Law Of Sines
Calculator
Results
- Side b
- 14.099646
- Side c
- 15.027138
- Angle C (degrees)
- 75
- Circumradius
- 7.778619
Results
| Side b | 14.099646 |
| Side c | 15.027138 |
| Angle C (degrees) | 75 |
| Circumradius | 7.778619 |
formula-map diagram
- Side b
- 14.099646
- Side c
- 15.027138
- Angle C (degrees)
- 75
- Circumradius
- 7.778619
Formula map
Formula
a ÷ sin A = b ÷ sin B = c ÷ sin C = 2R= 14.09964618762
Note
Simplified model: these are the exact results of the stated idealised geometric formulas, assuming perfect shapes, exact inputs and consistent units. Real measurements carry error, and results such as the cyclic-quadrilateral area only hold when the figure truly satisfies the stated condition. Verify independently before relying on these figures.
More in Advanced geometry
See all →Frequently asked questions
When can I use the law of sines to solve a triangle?+
The law of sines requires knowing at least one complete angle-side pair (the angle and the side directly opposite it) plus one additional piece of information, either another angle (AAS/ASA) or another side (SSA). Without at least one matched pair, you can't set up the required ratio.
What is the law of sines formula?+
The law of sines states a/sin(A) = b/sin(B) = c/sin(C), meaning the ratio of each side to the sine of its opposite angle is the same constant for all three sides of any triangle. Knowing any three of these six values (fitting the pattern) lets you solve for a fourth.
What is the 'ambiguous case' (SSA) and why does it sometimes produce two possible triangles?+
Given two sides and a non-included angle (SSA), it's possible for two different triangles to satisfy the same given values, because sin(θ) = sin(180°−θ), so an angle and its supplement give the same sine ratio. The calculator needs to check both the acute and obtuse possibilities for the unknown angle, and sometimes only one, both, or neither will produce a valid triangle.
How do I know if my SSA triangle has one solution, two solutions, or no solution at all?+
It depends on comparing the side opposite the given angle to the height of the triangle formed by the other side and angle: if it's shorter than that height, no triangle exists; if equal, exactly one (a right triangle); if longer than the height but shorter than the adjacent side, two triangles are possible; if longer than or equal to the adjacent side, exactly one triangle exists. This is why SSA is called 'ambiguous' unlike SSS, SAS, or ASA.
Why can't the law of sines be used directly when I only know three sides (SSS)?+
With only three sides and no known angle, there's no angle-side ratio available to start the law of sines calculation, since every term in the equation pairs a side with its opposite angle. In an SSS situation, the law of cosines must be used first to find one angle, after which the law of sines can find the remaining angles if desired.