Mathematics · Geometry
Polygon Interior Angle Calculator
Calculate the interior-angle sum and regular interior angle.
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
- (6−2)×180=720°.
- Regular angle=720÷6=120°.
Understand Polygon interior angles
One idea, three depths
Choose how deeply to explain Polygon interior angles
Polygon interior angles: Calculate the interior-angle sum and regular interior angle.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Polygon interior angles to answer this question: calculate the interior-angle sum and regular interior angle? Enter Number of sides n; the calculator shows Interior angle sum. For example: A regular hexagon has 720° total and 120° per angle. The answer tells you Interior angle sum.
Age 15Explain it to a 15-year-oldConnect it to the formula
Triangulating an n-gon from one vertex produces n−2 triangles. The rule is S=(n−2)180°; α=S/n. Its input values are Number of sides n, and the main result is Interior angle sum. For example: A regular hexagon has 720° total and 120° per angle.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated polygon interior angles relation over the valid integer domain stated below. The implemented relation is S=(n−2)180°; α=S/n, evaluated from Number of sides n to produce Interior angle sum. Triangulating an n-gon from one vertex produces n−2 triangles. The single-angle result assumes a regular polygon.
Inputs and valid domain
- Number of sides n must be an integer, at least 3.
Important boundary: The single-angle result assumes a regular polygon.
The formula
S=(n−2)180°; α=S/n
How the calculator works through it
It substitutes Number of sides n into the formula and exposes every numerical step above. The main output is Interior angle sum, accompanied by Each regular interior angle, Each exterior angle.
Read the result correctly
The Interior angle sum is the direct answer to “calculate the interior-angle sum and regular interior angle.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
A regular hexagon has 720° total and 120° per angle.
Where this model stops being reliable
The single-angle result assumes a regular polygon.
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 Polygon interior angles works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Polygon interior angles uses S=(n−2)180°; α=S/n. 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 Polygon interior angles.
Review this foundation about 4 min
Optional enrichment
- Angles and geometric relationships
Angle language provides useful geometric context for extending Polygon interior angles 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 Number of sides n.
- Evaluate the principal relationship: S=(n−2)180°; α=S/n.
- Return Interior angle sum and check the domain conditions described above.
Python
from math import *
def polygon_interior_angle(n) -> float:
return ((n - 2.0) * 180.0)
assert abs(polygon_interior_angle(6) - 720) < 1e-6 * max(1.0, abs(720))
C
#include <assert.h>
#include <math.h>
double polygon_interior_angle(double n) {
return ((n - 2.0) * 180.0);
}
int main(void) {
const double expected = 720;
const double actual = polygon_interior_angle(6);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double polygon_interior_angle(double n) {
return ((n - 2.0) * 180.0);
}
int main() {
constexpr double expected = 720;
const double actual = polygon_interior_angle(6);
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 polygon_interior_angle(double n)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global polygon_interior_angle
section .text
polygon_interior_angle:
push rbp
mov rbp, rsp
sub rsp, 48
movsd [rbp-8], xmm0
mov rax, 0x4000000000000000
movq xmm0, rax
movsd [rbp-32], xmm0
movsd xmm0, [rbp-8]
subsd xmm0, [rbp-32]
movsd [rbp-24], xmm0
mov rax, 0x4066800000000000
movq xmm0, rax
movsd [rbp-40], xmm0
movsd xmm0, [rbp-24]
mulsd xmm0, [rbp-40]
movsd [rbp-16], xmm0
movsd xmm0, [rbp-16]
leave
ret
MATLAB
function result = polygon_interior_angle(n)
result = ((n - 2.0) * 180.0);
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[n_] := ((n - 2.0) * 180.0);
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). Polygon Interior Angle Calculator. MW SysArc Tools. https://math.mwsysarc.com/geometry/polygon-interior-angles
MLA 9
MW SysArc. “Polygon Interior Angle Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/geometry/polygon-interior-angles. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Polygon Interior Angle Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/geometry/polygon-interior-angles.
Harvard
MW SysArc (2026) ‘Polygon Interior Angle Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/geometry/polygon-interior-angles (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_polygon_interior_angle_2026,
author = {{MW SysArc}},
title = {Polygon Interior Angle Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/geometry/polygon-interior-angles},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Polygon Interior Angle Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/geometry/polygon-interior-angles
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Polygon interior angles do?
Calculate the interior-angle sum and regular interior angle.
How does the Polygon interior angles work?
The calculator applies S=(n−2)180°; α=S/n. Triangulating an n-gon from one vertex produces n−2 triangles.
What can I learn from the Polygon interior angles?
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 .