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Formula used
This calculator assumes a standard triangle where side a is opposite angle A, side b is opposite B, and side c is opposite C.
- Angle sum: A + B + C = 180° (or π radians).
- Law of sines: a/sin(A) = b/sin(B) = c/sin(C).
- Heron’s formula (area): Area = √(s(s−a)(s−b)(s−c)), s=(a+b+c)/2.
- Altitudes: ha = 2·Area/a (similarly for hb, hc).
- Radii: r = Area/s, R = a/(2·sin(A)).
How to use this calculator
- Select your angle unit (degrees or radians).
- Choose which two angles you know (A and B, B and C, or A and C).
- Enter the two angles and pick the known side (a, b, or c).
- Enter the side length, choose a unit, and set rounding.
- Click Calculate to get all sides, the third angle, and extra measures.
- Use Download CSV or Download PDF to export results.
Example data table
| Given angles | Known side | Computed third angle | Computed sides (a, b, c) | Area |
|---|---|---|---|---|
| (A_B) 45°, 60° | a = 10 cm | 75.00° | a=10.00, b=12.25, c=13.66 | 59.15 cm² |
| (A_C) 35°, 75° | c = 12 m | 75.00° | a=7.13, b=11.67, c=12.00 | 40.18 m² |
| (B_C) 50°, 40° | b = 8 in | 40.00° | a=10.44, b=8.00, c=6.71 | 26.85 in² |
Examples are precomputed in degrees. Your results may differ with rounding.
Two angles and one side overview
Two angles and one side is the AAS case. The triangle’s shape is fixed when two angles are set, and one side length sets the scale. This tool accepts degrees or radians and lets you choose which side is known (a, b, or c).
Validity rules and constraints
Inputs must form a real triangle. Angle values must be greater than 0 and the sum of the two given angles must be less than 180° (or less than π radians). The calculator computes the third angle as 180° − (A + B), 180° − (A + C), or 180° − (B + C), then validates positivity.
Law of Sines scaling step
After the third angle is found, the Law of Sines provides the remaining sides. Using a known opposite pair, the scale factor is k = a/sin(A) (or b/sin(B), or c/sin(C)). Then the missing sides follow directly: b = k·sin(B) and c = k·sin(C). This is numerically stable for most common angle ranges.
Perimeter, area, and extra measures
Beyond side lengths, the results panel reports perimeter, semiperimeter, and area. Area is computed with Heron’s formula, s = (a+b+c)/2 and Area = √(s(s−a)(s−b)(s−c)). Extra geometry outputs include altitudes (2·Area/side), medians, angle bisectors, inradius (Area/s), and circumradius (a/(2·sin(A))).
Units, precision, and rounding
For reliable comparisons, keep consistent units on the known side; all computed lengths return in that same unit. The calculator uses floating‑point math, so very small angles (for example under 0.5°) can magnify rounding. Use more decimal places and prefer radians only when your input data is already in radians. for cleaner output.
Common input mistakes to avoid
If results look wrong, check the “which angles are known” selector and confirm the side letter matches the side opposite the corresponding angle. A common mistake is entering a side adjacent to an angle. Also verify you didn’t mix degrees and radians; 60 radians is enormous, while 60° is normal.
Where AAS shows up in practice
AAS triangle solving appears in surveying, roof framing, and navigation where angles are measured first and a baseline is known. For example, with A = 45°, B = 60°, and a = 10 cm, the third angle is 75° and the remaining sides scale proportionally. Exporting CSV or PDF makes field reporting faster.
FAQs
1) Can I enter any two angles?
Yes, as long as both angles are positive and their sum is less than 180° (or π). The calculator will compute the third angle automatically and reject impossible combinations.
2) Which side should I enter: a, b, or c?
Enter the side whose length you actually know. The letter must match the side opposite its angle (a opposite A, b opposite B, c opposite C). The solver uses that opposite pair to scale all other sides.
3) What happens if my angles add up to 180° or more?
No triangle can be formed. The third angle would be zero or negative, so the calculator shows an error and stops before applying the Law of Sines.
4) Does this work for right triangles?
Yes. If one angle is 90° (or π/2), the triangle is right‑angled. Provide the other known angle and any one side, and the tool will return the missing legs, hypotenuse, and area.
5) Why do I see small differences from my textbook answers?
Rounding and trigonometric precision can cause tiny differences, especially with very small or very large angles. Increase decimal places, keep consistent units, and compare using the same rounding rule as your reference.
6) Can I use the exports for reports or homework tables?
Yes. CSV is convenient for spreadsheets, while PDF is a ready‑to‑print summary of your inputs and computed values. Always cite your input units and rounding choice in your report.