Mathematics · Geometry
Torus Volume and Surface Area Calculator
Calculate the volume and surface area of a circular torus.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Change an input to reshape this diagram.
Calculation steps
- Volume=2π²×5×2²=394.78417604357435.
- Area=4π²×5×2=394.78417604357435.
Understand Torus geometry
One idea, three depths
Choose how deeply to explain Torus geometry
Torus geometry: Calculate the volume and surface area of a circular torus.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Torus geometry to answer this question: calculate the volume and surface area of a circular torus? Enter Major radius R and Minor radius r; the calculator shows Volume. For example: R=5 and r=2 gives volume 40π² and area 40π². The answer tells you Volume.
Age 15Explain it to a 15-year-oldConnect it to the formula
Pappus's centroid theorem sweeps a circular cross-section around the major axis. The rule is V=2π²Rr²; A=4π²Rr. Its input values are Major radius R, Minor radius r, and the main result is Volume. For example: R=5 and r=2 gives volume 40π² and area 40π².
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated torus geometry relation over the valid real-number domain stated below. The implemented relation is V=2π²Rr²; A=4π²Rr, evaluated from Major radius R, Minor radius r to produce Volume. Pappus's centroid theorem sweeps a circular cross-section around the major axis. The major radius runs from the torus center to the tube center, not to its outer edge.
Inputs and valid domain
- Major radius R must be a finite real number, at least 0.
- Minor radius r must be a finite real number, at least 0.
Important boundary: The major radius runs from the torus center to the tube center, not to its outer edge.
The formula
V=2π²Rr²; A=4π²Rr
How the calculator works through it
It substitutes Major radius R, Minor radius r into the formula and exposes every numerical step above. The main output is Volume, accompanied by Surface area, Outer radius, Inner radius.
Read the result correctly
The Volume is the direct answer to “calculate the volume and surface area of a circular torus.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
R=5 and r=2 gives volume 40π² and area 40π².
Where this model stops being reliable
The major radius runs from the torus center to the tube center, not to its outer edge.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Torus geometry works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Torus geometry uses V=2π²Rr²; A=4π²Rr. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Ratios between measured quantities
Ratios help you check the scale, units and proportional meaning of Torus geometry.
Review this foundation about 4 min
Optional enrichment
- Angles and geometric relationships
Angle language provides useful geometric context for extending Torus geometry to related shapes and constructions.
Review this foundation about 4 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Major radius R, Minor radius r.
- Evaluate the principal relationship: V=2π²Rr²; A=4π²Rr.
- Return Volume and check the domain conditions described above.
Python
from math import *
def torus_geometry(a, b) -> float:
return (2.0 * ((pi * pi) * (a * (b * b))))
assert abs(torus_geometry(5, 2) - 394.78417604357435) < 1e-6 * max(1.0, abs(394.78417604357435))
C
#include <assert.h>
#include <math.h>
double torus_geometry(double a, double b) {
return (2.0 * ((3.141592653589793 * 3.141592653589793) * (a * (b * b))));
}
int main(void) {
const double expected = 394.78417604357435;
const double actual = torus_geometry(5, 2);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double torus_geometry(double a, double b) {
return (2.0 * ((std::numbers::pi * std::numbers::pi) * (a * (b * b))));
}
int main() {
constexpr double expected = 394.78417604357435;
const double actual = torus_geometry(5, 2);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double torus_geometry(double a, double b)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global torus_geometry
section .text
torus_geometry:
push rbp
mov rbp, rsp
sub rsp, 80
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
mov rax, 0x4000000000000000
movq xmm0, rax
movsd [rbp-32], xmm0
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-56], xmm0
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-64], xmm0
movsd xmm0, [rbp-56]
mulsd xmm0, [rbp-64]
movsd [rbp-48], xmm0
movsd xmm0, [rbp-16]
mulsd xmm0, [rbp-16]
movsd [rbp-80], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-80]
movsd [rbp-72], xmm0
movsd xmm0, [rbp-48]
mulsd xmm0, [rbp-72]
movsd [rbp-40], xmm0
movsd xmm0, [rbp-32]
mulsd xmm0, [rbp-40]
movsd [rbp-24], xmm0
movsd xmm0, [rbp-24]
leave
ret
MATLAB
function result = torus_geometry(a, b)
result = (2.0 * ((pi * pi) * (a * (b * b))));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_] := (2.0 * ((Pi * Pi) * (a * (b * b))));
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Algebra and Trigonometry 2e
Read the related free OpenStax mathematics chaptersCite this book
- APA 7
- Abramson, J. (2021). Algebra and trigonometry 2e. OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
- MLA 9
- Abramson, Jay. Algebra and Trigonometry 2e. OpenStax, 2021, https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
- Chicago author-date
- Abramson, Jay. 2021. Algebra and Trigonometry 2e. Houston, TX: OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Torus Volume and Surface Area Calculator. MW SysArc Tools. https://math.mwsysarc.com/geometry/torus-volume-surface-area
MLA 9
MW SysArc. “Torus Volume and Surface Area Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/geometry/torus-volume-surface-area. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Torus Volume and Surface Area Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/geometry/torus-volume-surface-area.
Harvard
MW SysArc (2026) ‘Torus Volume and Surface Area Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/geometry/torus-volume-surface-area (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_torus_geometry_2026,
author = {{MW SysArc}},
title = {Torus Volume and Surface Area Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/geometry/torus-volume-surface-area},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Torus Volume and Surface Area Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/geometry/torus-volume-surface-area
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Torus geometry do?
Calculate the volume and surface area of a circular torus.
How does the Torus geometry work?
The calculator applies V=2π²Rr²; A=4π²Rr. Pappus's centroid theorem sweeps a circular cross-section around the major axis.
What can I learn from the Torus geometry?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .