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Algebra

Polar to Cartesian

Convert polar (r, θ) to (x, y). Free online Polar to Cartesian. Calculate polar to cartesian online — fast, accurate, mobile-friendly, no signup needed.

x
4.330127
y
2.5

Derivation

  1. ├── 01Givenr = 5, theta = 30
  2. ├── 02Formulax: t × cos(a × π / 180)
  3. ├── 03Compute x4.330127
  4. ├── 04Formulay: t × sin(a × π / 180)
  5. └── 05Compute y2.5
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§01What is

Understanding the Polar to Cartesian

The Polar to Cartesian computes x from 2 inputs: r, θ (°). Convert polar (r, θ) to (x, y).

Algebra is the art of solving for the unknown. Rearranging a formula to isolate the variable you actually need is the single most common real-world math skill — and doing it with real numbers under time pressure is where errors happen. The Polar to Cartesian sits in that toolkit — it convert polar (r, θ) to (x, y). Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

x = t × cos(a × π / 180) | y = t × sin(a × π / 180)

Where

r
r
theta
θ (°)
x
Output value
y
Output value
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: r = 5, θ (°) = 30.

  1. 01Start by noting the input — r: 5.
  2. 02Start by noting the input — θ (°): 30.
  3. 03Substitute these values into the formula: x = t × cos(a × π / 180) | y = t × sin(a × π / 180)
  4. 04Compute x: the calculator returns 4.33013.
  5. 05Compute y: the calculator returns 2.5.
  6. 06Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Polar to Cartesian Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

r halved

r = 2.5 (from 5)

Keep every other input at its default and halve the r. See how x responds.

  1. 01New r: 2.5
  2. 02Baseline x: 4.33013
  3. 03New x: 2.16506
  4. 04x decreases by 50% → use this sensitivity to plan for real-world variation.
02 · PATTERN

r doubled

r = 10 (from 5)

Keep every other input at its default and double the r. See how x responds.

  1. 01New r: 10
  2. 02Baseline x: 4.33013
  3. 03New x: 8.66025
  4. 04x increases by 100% → use this sensitivity to plan for real-world variation.
03 · PATTERN

θ (°) halved

theta = 15 (from 30)

Keep every other input at its default and halve the θ (°). See how x responds.

  1. 01New θ (°): 15
  2. 02Baseline x: 4.33013
  3. 03New x: 4.82963
  4. 04x increases by 11.5% → use this sensitivity to plan for real-world variation.
04 · PATTERN

θ (°) doubled

theta = 60 (from 30)

Keep every other input at its default and double the θ (°). See how x responds.

  1. 01New θ (°): 60
  2. 02Baseline x: 4.33013
  3. 03New x: 2.5
  4. 04x decreases by 42.3% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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