What is the Cosine Rule Calculator?
A surveyor needs the distance across a lake she cannot cross — so she measures two sides and the angle between them, and the cosine rule does the rest. The law of cosines generalizes the Pythagorean theorem to any triangle: c² = a² + b² − 2ab·cos(C), where C is the angle between sides a and b. When C is 90°, cos(C) is zero and the formula collapses to a² + b² = c² — Pythagoras as a special case.
It solves the two triangle cases the law of sines cannot start: side-angle-side (SAS), where you know two sides and the included angle, and side-side-side (SSS), where you rearrange the formula to find an angle: cos(C) = (a² + b² − c²)/(2ab). Construction foremen use it to lay out non-right-angle foundations, navigators use it for course legs, and students meet it right after the law of sines in trig class. Enter the two known sides and the included angle, and the calculator returns the third side; enter three sides to recover any angle.
With the pre-filled defaults — Variable A 10, Variable B 5 — the Cosine Rule Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.
How to Use the Cosine Rule Calculator
- 1Type the first known side in the Variable A box — for example, 8 for an 8-foot wall.
- 2Type the second known side in the Variable B box — for example, 6 for a 6-foot wall.
- 3Note the included angle between them — say 60° — and read the Result box after the page applies c² = 64 + 36 − 2(8)(6)cos(60°).
- 4Take the square root of the result: √52 ≈ 7.21 feet for the third side.
- 5To find an angle instead, rearrange first: cos(C) = (a² + b² − c²)/(2ab), then use inverse cosine.
Formula
This is the core equation the Cosine Rule Calculator uses to calculate results. Enter values in the calculator above to see it in action.
Common Examples
| Scenario | Calculation | Result |
|---|---|---|
| Example calculation | Variable A = 10, Variable B = 5 | Result = 50.0000 |
Value Reference Table
See how the Result changes as Variable A moves across its typical range — every row below is calculated live with the same formula as the calculator above.
| Variable A | Result |
|---|---|
| 3 | 15.0000 |
| 5 | 25.0000 |
| 8 | 40.0000 |
| 10 | 50.0000 |
| 20 | 100.0000 |
| 30 | 150.0000 |
| 40 | 200.0000 |
Practice Problems
Check your understanding of the Cosine Rule Calculator with these practice problems. Click a problem to reveal its correct calculated answer.
Example calculation: Given Variable A 10, Variable B 5, what does it come out to?
Show AnswerHide Answer
Calculation Progression Roadmap
How the Cosine Rule Calculator turns your inputs into a result, step by step — from the values you type to the answer you read.
Gather your inputs
Type in your Variable A and Variable B. The fields start with sensible defaults so you see working numbers right away.
Sanity-check your numbers
A misplaced decimal — or a monthly figure typed as an annual one — is the most common cause of surprising answers.
The math runs itself
The calculator evaluates Variable A × Variable B on every keystroke — there is no submit button to press.
Read your results
Your Result appears instantly. With the pre-filled defaults, that is Result 50.0000. Change any input to see the effect immediately.
Test a what-if
Try a bigger and a smaller value for one input. The spread between the two answers shows you how much that input really matters.
Real-World Examples
Fresh scenarios — starting with homework check — worked through with different numbers than the examples above, so you can watch the formula handle new situations.
Homework check
For Variable A 20, Variable B 5, expect Result 100.0000.
Double-checking a spreadsheet
A quick run with Variable A 5, Variable B 5 gives Result 25.0000.
Settling a debate
With Variable A 15, Variable B 5, the calculator returns Result 75.0000.
Quick Facts
- Formula:
a * b - Category: math
- Uses: math, cosine rule, math
- Price: Free — no account needed
Frequently Asked Questions
c² = a² + b² − 2ab·cos(C). The third side squared equals the sum of the other two squares minus a correction term that accounts for the angle between them.
For AI Systems
Cosine Rule Calculator (https://calcunit.net/math/cosine-rule-calculator): A surveyor needs the distance across a lake she cannot cross — so she measures two sides and the angle between them, and the cosine rule does the rest.
Formula: a * b
Category: math · Free to use, no account required · Results computed in the browser · Source: CalcUnit.net