Enter ASA triangle values
Provide two known angles and the included side. The form keeps a responsive three, two, or one column layout based on screen size.
Example data table
This worked example shows a complete ASA solution using angles A and B with the included side c.
| Item | Value | Note |
|---|---|---|
| Angle A | 52° | Known input |
| Angle B | 61° | Known input |
| Included side c | 10 cm | Known input |
| Angle C | 67° | Computed from 180° − A − B |
| Side a | 8.5606 cm | Law of sines result |
| Side b | 9.5015 cm | Law of sines result |
| Area | 37.4365 cm² | Using ½ab sin C |
| Perimeter | 28.0622 cm | Sum of all three sides |
Formula used
ASA means two angles and the included side are known. The solver first finds the third angle, then applies trigonometric identities.
1) Third angle
In degrees, C = 180° − A − B. In radians, C = π − A − B.
2) Unknown sides with the law of sines
a / sin(A) = b / sin(B) = c / sin(C). Therefore, a = c·sin(A)/sin(C) and b = c·sin(B)/sin(C).
3) Area and perimeter
Area K = ½ab sin(C). Perimeter P = a + b + c. Semiperimeter s = P / 2.
4) Altitudes and radii
ha = 2K / a, hb = 2K / b, hc = 2K / c. The inradius is r = K / s, while the circumradius is R = c / (2 sin C).
5) Medians and angle bisectors
The medians come from Apollonius-based formulas. Each angle bisector uses l = √(mn[1 − x²/(m + n)²]) with the matching adjacent sides.
How to use this calculator
- Enter the two known angles that belong to the triangle.
- Enter the included side between those two angles as side c.
- Choose whether your angles are in degrees or radians.
- Set the decimal precision and optional side unit label.
- Click Solve triangle to generate all remaining measures.
- Review side lengths, area, perimeter, altitudes, radii, medians, and classifications.
- Use the CSV or PDF buttons to save the generated report.
Frequently asked questions
1) What does ASA mean in triangle solving?
ASA means you know two angles and the included side between them. That information uniquely determines one triangle when the angle sum stays below a straight angle.
2) Why must the two known angles add to less than 180°?
A triangle’s interior angles always total 180°. If the first two already add to 180° or more, the third angle becomes zero or negative, so no valid triangle exists.
3) Which side should I enter as side c?
Enter the side located between the two known angles. In this naming convention, that included side is labeled c, and it is opposite the unknown third angle C.
4) Can I use radians instead of degrees?
Yes. Select radians in the angle unit field, enter both angles in radians, and the solver will still return the triangle measures correctly.
5) Why are there no ambiguous solutions here?
ASA does not create the classic ambiguous case seen in some SSA problems. Two angles already lock the third angle, so the shape becomes uniquely determined.
6) What extra values does this solver calculate?
Besides the missing sides and angle, it returns perimeter, semiperimeter, area, three altitudes, inradius, circumradius, medians, bisectors, and triangle classifications.
7) Are the exported CSV and PDF files based on my result?
Yes. Once a valid solution appears, the export buttons package the current result values shown on the page into downloadable summary files.
8) Is this useful for classroom and engineering work?
Yes. It is useful for homework checks, drafting geometry, trigonometry practice, quick design verification, and any workflow needing consistent triangle measurements.