Non-right Triangles with Law of Sines: Learn It 3

Finding the Area of an Oblique Triangle Using the Sine Function

Now that we can solve a triangle for missing values, we can use some of those values and the sine function to find the area of an oblique triangle. Recall that the area formula for a triangle is given as [latex]\text{Area}=\frac{1}{2}bh[/latex], where [latex]b[/latex] is base and [latex]h[/latex] is height. For oblique triangles, we must find [latex]h[/latex] before we can use the area formula. Observing the two triangles, one acute and one obtuse, we can drop a perpendicular to represent the height and then apply the trigonometric property [latex]\sin \alpha =\frac{\text{opposite}}{\text{hypotenuse}}[/latex] to write an equation for area in oblique triangles. In the acute triangle, we have [latex]\sin \alpha =\frac{h}{c}[/latex] or [latex]c\sin \alpha =h[/latex]. However, in the obtuse triangle, we drop the perpendicular outside the triangle and extend the base [latex]b[/latex] to form a right triangle. The angle used in calculation is [latex]{\alpha }^{\prime }[/latex], or [latex]180-\alpha[/latex].

Two oblique triangles with standard labels. Both have a dotted altitude line h extended from angle beta to the horizontal base side b. In the first, which is an acute triangle, the altitude is within the triangle. In the second, which is an obtuse triangle, the altitude h is outside of the triangle.

Thus,

[latex]\text{Area}=\frac{1}{2}\left(\text{base}\right)\left(\text{height}\right)=\frac{1}{2}b\left(c\sin \alpha \right)[/latex]

Similarly,

[latex]\text{Area}=\frac{1}{2}a\left(b\sin \gamma \right)=\frac{1}{2}a\left(c\sin \beta \right)[/latex]

area of an oblique triangle

The formula for the area of an oblique triangle is given by

[latex]\begin{align}\text{Area}&=\frac{1}{2}bc\sin \alpha \\ &=\frac{1}{2}ac\sin \beta \\ &=\frac{1}{2}ab\sin \gamma \end{align}[/latex]

This is equivalent to one-half of the product of two sides and the sine of their included angle.

Find the area of a triangle with sides [latex]a=90,b=52[/latex], and angle [latex]\gamma =102^\circ[/latex]. Round the area to the nearest integer.

Find the area of the triangle given [latex]\beta =42^\circ ,a=7.2\text{ft},c=3.4\text{ft}[/latex]. Round the area to the nearest tenth.