Enter SAS Triangle Values
Provide two sides and the included angle. The solver returns the third side, remaining angles, area, perimeter, radii, medians, and heights.
Example Data Table
This sample demonstrates a valid side-angle-side case using degree input.
| Side b | Side c | Angle A | Third Side a | Angle B | Angle C | Area | Perimeter |
|---|---|---|---|---|---|---|---|
| 7 cm | 10 cm | 48° | 7.4379 cm | 44.3786° | 87.6214° | 26.0101 cm² | 24.4379 cm |
Formula Used
The solver uses standard triangle identities for a side-angle-side configuration.
1) Law of Cosines
a² = b² + c² − 2bc cos(A)
This finds the unknown third side when sides b and c with included angle A are known.
2) Remaining Angles
cos(B) = (a² + c² − b²) / 2ac
cos(C) = (a² + b² − c²) / 2ab
These relationships avoid ambiguity and give stable angle results.
3) Area
K = 1/2 × b × c × sin(A)
4) Extra Measures
Perimeter = a + b + c
Semiperimeter = (a + b + c) / 2
Inradius = K / s
Circumradius = a / 2sin(A)
Height to a = 2K / a
How to Use This Calculator
- Enter the two known sides that touch the included angle.
- Type the included angle value and choose degrees or radians.
- Add a unit label such as cm, m, ft, or units.
- Select the number of decimal places for rounded output.
- Press Solve Triangle to show results above the form.
- Review the computed side, angles, area, perimeter, heights, medians, and radii.
- Use the export buttons to save the result as CSV or PDF.
FAQs
1) What does SAS mean in triangle solving?
SAS means side-angle-side. You know two sides and the angle between them. That information determines one unique triangle when the angle is strictly between 0° and 180°.
2) Why is the included angle important?
The included angle must be between the two known sides. If you use a different angle, the setup changes and the Law of Cosines may no longer match the intended triangle.
3) Can I enter radians instead of degrees?
Yes. Choose radians from the angle unit menu. The solver converts the value internally and still reports the main angle in both degrees and radians.
4) What formulas does the solver use first?
It first applies the Law of Cosines to find the third side. After that, it computes the other angles and then derives area, perimeter, heights, medians, and radii.
5) Does every valid SAS input create exactly one triangle?
Yes. Unlike some SSA cases, SAS produces a single triangle when both side lengths are positive and the included angle stays within a valid open interval.
6) What extra values are shown besides sides and angles?
The calculator also returns area, perimeter, semiperimeter, inradius, circumradius, heights to each side, medians, and basic triangle classification for sides and angles.
7) Why might the calculator reject my input?
Inputs are rejected when a side is zero or negative, when precision is outside the allowed range, or when the included angle is not strictly between 0 and 180 degrees.
8) When should I use the CSV or PDF options?
Use CSV when you want spreadsheet-friendly output. Use PDF when you need a clean shareable summary for notes, assignments, documentation, or printed study material.