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Side-Angle-Side (SAS) Triangle: Definition, Theorem & Formula

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  • 0:05 The SAS Theorem
  • 2:14 A Review of Trigonometry
  • 2:55 Using SAS to Calculate Area
  • 4:16 Lesson Summary
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Lesson Transcript
Instructor: Betty Bundly

Betty has a master's degree in mathematics and 10 years experience teaching college mathematics.

In this lesson we will learn about trigonometry and geometry. The SAS theorem ties together those two subjects into a formula which allows us to find the area of a triangle.

The SAS Theorem

Triangles may seem to be just another geometric shape, with three sides and three angles. However, they are special enough to have an entire subject devoted to them—trigonometry. What makes triangles so interesting is that certain sides and certain angles relative to those sides have a special relationship. One of these relationships is found in the SAS theorem and the SAS formula (which uses the principles of the theorem to calculate the area of a triangle).

The SAS theorem states that two triangles are equal if two sides and the angle between those two sides are equal. The angle between the two sides is also called the included angle. In this diagram, if angle C = angle X, and side a = side z and side b = side y, then by the SAS theorem, these two triangles would be equal. SAS stands for side angle side.

Triangles ABC and ZYX

In the diagram, notice that the angles are represented with capital letters and the sides of the triangle are represented with lowercase letters. If SAS is true for two triangles, this means that everything else about these triangles is equal—all the other angles, all the other sides, and the area of the triangles.

This theorem from geometry led to a formula in trigonometry, called the SAS area formula. Using the SAS area formula, you can find the area of a triangle if you know the length of two sides of a triangle and included angle. Specifically, if the sides of the included angle are symbolized as b and c, and the included angle is called A, then:

area of triangle ABC = (b*c*sin A) / 2

It's important that the angle you use for the area calculation lies between the two sides you use for the calculation. In fact, using this idea for any triangle, there are three ways to calculate the area.

Triangle ABC

Why? Looking at triangle ABC, we see there are three angles and each one is included between two different sides: angle C is the included angle for sides a and b; angle B is the included angle for sides a and c; angle A is the included angle for sides b and c. Any set of this trio of parts could be used to calculate the area, if you know the value of those three parts.

A Review of Trigonometry

Before we move forward with SAS triangles, let's review a little trigonometry. A common formula from geometry is the formula for the area of a triangle. This formula says that area = b*h / 2, where b is a side of the triangle called the base, and h is the height of the triangle, where the height is always at 90 degrees to the base. Using SAS and this area formula, we will see why the SAS area formula works.

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