Enter Known Triangle Values
Use degrees for angles. Enter at least three values, including one side. Leave unknown values blank.
Example Data Table
| Case | Angle A | Angle B | Angle C | Side a | Side b | Side c | Use Case |
|---|---|---|---|---|---|---|---|
| AAS | 40° | 65° | Blank | 12 | Blank | Blank | Find one angle and two sides |
| ASA | 50° | Blank | 70° | Blank | 18 | Blank | Find remaining parts |
| SSA | 35° | Blank | Blank | 10 | 8 | Blank | Check ambiguous triangle |
Formula Used
The calculator uses the law of sines:
a / sin(A) = b / sin(B) = c / sin(C) = 2R
Here, a, b, and c are sides. Angles A, B, and C are opposite those sides.
The third angle is found by:
A + B + C = 180°
Area is calculated with:
Area = 1/2 × b × c × sin(A)
Perimeter is calculated with:
P = a + b + c
The circumradius is calculated with:
R = common ratio / 2
How to Use This Calculator
- Enter known angles in degrees.
- Enter known side lengths in matching opposite positions.
- Use side a opposite angle A, side b opposite angle B, and side c opposite angle C.
- Leave unknown fields empty.
- Provide at least three values, including one side.
- Click the calculate button.
- Review solved sides, angles, perimeter, area, ratio, and circumradius.
- Use CSV or PDF export for reports and assignments.
Understanding the Triangle Law of Sines
What This Tool Solves
The triangle law of sines is useful when a triangle has a known side and its opposite angle. It also works well when two angles and one side are known. This calculator finds missing sides, missing angles, area, perimeter, semi perimeter, and circumradius. It helps students, teachers, designers, surveyors, and technical users check triangle measurements quickly.
Why the Law Works
Every triangle has a steady relationship between each side and the sine of its opposite angle. That relationship creates one common ratio. Once the ratio is known, unknown sides can be found by multiplying the ratio by the sine of their angles. Unknown angles can also be found with inverse sine. The calculator shows those steps so the process stays clear.
Supported Triangle Cases
The tool supports AAS, ASA, and many SSA inputs. AAS means two angles and a non-included side are known. ASA means two angles and the included side are known. SSA means two sides and a non-included angle are known. SSA can create an ambiguous case. That means two different triangles may satisfy the same input. The calculator checks this situation and shows more than one solution when possible.
Accuracy and Practical Use
Results are rounded for clean reading. Small decimal differences can appear because trigonometric functions use decimal approximations. Use consistent units for all sides. For example, do not mix meters with feet in one calculation. Angles must be entered in degrees. The calculator checks invalid angles, negative sides, impossible triangles, and missing data.
When to Use Another Method
The law of sines is not always the best method. If three sides are known, use the law of cosines first. If two sides and the included angle are known, law of cosines is usually stronger. After one opposite side and angle pair is found, law of sines can complete the triangle.
Frequently Asked Questions
1. What is the law of sines?
The law of sines states that each side divided by the sine of its opposite angle gives the same ratio in a triangle.
2. What values should I enter?
Enter at least three known triangle values. At least one value must be a side length, or the triangle scale cannot be solved.
3. Can I leave fields blank?
Yes. Leave unknown fields blank. The calculator uses the known values to solve missing angles, sides, and related measurements.
4. What is an SSA ambiguous case?
An SSA ambiguous case happens when two sides and a non-included angle can form two possible triangles with different angles.
5. Are angles entered in radians?
No. Enter all angles in degrees. The calculator converts degrees internally for sine and inverse sine operations.
6. Can this solve three-side triangles?
No. If only three sides are known, use the law of cosines first. Then law of sines can complete extra checks.
7. What does common ratio mean?
The common ratio is side divided by opposite angle sine. It also equals twice the circumradius of the triangle.
8. Why did I get no solution?
No solution appears when angles are invalid, sides are negative, data is incomplete, or the entered values cannot form a triangle.