Calculate+the+Area+of+a+Triangle+Using+Sine+Area+Formula

16. Calculate the Area of a Triangle Using Sine Area Formula
-Why Sine Area Formula? I really enjoyed the section on solving triangles that are not specifically right triangles. At first it was hard for me to memorize all the formulas like the Law of Cosines and this Sine Area Formula, but when I did I found that solving triangles was simple if you had the correct information. I like geometry, and since this incorporated some of the basics we learned from two years ago I found a lot of the things that I already knew connected. After I finally memorized all the formulas, this section involving solving triangles that are not right triangles became one of my favorite sections.

-What is the Sine Area Formula? The Sine Area Formula is 1/2bcsinA (and variations, see Example 2)

-How do you derive the Sine Area Formula?

First, you must think back to Geometry and recall that the area formula for a triangle is 1/2 x base x height or 1/2bh.

In this situation, the area formula can be modified for the specific triangle shown above, as 1/2ch.

1. In order to solve for the unknown, h, assuming d is given, you must first find the sine of angle A, which is opposite/hypotenuse, or h/b. Then state sinA=h/b. Solve for h and then you get h=bsinA. This is the formula for the altitude, or height, of a triangle.

2. Plug in this new formula for h into the 1/2ch equation. You get 1/2bcsinA.

-What are some examples of the Sine Area Formula usage?

Example 1:



Using the same triangle above, assume angle A is 65 degrees, c is 1.3 meters, and b is .5 meters.

Plug in A, c, and b into the Sine Area Formula like this: 1/2(.5)(1.3)sin(65).

You should get the area as 0.294 meters squared.

Example 2:

This example will be using the same triangle from Example 1, but will be a little more difficult because instead of giving angle A, side b, and side c, you will now be given angle B, side a, and also side c.

To make this work you need to modify the Sine Area Formula. As in the introduction, make an equation finding instead the sine of angle B. So, sinB=h/a. Therefore, h=asinB, and the equation to use will be 1/2acsinB.

Given angle B as 32 degrees, side a as 1.1 meters, and side c as 1.3 meters again, plug all into the equation like so: 1/2(1.1)(1.3)sin(32).

You should get 0.378 meters squared.

Sources: []

-Portfolio Problem: Using the triangle from above, calculate the area given angle B, side a, and side c as, respectively, 74 degrees, 0.2 meters, and 0.5 meters.

Please note: The picture used in the examples above is a right triangle, giving angle C as 90 degrees, but for the purpose of this demonstration please disregard that angle marking, as the Sine Area Formula is specifically used for triangles that are not right triangles.

Instructional Video: []