Law of Sines Calculator

Solve any triangle with the Law of Sines. Enter two angles and a side, or two sides and an angle, for a complete solution.

📐 Law of Sines Calculator

ASA: Enter angle A, its opposite side a, and angle B.

AAS: Enter angle A, angle B, and side b (opposite B).

SSA (Ambiguous Case): Enter side a, side b, and angle A (opposite a).

Side a
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Side b
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Side c
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Angle A
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Angle B
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Angle C
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Area
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Perimeter
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Sine Ratio
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Triangle Type
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What is the Law of Sines?

The Law of Sines (also called the Sine Rule) states that in any triangle, the ratio of each side length to the sine of its opposite angle is the same constant:

a / sin(A) = b / sin(B) = c / sin(C) = 2R

Here sides a, b, c are opposite to angles A, B, C respectively, and R is the circumradius (the radius of the circle passing through all three vertices).

Unlike the Pythagorean theorem (which only works for right triangles), the Law of Sines works for all triangles: acute, right, and obtuse. It allows you to solve a triangle whenever you know two angles and any one side (ASA or AAS), or two sides and the angle opposite one of them (SSA).

The law is essential in surveying (triangulation to measure inaccessible distances), navigation, astronomy, and engineering design. Whenever you have partial information about a triangle and need a complete solution, the Law of Sines is the first tool to reach for.

Formula and Cases

Law of Sines: a / sin(A) = b / sin(B) = c / sin(C) = 2R

ASA (Angle-Side-Angle):
Known: A, a, B. Solve: C = 180 - A - B; b = a * sin(B) / sin(A); c = a * sin(C) / sin(A)

AAS (Angle-Angle-Side):
Known: A, B, b. Solve: C = 180 - A - B; a = b * sin(A) / sin(B); c = b * sin(C) / sin(B)

SSA (Ambiguous Case):
Known: a, b, A. Solve: sin(B) = b * sin(A) / a. If sin(B) greater than 1 then no triangle.

Area: Area = (1/2) * a * b * sin(C)

Variables:

  • a, b, c — side lengths, each opposite its corresponding angle
  • A, B, C — interior angles in degrees (must sum to 180°)
  • R — circumradius of the triangle
  • sin(X) — trigonometric sine of angle X

The SSA ambiguous case arises because arcsin is multi-valued: for a given sin(B) there may be two angles (B and 180°-B) that both produce valid triangles with different shapes.

📖 How to Use the Law of Sines Calculator

Steps to Solve

1
Select a mode — Choose ASA (two angles + opposite side), AAS (two angles + non-included side), or SSA (two sides + opposite angle) depending on which values you know.
2
Enter the known values — Type in the angles (in degrees) and side lengths. All sides must be positive; each angle must be between 0° and 179°; and two angles together must be less than 180°.
3
Click Solve Triangle — The calculator applies the Law of Sines to find all remaining values.
4
Read the results — You get all three sides, all three angles, area, perimeter, the common sine ratio (a/sin A), and the triangle classification (acute/right/obtuse).
5
Watch for SSA ambiguity — In SSA mode, if a second valid triangle exists, the calculator will display its angles as an informational note.

Example Calculations

Example 1 — ASA: A = 30°, a = 5, B = 60°
1
C = 180 - 30 - 60 = 90°
2
Common ratio = 5 / sin(30°) = 5 / 0.5 = 10
3
b = 10 x sin(60°) = 10 x 0.8660 = 8.6603
4
c = 10 x sin(90°) = 10
5
Area = 0.5 x 5 x 8.6603 x sin(90°) = 21.6506 sq units
Try this example
a=5, b=8.6603, c=10 | A=30°, B=60°, C=90° | Area=21.6506 sq units
Example 2 — AAS: A = 45°, B = 60°, b = 8
1
C = 180 - 45 - 60 = 75°
2
Common ratio = 8 / sin(60°) = 9.2376
3
a = 9.2376 x sin(45°) = 6.5321
4
c = 9.2376 x sin(75°) = 8.9233
5
Area = 0.5 x 6.5321 x 8 x sin(75°) = 25.2376 sq units
Try this example
a=6.5321, b=8, c=8.9233 | A=45°, B=60°, C=75° | Area=25.2376 sq units
Example 3 — SSA: a = 7, b = 10, A = 40°
1
sin(B) = 10 x sin(40°) / 7 = 0.9184
2
B = arcsin(0.9184) = 66.6746°
3
C = 180 - 40 - 66.6746 = 73.3254°
4
c = (7 / sin(40°)) x sin(73.3254°) = 10.4322
Try this example
a=7, b=10, c=10.4322 | A=40°, B=66.6746°, C=73.3254°
Example 4 — Equilateral triangle: A = 60°, a = 6, B = 60°
1
C = 180 - 60 - 60 = 60°
2
Common ratio = 6 / sin(60°) = 6.9282
3
b = c = 6.9282 x sin(60°) = 6
4
Area = 0.5 x 6 x 6 x sin(60°) = 15.5885 sq units
Try this example
a=b=c=6 | A=B=C=60° | Area=15.5885 sq units

Frequently Asked Questions

What is the Law of Sines?+
The Law of Sines states that in any triangle the ratio of each side to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). This common ratio equals twice the circumradius R of the triangle. The law holds for all triangles regardless of shape or size.
When should I use the Law of Sines?+
Use the Law of Sines when you know two angles and any one side (ASA or AAS), or when you know two sides and the angle opposite one of them (SSA). For two sides and the included angle (SAS) or all three sides (SSS), use the Law of Cosines instead.
What is the ambiguous SSA case?+
The SSA case is called ambiguous because knowing two sides and a non-included angle can produce two different valid triangles, one triangle, or no triangle at all. This happens because arcsin returns a value between 0 and 90 degrees, but 180 minus that value may also satisfy the equation and form a second valid triangle. This calculator detects and flags that scenario.
What is the difference between ASA and AAS?+
In ASA you know two angles and the side that lies between them. In AAS you know two angles and a side that is not between them (it is opposite one of the known angles). Both are solved with the Law of Sines but with different starting ratios depending on which side and angle pair you enter.
Does the Law of Sines work for obtuse triangles?+
Yes. The Law of Sines applies to all triangles including those with an obtuse angle (greater than 90 degrees). The sine function returns a positive value for both acute and obtuse angles, so the ratio a/sin(A) remains valid in all cases.
What is the circumradius and how does it relate to the Law of Sines?+
The circumradius R is the radius of the circle that passes through all three vertices of the triangle (the circumscribed circle or circumcircle). The Law of Sines shows that a/sin(A) = 2R, so dividing the sine ratio by 2 gives R. Larger triangles with wider angles have larger circumradii.
How do I find the area using the Law of Sines?+
Once all sides and angles are known, area = (1/2) times a times b times sin(C), where C is the angle between sides a and b. This formula works for all triangles and is derived from the base-times-height formula by expressing the height as b times sin(C).
What happens when sin(B) is greater than 1 in the SSA case?+
If the calculation b times sin(A) divided by a gives a result greater than 1, no real angle B exists because the sine function only outputs values between -1 and 1. This means no triangle can be formed with those inputs. The calculator displays an error and you should adjust the side or angle values.
How is the Law of Sines used in real life?+
The Law of Sines is used in land surveying and triangulation (measuring inaccessible distances by forming triangles), GPS positioning systems, navigation (calculating headings and positions at sea or in air), astronomy (measuring distances to nearby stars via parallax triangles), and structural engineering for non-rectangular roof and truss designs.
What is the difference between the Law of Sines and the Law of Cosines?+
The Law of Sines (a/sin A = b/sin B = c/sin C) is used when you know two angles and a side or two sides and a non-included angle. The Law of Cosines (c^2 = a^2 + b^2 - 2ab cos C) is used when you know all three sides or two sides and their included angle. Together they cover every possible triangle-solving scenario.
Must the angles sum to exactly 180 degrees?+
Yes. In any flat (Euclidean) triangle the three interior angles always sum to exactly 180 degrees. This is why knowing two angles determines the third: C = 180 - A - B. The calculator validates that the given angles are positive and sum to less than 180 before computing the third angle.
Can I use this calculator for right triangles?+
Yes. The Law of Sines works perfectly for right triangles. When one angle is 90 degrees, sin(90) = 1, so the formula simplifies to a/sin(A) = c, meaning the hypotenuse c equals the common ratio. You will get the same result as the Pythagorean theorem in this special case.
Tags: Law of Sines Calculator Sine Rule Triangle Solver Asa Triangle Aas Triangle Ssa Ambiguous Case Trigonometry Calculator Free

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What is the Law of Sines?

The Law of Sines states that in any triangle, the ratio of each side to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). This ratio equals twice the circumradius. The law works for all triangles.

When do you use the Law of Sines?

Use it when you know two angles and any one side (AAS or ASA), or when you know two sides and the angle opposite one of them (SSA). For SAS or SSS cases, use the Law of Cosines.

What is the ambiguous SSA case in the Law of Sines?

The ambiguous case (SSA) occurs when you know two sides and an angle opposite one of them. Depending on the values, there may be zero, one, or two valid triangles. If the given side opposite the known angle is shorter than the other given side, two different triangles can satisfy the conditions. This calculator detects and reports both solutions when they exist.

How do you find all sides and angles of a triangle using the Law of Sines?

In ASA mode, enter the two known angles and the side between them. The third angle is found as 180 minus the sum of the two known angles. The remaining sides are then computed using the sine ratios. In AAS mode, enter two angles and a non-included side. In SSA mode, enter two sides and the angle opposite the first side, then use the sine ratio to find the second angle.

Can the Law of Sines be used for right triangles?

Yes. The Law of Sines works for any triangle, including right triangles. For a right triangle with the 90-degree angle at C, sin(C) = 1, so the ratio a/sin(A) = b/sin(B) = c. However, for right triangles it is usually simpler to use basic trigonometric ratios (sin, cos, tan) or the Pythagorean theorem directly.

What is the circumradius of a triangle and how does it relate to the Law of Sines?

The circumradius R is the radius of the circle that passes through all three vertices of a triangle. The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C) = 2R. Knowing any side and its opposite angle therefore lets you compute R directly. The circumradius appears in the output of this calculator.

What is the difference between ASA, AAS, and SSA triangle configurations?

ASA (Angle-Side-Angle) means you know two angles with the side between them. AAS (Angle-Angle-Side) means you know two angles and a side that is not between them. SSA (Side-Side-Angle) means you know two sides and the angle opposite one of them. ASA and AAS always produce a unique triangle; SSA may produce zero, one, or two triangles (the ambiguous case).

What triangle area formula does the Law of Sines give?

Once all three sides and angles are known, the area can be calculated as Area = (1/2) x a x b x sin(C), where a and b are any two sides and C is the included angle between them. This calculator shows the area after solving the triangle.

How accurate is this Law of Sines calculator?

Calculations use JavaScript double-precision floating point (IEEE 754), giving accuracy to about 15 significant digits. Results are rounded to 4 decimal places for display. Input angles must be in degrees and must sum to less than 180 for a valid triangle.

What happens if the angles entered do not form a valid triangle?

If the two known angles already sum to 180 degrees or more, the third angle would be zero or negative, which is impossible. The calculator detects this and displays an error. Similarly, if an SSA configuration produces no valid triangle (the given side is too short to reach the base), the calculator reports no solution.

How is the perimeter of a triangle calculated after using the Law of Sines?

Once all three sides are known, the perimeter is simply the sum of all three sides: P = a + b + c. This calculator displays the perimeter alongside the individual side lengths and angles after solving the triangle.

📌 Quick Tips

💡Use ASA mode when you know two angles and their opposite side.
💡Use AAS mode when you know two angles and a non-included side opposite one of them.
💡The SSA case (ambiguous case) can have two valid triangle solutions -- the calculator flags this.
💡All three interior angles of any triangle must sum to exactly 180 degrees.
💡The Law of Sines ratio a/sin(A) equals the diameter of the triangle's circumscribed circle (2R), a useful check for your results.