Law Of Sines CalculatorFree online math calculator — instant results, no sign-up
Find a triangle side using two angles and a known opposite side.
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About the Law Of Sines Calculator
The Law Of Sines Calculator finds side b of a triangle from a known side a, its opposite angle A, and a second angle B. It uses the ratio of each side to the sine of its opposite angle, with both angles entered in degrees. With a = 5, A = 30°, and B = 45°, it returns b = 7.0710678119.
- Known side a
- The length of the side opposite angle A, entered in any consistent length unit; the default is 5.
- Opposite angle A (degrees)
- The interior angle opposite known side a, measured in degrees; the default is 30.
- Angle B (degrees)
- The interior angle opposite the side b being calculated, measured in degrees; the default is 45.
Formula and method
a / sin(A) = b / sin(B), so b = a × sin(B) / sin(A). In degree-based inputs: b = a × sin(B × π / 180) / sin(A × π / 180). With the defaults: b = 5 × sin(45°) / sin(30°) ≈ 7.0710678119.
- aKnown side length opposite angle A.
- bCalculated side length opposite angle B, in the same unit as a.
- AInterior angle opposite side a, entered in degrees.
- BInterior angle opposite side b, entered in degrees.
- sinThe sine function, evaluated using the angle's correct degree or radian representation.
- πPi; multiplying a degree value by π / 180 converts it to radians.
First check that a > 0, A > 0°, B > 0°, and A + B < 180° so the inputs describe a nondegenerate triangle. Convert the angles to radians if required by the sine implementation, then multiply a by sin(B) and divide by sin(A). For the defaults, sin(30°) = 0.5 and sin(45°) = √2 / 2, giving b = 5√2 ≈ 7.0710678119.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Known side a5
- Opposite angle A (degrees)30
- Angle B (degrees)45
Results
- Result7.0710678119
How to use the Law Of Sines Calculator
- 1Enter the known side length in Known side a; use 5 for the default example.
- 2Enter the angle opposite that known side in Opposite angle A (degrees); use 30 for the default example.
- 3Enter the angle opposite the desired side in Angle B (degrees); use 45 for the default example.
- 4Confirm that the side length and both angles are positive and that the angles add to less than 180°, then run the calculation.
- 5Read the Result as side b: the default inputs give 7.0710678119 in the same length unit as side a.
Reading your result
The Result row reports b, the side opposite angle B—not the remaining third side. With the default inputs, Result = 7.0710678119 means that b is approximately 7.0710678119 units long when a is 5 units long. Since B = 45° is greater than A = 30°, b should be longer than a, which agrees with this result.
Limitations
This calculation assumes a flat, Euclidean triangle with correctly matched sides and opposite angles. It requires a positive known side and positive angles whose sum is less than 180°; otherwise, the inputs do not define a valid nondegenerate triangle. Angles must be entered in degrees, not radians. Very small values of A make division by sin(A) sensitive to input errors, and the displayed decimal result is an approximation rather than a guarantee of measurement accuracy.
Common questions
Which side does the Law Of Sines Calculator find?+
It finds side b, opposite the entered angle B. The known side a must be opposite angle A for the formula to apply correctly.
Why do the default inputs produce 7.0710678119?+
Using a = 5, A = 30°, and B = 45°, the formula gives b = 5 × sin(45°) / sin(30°) = 5√2, which rounds to 7.0710678119 at ten decimal places.
Do I need to enter the third angle?+
No. The two entered angles determine it as 180° − A − B. With A = 30° and B = 45°, the third angle is 105°, but it is not needed to compute b.
Can these inputs produce two different triangles?+
No. Two valid angles and one known side determine a unique triangle up to position and orientation. The ambiguous law-of-sines case involves two sides and an angle opposite one of them, not the two-angle inputs used here.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
Further reading
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Results are estimates for informational purposes and are not financial, medical or legal advice.