Triangle Area Calculator — SAS (Two Sides & Included Angle)

About the Triangle Area Calculator — SAS (Two Sides & Included Angle)

The SAS (Side-Angle-Side) method is one of the most reliable ways to solve a triangle: given two sides and the angle between them, the triangle is uniquely determined. This calculator takes side a, side b, and the included angle C, then computes everything else — the third side via the law of cosines, the remaining two angles, and all key properties.

The area formula for SAS is elegantly simple: Area = ½ · a · b · sin(C). This is derived from the standard base-times-height formula by expressing the height in terms of the included angle. The law of cosines c² = a² + b² − 2ab·cos(C) gives the third side, and then the law of sines or additional cosine applications yield the remaining angles.

Beyond the basics, this calculator reports the circumradius R = abc / (4·Area), the inradius r = Area / s (where s is the semi-perimeter), all three altitudes, and classifies the triangle by its angles (acute, right, or obtuse) and sides (scalene, isosceles, or equilateral). Preset buttons let you explore classic triangles — from the 3-4-5 right triangle to equilateral and obtuse cases.

SAS problems appear constantly in surveying, navigation, physics (force resolution), and construction. Any time you can measure two lengths and the angle between them, this calculator gives you a complete picture of the triangle.

When This Page Helps

This calculator is useful when two measured sides and the included angle are the only reliable field or drawing dimensions available. From that SAS input it immediately produces the third side, full angle set, and area, so you can move from partial measurements to a complete triangle without chaining several separate formulas by hand. That is especially helpful in surveying layouts, construction geometry, truss checks, and classroom problems where the included angle is the cleanest known value.

How to Use the Inputs

1. Enter side a in the chosen unit (mm, cm, in, m, or ft).
2. Enter side b in the same unit.
3. Enter the included angle C between sides a and b (degrees or radians).
4. Or click a preset to load a classic SAS triangle.
5. View the computed third side c, all angles, area, and perimeter.
6. Check altitudes, circumradius, inradius, and classification.
7. Use the bar charts to visualize side and angle proportions.

Example Calculation

Tips & Best Practices

• The SAS case always produces exactly one triangle — there is no ambiguity, unlike the SSA case.
• When angle C = 90°, the formula simplifies to Area = ½ab and c = √(a² + b²) — the Pythagorean theorem.
• To convert from radians to degrees, multiply by 180/π. This calculator supports both input formats.
• If the included angle is obtuse (>90°), cos(C) is negative, making c larger than √(a² + b²).
• The circumradius equals the hypotenuse divided by 2 for a right triangle.

Area from the Included Angle

The SAS area formula works because the included angle determines the triangle height implicitly. If side a is used as the base, the height contributed by side b is b·sin(C), so the area becomes 1/2 · a · b · sin(C). That is why SAS is one of the most efficient triangle setups: you do not need to construct or measure an altitude separately, and the same formula works for acute, right, and obtuse included angles.

Why SAS Gives One Unique Triangle

Unlike SSA, the SAS configuration does not branch into multiple possibilities. Once two sides and the angle between them are fixed, the third vertex can only sit in one position, so the triangle is uniquely determined. This makes SAS especially reliable in engineering and fabrication workflows where you want a single unambiguous result from limited measurements.

From Partial Measurements to Full Properties

After the area is found, the same SAS inputs support a full triangle solve. The third side comes from the law of cosines, the other two angles follow from trigonometry, and then perimeter, altitudes, inradius, and circumradius all fall out from standard formulas. In practice, that means a drawing that starts with just two members and the joint angle can be expanded into a complete geometric description from the same measurement set.

Frequently Asked Questions

• What is the SAS method for triangles?

  SAS (Side-Angle-Side) means you know two sides and the angle between them. This uniquely determines the triangle, and you can find all other properties using the law of cosines and the SAS area formula.

• What is the formula for the area of a triangle given two sides and the included angle?

  Area = ½ × a × b × sin(C), where a and b are the two known sides and C is the angle between them.

• How do I find the third side using SAS?

  Use the law of cosines: c = √(a² + b² − 2ab·cos C). This works for any included angle.

• What is the difference between SAS and SSA?

  SAS specifies the angle between the two known sides (included angle) and always gives one unique solution. SSA specifies the angle opposite one of the known sides and can give 0, 1, or 2 solutions (the ambiguous case).

• What is the circumradius of a triangle?

  The circumradius R is the radius of the circle passing through all three vertices. R = abc / (4·Area).

• Can SAS give an obtuse triangle?

  Yes. If the included angle C is greater than 90°, the triangle is obtuse at C. Even if C is acute, one of the other angles could be obtuse depending on the side lengths.