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Using Sine to Calculate the Area of a Triangle - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

Using the standard formula for the area of a triangle, we can derive a formula for using sine to calculate the area of a triangle. Using sine to calculate the area of a triangle means that we can find the area knowing only the measures of two sides and an angle of the triangle. There is no need to know the height of the triangle, only how to calculate using the sine function.

We can use sine to calculate the area of any triangle. It doesn't have to be a right triangle so let's say if I knew an angle c and the 2 sides that were adjacent to it. How would I calculate the area? Well to do that we're going to have to go back way back to the beginning of Geometry, when we said that the area of a triangle is its base times its height divided by 2. Well we can see that if I dropped and altitude down right here and if I called that my height h, we can see that our base is actually a in terms of this drawing and our height well we're going to have to use trigonometry to find our height.
If I look at angle c which apparently is the only that I know, if I look at the opposite and hypotenuses that's going to be sine. So I could say that the sine of angle c is equal to our height divided by the hypotenuse of this right triangle which is b. So if I want to solve for h because again I want to substitute in here b and h I would multiply both sides by the variable b. So b divided by b is 1 so we get our height is equal to b times the sine of the angle c. So substituting in b times the sine of c for h and a for b into this equation we're going to find that the area of this triangle is a times b times the sine of c all divided by 2. So if you want to calculate the area of any triangle and you don't know the height all you're needed to know is 1 angle and 2 included sides and you can use this formula a times b times the sine of c divided by 2.

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