SAS Triangle Calculator

Given two sides and the included angle (SAS), calculate the third side, all angles, area, and perimeter.

Enter Two Sides & Included Angle

Result

Area
--
square units
Side a--
Side b--
Side c--
Angle A--
Angle B--
Angle C--
Perimeter--

Step-by-Step Solution

What is the SAS Method?

SAS stands for Side-Angle-Side. When you know two sides of a triangle and the angle between them (the included angle), you can determine all remaining properties of the triangle. This is one of the fundamental triangle congruence criteria and a common scenario in trigonometry.

Formulas Used

Law of Cosines

Find the third side from two sides and the included angle.

c = sqrt(a^2 + b^2 - 2ab*cos(C))

Law of Sines

Find remaining angles after computing the third side.

sin(A)/a = sin(B)/b = sin(C)/c

Area (SAS)

Direct area formula using two sides and included angle.

Area = (1/2) * a * b * sin(C)

Step-by-Step Process

  1. Use the Law of Cosines to find the unknown side c.
  2. Use the Law of Sines to find angle A (opposite side a).
  3. Calculate angle B = 180 - A - C.
  4. Compute the area using Area = (1/2)ab sin(C).
  5. Sum all three sides for the perimeter.

When to Use SAS

The SAS method is applicable whenever you have measurements for two sides and the angle between them. This commonly occurs in surveying, navigation, engineering design, and physics problems involving force triangles.