Mathematics · Trigonometry
Law of Sines Side Calculator
Find a corresponding triangle side from one side and two angles.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- 10sin(45°)/sin(30°)=14.142135623730951.
- Report Target side b=14.142135623730951, Third angle=105.
Understand Sine law side
One idea, three depths
Choose how deeply to explain Sine law side
Sine law side: Find a corresponding triangle side from one side and two angles.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Sine law side to answer this question: find a corresponding triangle side from one side and two angles? Enter Known side a, Opposite angle A, Target angle B; the calculator shows Target side b. For example: a=10, A=30°, B=45° gives b≈14.142. The answer tells you Target side b.
Age 15Explain it to a 15-year-oldConnect it to the formula
Every side-to-opposite-sine ratio equals the triangle's circumdiameter. The rule is b=a sin B/sin A. Its input values are Known side a, Opposite angle A (°), Target angle B (°), and the main result is Target side b. For example: a=10, A=30°, B=45° gives b≈14.142.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated sine law side relation over the valid real-number domain stated below. The implemented relation is b=a sin B/sin A, evaluated from Known side a, Opposite angle A (°), Target angle B (°) to produce Target side b. Every side-to-opposite-sine ratio equals the triangle's circumdiameter. Match every side with its opposite angle; ambiguous SSA cases may have another triangle.
Inputs and valid domain
- Known side a must be a finite real number, at least 0.
- Opposite angle A must be a finite real number in °.
- Target angle B must be a finite real number in °.
Important boundary: Match every side with its opposite angle; ambiguous SSA cases may have another triangle.
The formula
b=a sin B/sin A
How the calculator works through it
It substitutes Known side a, Opposite angle A, Target angle B into the formula and exposes every numerical step above. The main output is Target side b, accompanied by Third angle.
Read the result correctly
The Target side b is the direct answer to “find a corresponding triangle side from one side and two angles.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
a=10, A=30°, B=45° gives b≈14.142.
Where this model stops being reliable
Match every side with its opposite angle; ambiguous SSA cases may have another triangle.
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 Sine 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
Sine law side uses b=a sin B/sin A. 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 Sine law side.
Review this foundation about 5 min
Optional enrichment
- Functions and their graphs
Function graphs show how the Sine 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 Known side a, Opposite angle A, Target angle B.
- Evaluate the principal relationship: b=a sin B/sin A.
- Return Target side b and check the domain conditions described above.
Python
from math import *
def law_sines_side(a, b, c) -> float:
return ((a * sin(((c * pi) / 180.0))) / sin(((b * pi) / 180.0)))
assert abs(law_sines_side(10, 30, 45) - 14.142135623730951) < 1e-6 * max(1.0, abs(14.142135623730951))
C
#include <assert.h>
#include <math.h>
double law_sines_side(double a, double b, double c) {
return ((a * sin(((c * 3.141592653589793) / 180.0))) / sin(((b * 3.141592653589793) / 180.0)));
}
int main(void) {
const double expected = 14.142135623730951;
const double actual = law_sines_side(10, 30, 45);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double law_sines_side(double a, double b, double c) {
return ((a * std::sin(((c * std::numbers::pi) / 180.0))) / std::sin(((b * std::numbers::pi) / 180.0)));
}
int main() {
constexpr double expected = 14.142135623730951;
const double actual = law_sines_side(10, 30, 45);
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_sines_side(double a, double b, double c)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern sin
global law_sines_side
section .text
law_sines_side:
push rbp
mov rbp, rsp
sub rsp, 128
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-72], xmm0
movsd xmm0, [rbp-24]
mulsd xmm0, [rbp-72]
movsd [rbp-64], xmm0
mov rax, 0x4066800000000000
movq xmm0, rax
movsd [rbp-80], xmm0
movsd xmm0, [rbp-64]
divsd xmm0, [rbp-80]
movsd [rbp-56], xmm0
movsd xmm0, [rbp-56]
call sin wrt ..plt
movsd [rbp-48], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-48]
movsd [rbp-40], xmm0
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-112], xmm0
movsd xmm0, [rbp-16]
mulsd xmm0, [rbp-112]
movsd [rbp-104], xmm0
mov rax, 0x4066800000000000
movq xmm0, rax
movsd [rbp-120], xmm0
movsd xmm0, [rbp-104]
divsd xmm0, [rbp-120]
movsd [rbp-96], xmm0
movsd xmm0, [rbp-96]
call sin wrt ..plt
movsd [rbp-88], xmm0
movsd xmm0, [rbp-40]
divsd xmm0, [rbp-88]
movsd [rbp-32], xmm0
movsd xmm0, [rbp-32]
leave
ret
MATLAB
function result = law_sines_side(a, b, c)
result = ((a * sin(((c * pi) / 180.0))) / sin(((b * pi) / 180.0)));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_, c_] := ((a * Sin[((c * Pi) / 180.0)]) / Sin[((b * 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 Sines Side Calculator. MW SysArc Tools. https://math.mwsysarc.com/trigonometry/law-of-sines-side
MLA 9
MW SysArc. “Law of Sines Side Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/trigonometry/law-of-sines-side. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Law of Sines Side Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/trigonometry/law-of-sines-side.
Harvard
MW SysArc (2026) ‘Law of Sines Side Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/trigonometry/law-of-sines-side (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_law_sines_side_2026,
author = {{MW SysArc}},
title = {Law of Sines Side Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/trigonometry/law-of-sines-side},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Law of Sines Side Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/trigonometry/law-of-sines-side
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Sine law side do?
Find a corresponding triangle side from one side and two angles.
How does the Sine law side work?
The calculator applies b=a sin B/sin A. Every side-to-opposite-sine ratio equals the triangle's circumdiameter.
What can I learn from the Sine 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 .