Finding the Area of a Triangle Using Sine
You are familiar with the formula to find the area of a triangle where is the length of a base of the triangle and is the height, or the length of the perpendicular to the base from the opposite vertex.
Suppose has side lengths , , and . Let be the length of the perpendicular to the side of length from the vertex that meets the side at .
Then, the area of the triangle is .
Now, look at . It is a right triangle with hypotenuse that has a length of units.
Consider the sine of .
Substituting the value of in the formula for the area of a triangle, you get
Similarly, you can write formulas for the area in terms of or .
Find the area of .
You have the lengths of two sides and the measure of the included angle. So, you can use the formula where and are the lengths of the sides opposite to the vertices and respectively.
Using the formula the area, .
Therefore, the area of is about sq.cm.
The area of the right with the right angle at the vertex is sq. units. If and , solve the triangle.
First, draw a figure with the given measures.
Use the Pythagorean Theorem to find the length of the third side of the triangle.
Now, you have lengths of the three sides and the area of the triangle.
Substitute in the area formula.
Solve for .
Taking the inverse,
That is, .
Given that the angle at the vertex is a right angle. Therefore, .
Using the Triangle Angle Sum Theorem , the measure of the third angle is,
Therefore, the measure of is .