Mathematics · Trigonometry
Law of Cosines Side Calculator
Find a triangle side from two sides and their included angle.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- c=√(3²+4²−2×3×4cos90°)=5.
- Report Opposite side c=5.
Understand Cosine law side
One idea, three depths
Choose how deeply to explain Cosine law side
Cosine law side: Find a triangle side from two sides and their included angle.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Cosine law side to answer this question: find a triangle side from two sides and their included angle? Enter Side a, Side b, Included angle C; the calculator shows Opposite side c. For example: Sides 3 and 4 with 90° between them give side 5. The answer tells you Opposite side c.
Age 15Explain it to a 15-year-oldConnect it to the formula
The cosine law generalizes the Pythagorean theorem to non-right triangles. The rule is c=√(a²+b²−2ab cos C). Its input values are Side a, Side b, Included angle C (°), and the main result is Opposite side c. For example: Sides 3 and 4 with 90° between them give side 5.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated cosine law side relation over the valid real-number domain stated below. The implemented relation is c=√(a²+b²−2ab cos C), evaluated from Side a, Side b, Included angle C (°) to produce Opposite side c. The cosine law generalizes the Pythagorean theorem to non-right triangles. Use the angle between the two entered sides.
Inputs and valid domain
- Side a must be a finite real number, at least 0.
- Side b must be a finite real number, at least 0.
- Included angle C must be a finite real number in °.
Important boundary: Use the angle between the two entered sides.
The formula
c=√(a²+b²−2ab cos C)
How the calculator works through it
It substitutes Side a, Side b, Included angle C into the formula and exposes every numerical step above. The main output is Opposite side c.
Read the result correctly
The Opposite side c is the direct answer to “find a triangle side from two sides and their included angle.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
Sides 3 and 4 with 90° between them give side 5.
Where this model stops being reliable
Use the angle between the two entered sides.
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 Cosine law side works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Cosine law side uses c=√(a²+b²−2ab cos C). 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
- Angles in degrees and radians
Interpreting the angle convention is essential for understanding the inputs and output of Cosine law side.
Review this foundation about 5 min
Optional enrichment
- Functions and their graphs
Function graphs show how the Cosine law side relationship changes across a full angle or period.
Review this foundation about 6 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 Side a, Side b, Included angle C.
- Evaluate the principal relationship: c=√(a²+b²−2ab cos C).
- Return Opposite side c and check the domain conditions described above.
Python
from math import *
def law_cosines_side(a, b, c) -> float:
return sqrt((((a * a) + (b * b)) - (2.0 * ((a * b) * cos(((c * pi) / 180.0))))))
assert abs(law_cosines_side(3, 4, 90) - 5) < 1e-6 * max(1.0, abs(5))
C
#include <assert.h>
#include <math.h>
double law_cosines_side(double a, double b, double c) {
return sqrt((((a * a) + (b * b)) - (2.0 * ((a * b) * cos(((c * 3.141592653589793) / 180.0))))));
}
int main(void) {
const double expected = 5;
const double actual = law_cosines_side(3, 4, 90);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double law_cosines_side(double a, double b, double c) {
return std::sqrt((((a * a) + (b * b)) - (2.0 * ((a * b) * std::cos(((c * std::numbers::pi) / 180.0))))));
}
int main() {
constexpr double expected = 5;
const double actual = law_cosines_side(3, 4, 90);
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 law_cosines_side(double a, double b, double c)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern cos
global law_cosines_side
section .text
law_cosines_side:
push rbp
mov rbp, rsp
sub rsp, 144
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-8]
movsd [rbp-56], xmm0
movsd xmm0, [rbp-16]
mulsd xmm0, [rbp-16]
movsd [rbp-64], xmm0
movsd xmm0, [rbp-56]
addsd xmm0, [rbp-64]
movsd [rbp-48], xmm0
mov rax, 0x4000000000000000
movq xmm0, rax
movsd [rbp-80], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-16]
movsd [rbp-96], xmm0
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-128], xmm0
movsd xmm0, [rbp-24]
mulsd xmm0, [rbp-128]
movsd [rbp-120], xmm0
mov rax, 0x4066800000000000
movq xmm0, rax
movsd [rbp-136], xmm0
movsd xmm0, [rbp-120]
divsd xmm0, [rbp-136]
movsd [rbp-112], xmm0
movsd xmm0, [rbp-112]
call cos wrt ..plt
movsd [rbp-104], xmm0
movsd xmm0, [rbp-96]
mulsd xmm0, [rbp-104]
movsd [rbp-88], xmm0
movsd xmm0, [rbp-80]
mulsd xmm0, [rbp-88]
movsd [rbp-72], xmm0
movsd xmm0, [rbp-48]
subsd xmm0, [rbp-72]
movsd [rbp-40], xmm0
sqrtsd xmm0, [rbp-40]
movsd [rbp-32], xmm0
movsd xmm0, [rbp-32]
leave
ret
MATLAB
function result = law_cosines_side(a, b, c)
result = sqrt((((a * a) + (b * b)) - (2.0 * ((a * b) * cos(((c * pi) / 180.0))))));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_, c_] := Sqrt[(((a * a) + (b * b)) - (2.0 * ((a * b) * Cos[((c * Pi) / 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). Law of Cosines Side Calculator. MW SysArc Tools. https://math.mwsysarc.com/trigonometry/law-of-cosines-side
MLA 9
MW SysArc. “Law of Cosines Side Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/trigonometry/law-of-cosines-side. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Law of Cosines Side Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/trigonometry/law-of-cosines-side.
Harvard
MW SysArc (2026) ‘Law of Cosines Side Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/trigonometry/law-of-cosines-side (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_law_cosines_side_2026,
author = {{MW SysArc}},
title = {Law of Cosines Side Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/trigonometry/law-of-cosines-side},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Law of Cosines Side Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/trigonometry/law-of-cosines-side
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Cosine law side do?
Find a triangle side from two sides and their included angle.
How does the Cosine law side work?
The calculator applies c=√(a²+b²−2ab cos C). The cosine law generalizes the Pythagorean theorem to non-right triangles.
What can I learn from the Cosine law side?
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 .