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Area of a Triangle Using Sine

Rule

Area of a Triangle Using Sine

The area of a triangle is half the product of the lengths of any two sides and the sine of the included angle.

Triangle

Based on the diagram above, the area of can be found using any of the three formulas below.

Proof

Proof

To find the first formula, start by drawing the altitude from and let be its length. Since the altitude is perpendicular to the base, it generates two right triangles.

Triangle with one altitude drawn

Because is a right triangle, the height of the triangle can be related to the sine of using the sine ratio. Finally, substitute the expression found for into the general formula for finding the area of a triangle.

To obtain the second formula, notice that is also a right triangle and then, the sine ratio can be applied. Substituting this expression into the general formula for finding the area of a triangle gives the first formula.

To deduce the third formula, it is needed to draw the height from or

In this case, the length of the base is and the height is Since is a right triangle, the sine ratio can be used. Lastly, substitute the expression for into the formula to find the area of