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How to Find the Area of a Triangle with SAS

When you know the lengths of two of a triangle’s sides plus the measure of the angle between those sides (SAS), you can find the area of the triangle. This method requires a little trigonometry — you have to find the sine of the angle involved. But the formula is really straightforward.

If triangle ABC has sides measuring a, b, and c opposite the respective angles, then you can find the area with one of these formulas:

image0.png

For example, look at the 30-60-90 right triangle in the following figure.

Finding the area of the 30-60-90 triangle.
Finding the area of the 30-60-90 triangle.

First, find the area by using angle B and the two sides forming it.

  1. Choose the correct version of the formula.

    image2.png
  2. Find the sine of the angle.

    image3.png
  3. Substitute the values into the formula and simplify.

    image4.png

Now find the area by using angle C and the two sides forming it.

  1. Choose the correct version of the formula.

    image5.png
  2. Find the sine of the angle.

    image6.png
  3. Substitute the values into the formula and simplify.

    image7.png

Using the method involving angle A gives you the same result, of course. For a quick comparison, just use the formula

image8.png

to find the area, because you’re dealing with a right triangle:

image9.png

The methods all produce the same result.

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