Answer first
What this calculator tells you
Find the two unknown sides and the third angle of a triangle from two angles and one side. Solve a triangle when you know two angles and the side opposite one of them. Formula: a ÷ sin A = b ÷ sin B = c ÷ sin C, with C = 180° − A − B. At the worked-example inputs, the side b is 10.8. Holding every other input steady, moving angle a (degrees) from 32 to 48 moves the result from 9.3 to 13.1.
Transparent method
The formula
Solve a triangle when you know two angles and the side opposite one of them.
Worked example
Example inputs
How to interpret the result
In any triangle the ratio of a side to the sine of its opposite angle is the same for all three sides. Knowing two angles gives you the third, since they total 180, and one side sets the scale. With angles of 40 and 60 degrees and a side of 8 opposite the 40, the other sides come to about 10.78 and 12.26, and the third angle is 80.
At the worked-example inputs the side b is 10.8. It rises with side a (opposite angle a) and angle b (degrees) and falls as angle a (degrees) increases.
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
Match each side to its opposite angle. Side a sits across from angle A, and swapping them scrambles the whole ratio.
The common error
Where people go wrong with law of sines calculator
Entering two angles that add to 180 or more. The third angle would be zero or negative, so no triangle exists, and the page shows a dash instead of a number.
Sensitivity evidence
How angle a (degrees) changes the side b
Holding every other input at the worked-example value, moving angle a (degrees) from 32 to 48 moves the side b from 9.3 to 13.1: a spread of 3.8, or 35% of the worked-example result.
| Angle A (degrees) | Side b | Side c | Angle C (degrees) |
|---|---|---|---|
| 32 | 13.1 | 15.1 | 88 |
| 36 | 11.8 | 13.5 | 84 |
| 40worked example | 10.8 | 12.3 | 80 |
| 44 | 10 | 11.2 | 76 |
| 48 | 9.3 | 10.2 | 72 |
Every input, tested
Which input moves the side b most
Of the 3 inputs, side a (opposite angle a) moves the side b most (2.7 across the range tested) and angle b (degrees) moves it least (1.3).
| Input | Tested from | To | Side b at each end | Swing |
|---|---|---|---|---|
| Side a (opposite angle A) | 7 | 9 | 9.4 to 12.1 | 2.7 (25%) |
| Angle A (degrees) | 36 | 44 | 11.8 to 10 | 1.8 (17%) |
| Angle B (degrees) | 54 | 66 | 10.1 to 11.4 | 1.3 (12%) |
Two variables at once
Side b by angle a (degrees) and angle b (degrees)
Across the grid the side b runs from 8 to 14.4. Moving angle a (degrees) from 32 to 48 shifts it by 3.8 at the middle column, and moving angle b (degrees) from 48 to 72 shifts it by 2.6 at the middle row, so angle a (degrees) is the bigger lever here.
| Angle A (degrees) \ Angle B (degrees) | 48 | 60 | 72 |
|---|---|---|---|
| 32 | 11.2 | 13.1 | 14.4 |
| 36 | 10.1 | 11.8 | 12.9 |
| 40 | 9.2 | 10.8 | 11.8 |
| 44 | 8.6 | 10 | 11 |
| 48 | 8 | 9.3 | 10.2 |
The highlighted cell is the worked example: 10.8.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Angle A (degrees) | 40 | Enter the angle a (degrees) used in this calculation. |
| Angle B (degrees) | 60 | Enter the angle b (degrees) used in this calculation. |
| Side a (opposite angle A) | 8 | Enter the side a (opposite angle a) used in this calculation. |
| Side b | 10.8 | |
| Side c | 12.3 | |
| Angle C (degrees) | 80 | |
Inputs, definitions and assumptions
Angle A (degrees)
Enter the angle a (degrees) used in this calculation. The prefilled worked-example value is 40.
Angle B (degrees)
Enter the angle b (degrees) used in this calculation. The prefilled worked-example value is 60.
Side a (opposite angle A)
Enter the side a (opposite angle a) used in this calculation. The prefilled worked-example value is 8.
How to use this calculator
- 1Verify the inputs. Gather angle a (degrees), angle b (degrees) and side a (opposite angle a) from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the side b at 10.8. Store your own version of it as Scenario A.
- 3Test one change. Start with side a (opposite angle a), the input with the biggest effect here: moving side a (opposite angle a) from 7 to 9 takes the side b from 9.4 to 12.1, a swing of 25% of the worked-example figure.
- 4Check the extremes. At half the example side a (opposite angle a) (4) the side b is 5.4; at double (16) it is 21.6.
People also ask
Frequently asked questions
How do you calculate law of sines?
a ÷ sin A = b ÷ sin B = c ÷ sin C, with C = 180° − A − B. At the worked-example inputs the side b is 10.8.
What does the law of sines result mean?
Solve a triangle when you know two angles and the side opposite one of them. At the worked-example inputs the side b is 10.8. It rises with side a (opposite angle a) and angle b (degrees) and falls as angle a (degrees) increases.
How much does angle a (degrees) change the side b?
Holding every other input at the worked-example value, moving angle a (degrees) from 32 to 48 moves the side b from 9.3 to 13.1, a spread of 3.8.
What are the limits of this law of sines 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 angle a (degrees) only from 32 to 48; a value outside that range is not tabulated here.
Which input moves the side b most in the law of sines calculator?
Ranked by how far each moves the side b across the range tested: side a (opposite angle a) (2.7, 25%), angle a (degrees) (1.8, 17%) and angle b (degrees) (1.3, 12%).
If I double side a (opposite angle a) in the law of sines calculator, does the side b double?
Doubling it from 8 to 16 takes the side b from 10.8 to 21.6, which is 2.00 times the worked-example figure. So the result scales almost exactly in proportion. Halving it to 4 gives 5.4.
How much does angle b (degrees) matter in the law of sines calculator?
The worked example uses 60. With the other inputs left at the worked example, moving angle b (degrees) from 54 to 66 takes the side b from 10.1 to 11.4, a swing of 12% of the worked-example figure.
How much does side a (opposite angle a) matter in the law of sines calculator?
The worked example uses 8. Holding every other input at its worked-example value, moving side a (opposite angle a) from 7 to 9 takes the side b from 9.4 to 12.1, a swing of 25% of the worked-example figure.
Which inputs change the side c in the law of sines calculator?
At the worked-example inputs it is 12.3. Angle a (degrees) takes it from 13.5 to 11.2, angle b (degrees) takes it from 12.4 to 12 and side a (opposite angle a) takes it from 10.7 to 13.8.
Which inputs change the angle c (degrees) in the law of sines calculator?
At the worked-example inputs it is 80. Angle a (degrees) takes it from 84 to 76 and angle b (degrees) takes it from 86 to 74.
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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