Answer first
What this calculator tells you
Find the third side, area and perimeter of a triangle from two sides and the angle between them. Use it when you can measure two sides and the angle between them but cannot reach the third side. Formula: c² = a² + b² − 2ab·cos(C); area = ½ab·sin(C). At the worked-example inputs, the third side (c) is 8.2. Holding every other input steady, moving side a from 5.6 to 8.4 moves the result from 7.9 to 8.7.
Transparent method
The formula
Use it when you can measure two sides and the angle between them but cannot reach the third side.
Worked example
Example inputs
How to interpret the result
Two sides and the angle between them lock a triangle in place, so the third side is fixed and only needs the law of cosines to find it. When the angle is 90 degrees the cosine term vanishes and you get the Pythagorean theorem. Any other angle bends that result up or down. Sides of 7 and 9 meeting at 60 degrees close at about 8.19, shorter than the 11.4 a right angle would give.
At the worked-example inputs the third side (c) is 8.2. It rises with angle between them (degrees), side b and side a.
Formulas assume ideal geometric shapes. Real-world measurements carry their own error, and every input must use the same unit: mixing feet and inches silently produces a wrong answer.
Before you rely on it
What to check
The angle must sit between the two sides you entered. An angle elsewhere in the triangle needs the law of sines, not this calculation.
The common error
Where people go wrong with triangle third side calculator
Entering the angle in radians or mixing it with degrees. Sixty in the wrong mode gives a nonsense side length, so confirm the field is in degrees.
Sensitivity evidence
How side a changes the third side (c)
Holding every other input at the worked-example value, moving side a from 5.6 to 8.4 moves the third side (c) from 7.9 to 8.7: a spread of 0.844, or 10% of the worked-example result.
| Side a | Third side (c) | Triangle area | Perimeter |
|---|---|---|---|
| 5.6 | 7.9 | 21.8 | 22.5 |
| 6.3 | 8 | 24.6 | 23.3 |
| 7worked example | 8.2 | 27.3 | 24.2 |
| 7.7 | 8.4 | 30 | 25.1 |
| 8.4 | 8.7 | 32.7 | 26.1 |
Every input, tested
Which input moves the third side (c) most
Of the 3 inputs, angle between them (degrees) moves the third side (c) most (1.4 across the range tested) and side a moves it least (0.426).
| Input | Tested from | To | Third side (c) at each end | Swing |
|---|---|---|---|---|
| Angle between them (degrees) | 54 | 66 | 7.5 to 8.9 | 1.4 (17%) |
| Side b | 8 | 10 | 7.5 to 8.9 | 1.3 (16%) |
| Side a | 6.3 | 7.7 | 8 to 8.4 | 0.426 (5.2%) |
Two variables at once
Third side (c) by side a and side b
Across the grid the third side (c) runs from 6.4 to 10. Moving side a from 5.6 to 8.4 shifts it by 0.844 at the middle column, and moving side b from 7 to 11 shifts it by 2.6 at the middle row, so side b is the bigger lever here.
| Side a \ Side b | 7 | 9 | 11 |
|---|---|---|---|
| 5.6 | 6.4 | 7.9 | 9.5 |
| 6.3 | 6.7 | 8 | 9.6 |
| 7 | 7 | 8.2 | 9.6 |
| 7.7 | 7.4 | 8.4 | 9.8 |
| 8.4 | 7.8 | 8.7 | 10 |
The highlighted cell is the worked example: 8.2.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Side a | 7 | One side of the triangle. |
| Side b | 9 | A second side of the triangle. |
| Angle between them (degrees) | 60 | The angle formed where side a meets side b, from 0 to 180 degrees. |
| Third side (c) | 8.2 | |
| Triangle area | 27.3 | |
| Perimeter | 24.2 | |
Inputs, definitions and assumptions
Side a
One side of the triangle. The prefilled worked-example value is 7.
Side b
A second side of the triangle. The prefilled worked-example value is 9.
Angle between them (degrees)
The angle formed where side a meets side b, from 0 to 180 degrees. The prefilled worked-example value is 60.
How to use this calculator
- 1Verify the inputs. Gather side a, side b and angle between them (degrees) from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the third side (c) at 8.2. Store your own version of it as Scenario A.
- 3Test one change. Start with angle between them (degrees), the input with the biggest effect here: moving angle between them (degrees) from 54 to 66 takes the third side (c) from 7.5 to 8.9, a swing of 17% of the worked-example figure.
- 4Check the extremes. At half the example angle between them (degrees) (30) the third side (c) is 4.6; at double (120) it is 13.9.
People also ask
Frequently asked questions
How do you calculate triangle third side?
c² = a² + b² − 2ab·cos(C); area = ½ab·sin(C). At the worked-example inputs the third side (c) is 8.2.
What does the triangle third side result mean?
Use it when you can measure two sides and the angle between them but cannot reach the third side. At the worked-example inputs the third side (c) is 8.2. It rises with angle between them (degrees), side b and side a.
How much does side a change the third side (c)?
Holding every other input at the worked-example value, moving side a from 5.6 to 8.4 moves the third side (c) from 7.9 to 8.7, a spread of 0.844.
What are the limits of this triangle third side calculator?
Formulas assume ideal geometric shapes. Real-world measurements carry their own error, and every input must use the same unit: mixing feet and inches silently produces a wrong answer. The tables on this page test side a only from 5.6 to 8.4; a value outside that range is not tabulated here.
Which input moves the third side (c) most in the triangle third side calculator?
Ranked by how far each moves the third side (c) across the range tested: angle between them (degrees) (1.4, 17%), side b (1.3, 16%) and side a (0.426, 5.2%).
If I double angle between them (degrees) in the triangle third side calculator, does the third side (c) double?
Doubling it from 60 to 120 takes the third side (c) from 8.2 to 13.9, which is 1.70 times the worked-example figure. So it grows, but by less than double. Halving it to 30 gives 4.6.
How much does side b matter in the triangle third side calculator?
The worked example uses 9. Holding every other input at its worked-example value, moving side b from 8 to 10 takes the third side (c) from 7.5 to 8.9, a swing of 16% of the worked-example figure.
How much does angle between them (degrees) matter in the triangle third side calculator?
The worked example uses 60. With the other inputs left at the worked example, moving angle between them (degrees) from 54 to 66 takes the third side (c) from 7.5 to 8.9, a swing of 17% of the worked-example figure.
Which inputs change the triangle area in the triangle third side calculator?
At the worked-example inputs it is 27.3. Side a takes it from 24.6 to 30, side b takes it from 24.2 to 30.3 and angle between them (degrees) takes it from 25.5 to 28.8.
Which inputs change the perimeter in the triangle third side calculator?
At the worked-example inputs it is 24.2. Side a takes it from 23.3 to 25.1, side b takes it from 22.5 to 25.9 and angle between them (degrees) takes it from 23.5 to 24.9.
Why does volume grow so much faster than a shape's dimensions?
Volume scales with the cube of a linear dimension, so doubling a sphere's radius multiplies its volume by eight, not two. This is why a slightly bigger container often holds far more than intuition suggests.
Sources and evidence
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