SAS Triangle Solver — Law of Cosines, All Sides, Angles & Properties

About the SAS Triangle Solver — Law of Cosines, All Sides, Angles & Properties

The SAS (Side-Angle-Side) triangle solver is the definitive tool for determining every property of a triangle when you know two sides and the angle between them. This configuration uniquely determines a triangle — no ambiguity, no multiple solutions — making SAS one of the most reliable input sets in triangle geometry.

The cornerstone formula is the Law of Cosines: c² = a² + b² − 2ab·cos C, which generalizes the Pythagorean theorem to non-right triangles. Once the third side c is known, the remaining angles follow from additional applications of the cosine law or from the identity A + B + C = 180°.

The area is computed directly from the SAS inputs using Area = ½ · a · b · sin C — no need to find the third side first. From the area and sides, every secondary property can be derived: the three altitudes (hₐ = 2·Area/a), the three medians (mₐ = ½√(2b²+2c²−a²)), the inradius r = Area/s (where s is the semi-perimeter), and the circumradius R = abc/(4·Area).

The calculator also classifies the triangle by angle type (acute, right, or obtuse) and side type (scalene, isosceles, or equilateral). This classification is often needed in homework, standardized tests, and engineering applications. Preset buttons let you explore everything from equilateral triangles to obtuse scalene configurations without manual entry.

SAS problems arise naturally in surveying (two measured distances and a theodolite angle), robotics (two arm segments and a joint angle), and structural engineering (brace lengths with known connection angles).

When This Page Helps

SAS is one of the most dependable triangle input sets because two sides and their included angle determine exactly one triangle. This calculator turns that strong starting point into a full geometry report: it finds the third side with the Law of Cosines, resolves the remaining angles, and then extends the work into area, altitudes, medians, inradius, circumradius, and triangle classification.

That makes it useful well beyond a single textbook answer. Students can verify every stage of a triangle problem, teachers can demonstrate how secondary properties depend on the same core inputs, and practitioners in surveying, layout, or mechanical design can confirm whether an included-angle measurement produces the expected triangle before using it elsewhere.

How to Use the Inputs

1. Enter side a and side b with the chosen unit.
2. Enter the included angle C between those two sides.
3. Select degrees or radians for the angle input.
4. Or click a preset to load a common SAS triangle.
5. Review side c, angles A and B, area, and perimeter.
6. Check heights, medians, circumradius, inradius, and classification.
7. Use the bar charts to compare side lengths and angles visually.

Example Calculation

Tips & Best Practices

• SAS always produces a unique triangle — there is never an ambiguous case (unlike SSA).
• When C = 90°, the law of cosines reduces to the Pythagorean theorem: c² = a² + b².
• An obtuse included angle (C > 90°) always gives c as the longest side of the triangle.
• For equilateral triangles, set a = b and C = 60° — the calculator will confirm c = a = b.
• The sum of any two sides must exceed the third side (triangle inequality). This calculator checks automatically.

Why SAS Produces a Unique Triangle

In SAS data, the included angle locks the two known sides into one exact configuration, so there is no ambiguous case. That is one reason geometry courses rely so heavily on this setup when introducing the Law of Cosines. Once the included angle is fixed, the third side is forced, and every other triangle property follows from that same structure. This makes SAS one of the cleanest ways to move from partial measurements to a complete solved triangle.

Connecting the Law of Cosines and Area

One of the most useful features of SAS problems is that you can compute different properties from the same input without first solving everything else. The Law of Cosines gives the missing side, while the area comes directly from $ frac12 absin C$. From there, altitudes, medians, inradius, and circumradius can all be derived. Seeing those results together helps show how one triangle can be analyzed from multiple equivalent starting points.

Practical Uses of Included-Angle Measurements

SAS measurements appear naturally when two lengths can be measured directly and the angle between them is known from an instrument or drawing. Surveying, truss layout, robotic linkages, and machine parts all produce that pattern. A dedicated SAS solver helps confirm whether the resulting triangle is acute, right, or obtuse and whether the derived lengths are reasonable before those numbers are carried into a design, report, or proof.

Frequently Asked Questions

• What is the Law of Cosines?

  c² = a² + b² − 2ab·cos C. It relates the three sides of a triangle to one of its angles. It generalizes the Pythagorean theorem (which is the special case when C = 90°).

• Why does SAS uniquely determine a triangle?

  Two sides and their included angle fix the position of all three vertices. There is only one possible configuration — unlike SSA, which can produce 0, 1, or 2 triangles.

• What if the included angle is exactly 90°?

  The triangle is a right triangle. The law of cosines simplifies to c² = a² + b² (Pythagorean theorem), and the area is simply ½ab.

• What is the circumradius?

  The circumradius R is the radius of the circle that passes through all three vertices. For any triangle, R = abc/(4·Area). For a right triangle, R = c/2.

• What is the inradius?

  The inradius r is the radius of the largest circle that fits inside the triangle, tangent to all three sides. r = Area / s, where s = (a+b+c)/2 is the semi-perimeter.

• How are medians different from altitudes?

  An altitude is a perpendicular segment from a vertex to the opposite side (giving the "height"). A median connects a vertex to the midpoint of the opposite side. They coincide only in equilateral triangles.