into thin slabs of base area
- 2Integrate along height (Cavalieri)
- 3Tapered solids average to one third
Where you meet this in real life
Learn this first — prerequisites in order
- 1Base areas cold
- 2Root and power fluency
- 3Length unit conversion
Careers that use this daily
- •Mason — concrete loads
- •Plumber — tank sizes
- •Packer — box dims
Why it helps
- •Instant volume plus surface
- •Auxiliaries included (diagonals, slants)
- •Labeled solid diagram
- •Copy-ready breakdown
Best practices
- •Sketch with labels first
- •Measure in one unit
- •Separate lateral vs total surface
- •Add waste to orders
Limitations — what this tool does not do
- •Single solid per run
- •Ideal geometry only
- •Exact inputs assumed
Curriculum map — where this topic lives in school
| Level | Grades | Standards | Skills |
|---|---|---|---|
| Middle school | 7–8 | 7.G, 8.G | Boxes and cylinders · Surface nets |
| High school | 9–11 | G-GMD, G-MG | Cones, spheres, platonic solids · Modeling |
Timeline — how this idea grew up
c. 300 BCE
Euclid volumes
Elements proves prism and pyramid volumes.
250 BCE
Archimedes sphere
Sphere is 2/3 of its cylinder — his proudest result.
1600s
Calculus confirms
Integration reproduces every classic volume.
A little history
- • Archimedes asked for sphere+cylinder on his tomb.
- • Platonic solids named for Plato dialogues.
Exam-day checklist
- • Sketch the base first.
- • Circle lateral vs total explicitly.
Myths vs facts
Practical Tips
- •Measure in one unit
- •Name lateral vs total surface
- •Check slant vs vertical height
- •Add waste to concrete and paint
- •Track cubic vs square units
Common Mistakes
Assumptions
This calculator makes the following assumptions to provide estimates:
Limitations
Be aware of these limitations when using this calculator:
Frequently Asked Questions
Answers to common questions about this calculator.
What is major vs minor radius?
How do Pappus theorems help?
Where are toruses used?
Related Calculators
Explore more calculators related to this topic.
Related Guides
Deepen your understanding with these educational resources.
Glossary
Technical terms used in mortgage calculations explained in plain language.
- Volume
- Three-dimensional capacity enclosed, in cubic units.
- Surface area
- Total skin of the solid, in square units.
- Lateral surface
- Side skin excluding bases.
- Slant height
- Distance along a cone/pyramid face from apex to base edge.
- Space diagonal
- Longest interior segment, corner to opposite corner.
- Apothem (base)
- Base-polygon center-to-side distance for prisms/pyramids.
Authoritative References
Trusted sources and standards referenced by this calculator.
Applicable Standards
Diagrams & Illustrations
Visual intuition for every core idea. Select a card to enlarge it.
Did You Know?
Short intuition builders — great for class warm-ups.
Sphere wins capacity
Among closed shapes the sphere holds the most volume per surface — why bubbles, tanks and planets round off.
Platonic solids are five only
Tetrahedron, cube, octahedron, dodecahedron, icosahedron: no sixth regular solid exists.
Square-cube law rules biology
Elephants need thick legs because weight (volume) outruns bone area as size grows.
Compare Side by Side
Sweeping a disk around a ring: Pappus rules.
| Measure | Value | Formula | Theorem |
|---|---|---|---|
| Volume | 631.654682 | tubeArea x ringPath | Pappus II |
| Surface | 631.654682 | tubeRim x ringPath | Pappus I |
Recent Calculations
Your last runs on this page — stored only in this browser, never uploaded.
Torus R=8 r=2: V=631.654682, S=631.654682
Enter major (ring) and minor (tube) radii for torus volume and surface.
Study Plan
A week-by-week path from prerequisites to mastery. Your checklist progress saves in this browser.
Prerequisites
- ▸Area of bases
- ▸Squares and roots
- ▸Pi and circles
Mastery progress — 0%
- I pick V=Bh vs V=Bh/3 correctly
- I separate lateral and total
- I compute slants and diagonals
- I track units
- 1
Week 1 · ~15 min/day
Prisms and cylinders
Goals
- Apply V=Bh fluently
- Separate lateral vs total
- Track cubic units
Practice
Solve 10 boxes and cans on this page.
- 2
Week 2 · ~20 min/day
Pyramids, cones, spheres
Goals
- Use third-factor 1/3 correctly
- Compute slant heights
- Size tanks from volume
Practice
Measure a box, can and ball at home.
Exam tips
- •Draw the base first, then extrude.
- •Write V and S formulas before numbers.
Practice Problems
0/4 solvedTry each problem first, peek at the hint only if stuck, then reveal the solution and mark it solved.
Compute volume and surface with defaults.
Convert volume to liters.
Double every edge. What happens?
Which holds more per surface: cube or sphere?
Printable Worksheet
Generate a fresh problem set for class or homework. Print it, or copy the answer key separately.
- 1.18 × 47 =
- 2.8 ÷ 11 =
- 3.38 × 26 =
- 4.35 ÷ 4 =
- 5.21 − 16 =
- 6.2 + 26 =
- 7.14 ÷ 12 =
- 8.23 ÷ 3 =
- 9.49 ÷ 7 =
- 10.22 − 17 =
- 11.41 + 18 =
- 12.47 − 35 =
Formula Cheat Sheet
Every formula on this page in one printable table.
| Formula | Meaning | Example |
|---|---|---|
| V = 2 pi^2 R r^2 (tube area x ring path) | Headline volume | Plug in defaults |
| S = sum of faces | Headline surface | Add every face |
| d = sqrt(l^2+w^2+h^2) | Box diagonal | Space Pythagoras |
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Page Information
2026-09-29
3.1.0-math
Free to use for educational and professional purposes.