What is the Sine Law used for?

The law of sines can be used to solve for a triangle when one angle and the opposite side are known, plus either another angle or another side. It is applicable in cases ASA (two angles and the included side), AAS (two angles and a non-included side) and SSA (two sides and an angle not between them), using the ratio a/sin(A) = b/sin(B) = c/sin(C).

What are ambiguous cases in the law of sines?

An ambiguous case is a SSA situation, i.e., when two sides and the angle that belongs to one of those sides are known. Since sine cannot distinguish between an acute or its obtuse supplement, the same data can result in zero, one, or two solutions triangles that satisfy the conditions. This calculator determines which case it is and shows all triangles that meet the requirements.

How do you determine if two triangles can be formed with an SSA triangle?

Based on the known side a and opposite angle A, sin(B) = b*sin(A)/a is calculated. If sin(B) is greater than 1, no triangle exists. If it equals 1, exactly one right triangle exists. If it's less than 1, both acute angle B and its obtuse supplement 180-B are potential angles. Only if a positive third angle results from 180-B do two triangles exist.

When should you use the law of sines instead of the law of cosines?

The law of sines is used when a side and its opposite angle are known (ASA, AAS, SSA). If only the three sides are known (SSS) or two sides and an included angle (SAS), then the law of cosines must be used first because there is no unique mapping between angles and sides. In many cases, the law of cosines will be used first followed by the law of sines to solve the problem.

Can you solve a right triangle with the law of sines?

Yes it is possible. If one angle is 90 degrees then sin(90) = 1 which means that a/sin(A) = b/sin(B) = c. That is, the hypotenuse c is equal to this common ratio. Although applicable, for right triangles the basic SOH-CAH-TOA formula is usually faster.

Why is the Sine Rule true?

The area of each triangle can be expressed in three different ways: half of a*b*sin of C, half of b.*c*sin(A), or half of a.*c*sin(B). If you equate and combine these expressions, you get a/sin(A) = b/sin(B) = c/sin(C). This ratio is also equal to the diameter of the circumcircle of the triangle, which is why it's denoted as 2R.

Law of Sines Calculator

Choose your known values

What do you know?

Pick the parts of the triangle you already have. Two angles and any one side give a unique triangle; two sides and an angle opposite one of them is the ambiguous case, which can have zero, one, or two solutions.

Two angles and a side (AAS / ASA)

Two sides and an angle (SSA - ambiguous)

Your known values

Angle A is opposite side a, B opposite b, C opposite c. Keep those pairs straight and the law of sines does the rest.

Angle A

Angle B

Side a (opposite A)

Side b (opposite B)

Side c (opposite C)

Solution

Angle C

Area

Perimeter

Circumradius R

Inradius r

Solved triangle: angles A = 40 deg (°), B = 60 deg (°), C = 80 deg (°); sides a = 6.53 m, b = 8.79 m, c = 10 m. Area 28.26, perimeter 25.32.

Steps and chart

Show the calculation steps

Walk through how the law of sines produces the answer above.

Show the side-length bar chart

Compare the three side lengths of the solved triangle at a glance.

The law of sines is a powerful tool for solving triangles that do not have right angles. It establishes a proportional relationship between each side of any triangle and the sine of its opposite angle, so knowing just one of these proportions allows you to determine the entire triangle with three known pieces of information.

This calculator considers three cases in which the law of sines can be used: when two angles and a side are known (AAS or ASA), and when two sides and an opposite angle to one of them is known (SSA). In the SSA case, there is the familiar ambiguous result that no, one, or two triangles may exist.

Formula for the law of sines:

In a triangle where the sides a, b and c are opposite the angles A, B and C respectively, the ratio of any side to the sine of the opposite angle is equal for all three sides.

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

This common ratio is not just a number. It corresponds to 2R, which is the double radius of a circle that passes through all three vertices. Therefore, in addition to measuring the lengths of the sides, the sine law also measures the circumcircle of the triangle.

It is important to note the corresponding relationships: Angle A is opposite side a, angle B is opposite side b and angle C is opposite side c. Mixing up the sides with the wrong angles will lead to errors in all subsequent steps.

When to use the Sine Rule:

To use the law of sines, you need at least one complete pair of a side and its opposite angle. This determines which cases the law of sines can be used in and also shows that it is not suitable for two cases where there is no corresponding relationship.

You know	Case	Use	Solutions
Two angles + the included side	ASA	Law of sines	Always 1
Two angles + a non-included side	AAS	Law of sines	Always 1
Two sides + an angle opposite one	SSA	Law of sines	0, 1, or 2
Two sides + the included angle	SAS	Law of cosines	Always 1
All three sides	SSS	Law of cosines	Always 1

If two angles are known, the third angle can be directly calculated since the sum of the interior angles of any triangle is 180 degrees. The third angle C is given by: C = 180 - A - B. If in addition one side is known, then the size of the triangle is fixed and the remaining two sides may be determined using the law of sines to give their ratio.

Example: AAS (Angle-Angle-Side)

We are solving for a triangle. Given side b = 3, angle A = 60 degrees and angle B = 45 degrees. First we calculate the third angle: C = 180 - 60 - 45 = 75 degrees. Next we determine the ratio between known side b and angle B to solve for sides a and c.

\frac{a}{\sin 60^\circ} = \frac{3}{\sin 45^\circ} \;\Rightarrow\; a = \frac{3\,\sin 60^\circ}{\sin 45^\circ} \approx 3.6742

c = \frac{3\,\sin 75^\circ}{\sin 45^\circ} \approx 4.0981

The area is half of the product of a, b and sin C which is approximately 5.3236. The perimeter is approximately 10.7723. If you enter A = 60, B = 45 and side b = 3 in two angle mode this calculator will give exactly these values.

Unclear cases (SSA)

The Sine Rule gained its fame here. When two sides and the opposite angle of one of these sides are known, then with sine you cannot distinguish between an acute and obtuse angle. These data can therefore point to two completely different triangles, only one triangle or no triangle at all.

You start with the known side a and its opposite angle A, then calculate the sine of the second angle B based on another known side b.

\sin B = \frac{b\,\sin A}{a}

This result can be used to assess the overall situation.

Condition	Triangles	Why
sin B > 1	0	No angle has a sine above 1, so side b cannot reach
sin B = 1	1	B is exactly 90 degrees, a single right triangle
sin B < 1, and 180 - B still fits	2	Both the acute B and obtuse 180 - B work
sin B < 1, but 180 - B is too big	1	The obtuse option overflows 180 degrees

For example: let side a = 6, side b = 8 and angle A = 35 degrees. Then sin B = 8*sin(35)/6 is approximately equal to 0.7648, which means that B can be either approximately 49.89 degrees or its supplement, 130.11 degrees. In both cases the third angle is positive so there are two triangles. One has an angle C of approximately 95.11 degrees and a side c of approximately 10.42. The other has an angle C of approximately 14.89 degrees and a side c of approximately 2.69. This calculator will give both results, and warn immediately if there is a second triangle.

Step-by-step solution.

In any case, the same simple steps are followed and a calculation tool can show these steps below the result.

1. Assign each known value to its corresponding opposite side (A with a, B with b, and C with c). 2. If two angles are known, then the third angle can be calculated by subtracting the sum of the two known angles from 180 degrees. 3. Based on a complete correspondence, the ratio of side to sine of the opposite angle is determined. 4. Determine each unknown side or angle one at a time. 5. If there are problems with the congruence theorem SSA (side-side-angle), it must be checked whether the complementary angle of the given obtuse angle also fulfills the conditions. 6. Make sure all three angles are positive and that they add up to 180 degrees.

Area, perimeter, two circles.

Once the solution to the triangle has been found, other measurements can be calculated. The area is determined by two sides and the included angle, while the perimeter is simply the sum of all three sides.

\text{Area} = \tfrac{1}{2}ab\sin C \qquad P = a + b + c

Every triangle has two associated circles. The circumradius R is the radius of the circle that passes through all three vertices of the triangle and can be found from the law of sines as exactly half the diameter. The inradius r is the radius of the largest possible circle that fits inside the triangle and can be calculated from the area and semiperimeter.

R = \frac{a}{2\sin A} \qquad r = \frac{\text{Area}}{s}, \quad s = \frac{a+b+c}{2}

Sine and cosine rules.

These two theorems are not in conflict but rather complement each other. The sine theorem is faster and easier to use but requires knowledge of an angle and its opposite side. If only three sides (SSS) or two sides and the included angle (SAS) are known, then the cosine theorem must be used as there is no direct relationship between angles and sides that can be used.

A common technique is to use the law of cosines first to find the largest angle and then go back to the law of sines to find a second angle, then use angles of a triangle add to 180° to find the third.

Applications of the Sine Rule:

The law of sines is used in all areas that deal with triangulation because it converts measurable angles into distances that cannot be measured directly. Navigation and surveying engineers use two lines of sight to locate positions and identify points that cannot be reached directly.

Astronomers use parallax to measure the distance to stars while engineers split forces on girders and frames. Physicists break vectors into their individual components within triangles that have no right angles.

Symbols used:

The table below shows the symbols used in the above formulas.

Symbol	Meaning	Example
a, b, c	The three side lengths	6, 8, 10.42
A, B, C	The angles opposite a, b, c	35, 49.89, 95.11 deg
R	Circumradius (circle through the vertices)	5.23
r	Inradius (largest inscribed circle)	2.23
s	Semi-perimeter, (a + b + c) / 2	12.21

This calculator is a general tool for education and problem solving. Use the law of cosines calculator when you know three sides (SSS) or two sides and the included angle (SAS). In these cases there is no unique mapping between angles and sides to start with the law of sines.

Frequently asked questions

What is the Sine Law used for?: The law of sines can be used to solve for a triangle when one angle and the opposite side are known, plus either another angle or another side. It is applicable in cases ASA (two angles and the included side), AAS (two angles and a non-included side) and SSA (two sides and an angle not between them), using the ratio a/sin(A) = b/sin(B) = c/sin(C).
What are ambiguous cases in the law of sines?: An ambiguous case is a SSA situation, i.e., when two sides and the angle that belongs to one of those sides are known. Since sine cannot distinguish between an acute or its obtuse supplement, the same data can result in zero, one, or two solutions triangles that satisfy the conditions. This calculator determines which case it is and shows all triangles that meet the requirements.
How do you determine if two triangles can be formed with an SSA triangle?: Based on the known side a and opposite angle A, sin(B) = b*sin(A)/a is calculated. If sin(B) is greater than 1, no triangle exists. If it equals 1, exactly one right triangle exists. If it's less than 1, both acute angle B and its obtuse supplement 180-B are potential angles. Only if a positive third angle results from 180-B do two triangles exist.
When should you use the law of sines instead of the law of cosines?: The law of sines is used when a side and its opposite angle are known (ASA, AAS, SSA). If only the three sides are known (SSS) or two sides and an included angle (SAS), then the law of cosines must be used first because there is no unique mapping between angles and sides. In many cases, the law of cosines will be used first followed by the law of sines to solve the problem.
Can you solve a right triangle with the law of sines?: Yes it is possible. If one angle is 90 degrees then sin(90) = 1 which means that a/sin(A) = b/sin(B) = c. That is, the hypotenuse c is equal to this common ratio. Although applicable, for right triangles the basic SOH-CAH-TOA formula is usually faster.
Why is the Sine Rule true?: The area of each triangle can be expressed in three different ways: half of a*b*sin of C, half of b.*c*sin(A), or half of a.*c*sin(B). If you equate and combine these expressions, you get a/sin(A) = b/sin(B) = c/sin(C). This ratio is also equal to the diameter of the circumcircle of the triangle, which is why it's denoted as 2R.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Khan Academy: Law of sines
Introduction and proof of the law of sines, including the ambiguous case.
Wikipedia: Law of sines
The sine rule, its relation to the circumradius (2R), and the SSA ambiguous case.