Mathematics · Trigonometry
Polar to Cartesian Converter
Convert a radius and angle into x and y coordinates.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- x=5cos53.130102°=3.0000000247247516.
- y=5sin53.130102°=3.9999999814564364.
Understand Polar to Cartesian
One idea, three depths
Choose how deeply to explain Polar to Cartesian
Polar to Cartesian: Convert a radius and angle into x and y coordinates.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Polar to Cartesian to answer this question: convert a radius and angle into x and y coordinates? Enter Radius r and Angle θ; the calculator shows x coordinate. For example: r=5 at 53.13° is approximately (3,4). The answer tells you x coordinate.
Age 15Explain it to a 15-year-oldConnect it to the formula
Cartesian components are projections of a radial vector onto perpendicular axes. The rule is x=r cosθ; y=r sinθ. Its input values are Radius r, Angle θ (°), and the main result is x coordinate. For example: r=5 at 53.13° is approximately (3,4).
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated polar to cartesian relation over the valid real-number domain stated below. The implemented relation is x=r cosθ; y=r sinθ, evaluated from Radius r, Angle θ (°) to produce x coordinate. Cartesian components are projections of a radial vector onto perpendicular axes. This version expects the angle in degrees.
Inputs and valid domain
- Radius r must be a finite real number, at least 0.
- Angle θ must be a finite real number in °.
Important boundary: This version expects the angle in degrees.
The formula
x=r cosθ; y=r sinθ
How the calculator works through it
It substitutes Radius r, Angle θ into the formula and exposes every numerical step above. The main output is x coordinate, accompanied by y coordinate.
Read the result correctly
The x coordinate is the direct answer to “convert a radius and angle into x and y coordinates.” 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 at 53.13° is approximately (3,4).
Where this model stops being reliable
This version expects the angle in degrees.
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 Polar to Cartesian works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Polar to Cartesian uses x=r cosθ; y=r sinθ. 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 Polar to Cartesian.
Review this foundation about 5 min
Optional enrichment
- Functions and their graphs
Function graphs show how the Polar to Cartesian 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 Radius r, Angle θ.
- Evaluate the principal relationship: x=r cosθ; y=r sinθ.
- Return x coordinate and check the domain conditions described above.
Python
from math import *
def polar_to_cartesian(a, b) -> float:
return (a * cos(((b * pi) / 180.0)))
assert abs(polar_to_cartesian(5, 53.130102) - 3.0000000247247516) < 1e-6 * max(1.0, abs(3.0000000247247516))
C
#include <assert.h>
#include <math.h>
double polar_to_cartesian(double a, double b) {
return (a * cos(((b * 3.141592653589793) / 180.0)));
}
int main(void) {
const double expected = 3.0000000247247516;
const double actual = polar_to_cartesian(5, 53.130102);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double polar_to_cartesian(double a, double b) {
return (a * std::cos(((b * std::numbers::pi) / 180.0)));
}
int main() {
constexpr double expected = 3.0000000247247516;
const double actual = polar_to_cartesian(5, 53.130102);
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 polar_to_cartesian(double a, double b)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern cos
global polar_to_cartesian
section .text
polar_to_cartesian:
push rbp
mov rbp, rsp
sub rsp, 64
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-56], xmm0
movsd xmm0, [rbp-16]
mulsd xmm0, [rbp-56]
movsd [rbp-48], xmm0
mov rax, 0x4066800000000000
movq xmm0, rax
movsd [rbp-64], xmm0
movsd xmm0, [rbp-48]
divsd xmm0, [rbp-64]
movsd [rbp-40], xmm0
movsd xmm0, [rbp-40]
call cos wrt ..plt
movsd [rbp-32], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-32]
movsd [rbp-24], xmm0
movsd xmm0, [rbp-24]
leave
ret
MATLAB
function result = polar_to_cartesian(a, b)
result = (a * cos(((b * pi) / 180.0)));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_] := (a * Cos[((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). Polar to Cartesian Converter. MW SysArc Tools. https://math.mwsysarc.com/trigonometry/polar-to-cartesian
MLA 9
MW SysArc. “Polar to Cartesian Converter.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/trigonometry/polar-to-cartesian. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Polar to Cartesian Converter.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/trigonometry/polar-to-cartesian.
Harvard
MW SysArc (2026) ‘Polar to Cartesian Converter’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/trigonometry/polar-to-cartesian (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_polar_to_cartesian_2026,
author = {{MW SysArc}},
title = {Polar to Cartesian Converter},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/trigonometry/polar-to-cartesian},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Polar to Cartesian Converter
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/trigonometry/polar-to-cartesian
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Polar to Cartesian do?
Convert a radius and angle into x and y coordinates.
How does the Polar to Cartesian work?
The calculator applies x=r cosθ; y=r sinθ. Cartesian components are projections of a radial vector onto perpendicular axes.
What can I learn from the Polar to Cartesian?
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 .