Triangle Inputs
Formula Used
Missing side: c² = a² + b² − 2ab cos(C).
Missing angle: C = cos⁻¹((a² + b² − c²) ÷ 2ab).
The same pattern applies after rotating side and angle labels.
The Law of Cosines connects three sides with one angle. It extends the Pythagorean theorem beyond right triangles. Use matching opposite side and angle labels throughout calculations.
How to Use
- Select automatic, SAS, SSS, missing-side, or missing-angle mode.
- Enter known sides and matching opposite angles.
- Select the target side or target angle when needed.
- Choose angle units, length units, and decimal precision.
- Press Calculate Triangle and review every derived measurement.
- Copy, print, or export the completed result.
Example Data
| Mode | Known values | Target | Expected use |
|---|---|---|---|
| SAS | b = 7, c = 10, A = 48° | Side a | Find side a and solve everything. |
| SSS | a = 8, b = 9, c = 12 | All angles | Find every angle and measurement. |
| Missing angle | a = 5, b = 6, c = 7 | Angle C | Apply the inverse cosine formula. |
Frequently Asked Questions
1. When should I use the Law of Cosines?
Use it for SAS or SSS triangle information. SAS means two sides and their included angle. SSS means all three side lengths are known.
2. Can this calculator solve an entire triangle?
Yes, valid SAS or SSS inputs produce complete results. The calculator finds sides, angles, area, and perimeter. It also calculates advanced triangle measurements.
3. What does included angle mean?
The included angle lies between two known sides. Its opposite side is the unknown SAS side. Correct matching prevents common label errors.
4. How does this differ from Pythagoras?
Pythagoras applies only to right triangles. The cosine formula works with general triangle angles. A ninety-degree angle removes the cosine correction term.
5. Why are inverse cosine values clamped?
Floating-point arithmetic can slightly exceed valid cosine limits. Clamping keeps values between negative one and one. This avoids invalid inverse cosine errors.
6. Can I enter radians?
Yes, choose radians before calculating the triangle. Inputs convert internally for reliable trigonometric processing. Results return using the selected angle unit.
7. Why does triangle inequality matter?
Every pair of sides must exceed the remaining side. Otherwise, the sides cannot close into a triangle. The calculator rejects impossible combinations immediately.
8. Does the selected length unit change calculations?
All side inputs must use one consistent unit. The selected unit labels every length and area. No automatic mixed-unit conversion is performed.
9. Are exported results reusable?
Yes, copy creates a reusable text summary. CSV supports spreadsheets and data storage. PDF provides a convenient printable calculation record.