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Geometry

Law of Sines

Law of Sines calculator. Free online Law of Sines. Calculate law of sines online — fast, accurate, mobile-friendly, no signup needed.

a / sin A = b / sin B = c / sin C
Side a
7.071068

Derivation

  1. ├── 01Givenb = 10, A = 30, B = 45
  2. ├── 02Formulat × sin(a × π / 180) / sin(n × π / 180)
  3. └── 03Compute Side a7.071068
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§01What is

Understanding the Law of Sines

The Law of Sines computes Side a from 3 inputs: side b, angle a (°), angle b (°). Law of Sines calculator.

Geometry is what turns raw measurements into useful answers about space — how much paint, how big a yard, how much material a project will need. Every craftsperson, architect, and DIYer reaches for these formulas regularly. The Law of Sines sits in that toolkit — it law of Sines calculator. 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

t × sin(a × π / 180) / sin(n × π / 180)

Where

b
Side b
A
Angle A (°)
B
Angle B (°)
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: Side b = 10, Angle A (°) = 30, Angle B (°) = 45.

  1. 01Start by noting the input — Side b: 10.
  2. 02Start by noting the input — Angle A (°): 30.
  3. 03Start by noting the input — Angle B (°): 45.
  4. 04Substitute these values into the formula: t × sin(a × π / 180) / sin(n × π / 180)
  5. 05Compute Side a: the calculator returns 7.07107.
  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 Law of Sines 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

Side b halved

b = 5 (from 10)

Keep every other input at its default and halve the side b. See how side a responds.

  1. 01New Side b: 5
  2. 02Baseline Side a: 7.07107
  3. 03New Side a: 3.53553
  4. 04Side a decreases by 50% → use this sensitivity to plan for real-world variation.
02 · PATTERN

Side b doubled

b = 20 (from 10)

Keep every other input at its default and double the side b. See how side a responds.

  1. 01New Side b: 20
  2. 02Baseline Side a: 7.07107
  3. 03New Side a: 14.1421
  4. 04Side a increases by 100% → use this sensitivity to plan for real-world variation.
03 · PATTERN

Angle A (°) halved

A = 15 (from 30)

Keep every other input at its default and halve the angle a (°). See how side a responds.

  1. 01New Angle A (°): 15
  2. 02Baseline Side a: 7.07107
  3. 03New Side a: 3.66025
  4. 04Side a decreases by 48.2% → use this sensitivity to plan for real-world variation.
04 · PATTERN

Angle A (°) doubled

A = 60 (from 30)

Keep every other input at its default and double the angle a (°). See how side a responds.

  1. 01New Angle A (°): 60
  2. 02Baseline Side a: 7.07107
  3. 03New Side a: 12.2474
  4. 04Side a increases by 73.2% → 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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