To find the area of a non-right triangle, let’s first review the standard area formula of a right triangle. A right triangle is made up of three sides: the base, the height, and the hypotenuse. To get the area of a triangle you must multiply the two adjacent side lengths of the 90° angle, which are the base and the height of the triangle, and divide this quantity by half. This is the formula for the area of a right triangle:
However, this formula doesn't work as effectively for acute and obtuse angles. So here's how to find the area of a non-right triangle.
Formulas for the Area of a Non-Right Triangle
When you use trigonometry, there's another group of formulas that can be used to find the area of a triangle with no right angles. Using these formulas, you can find the area of a non-right triangle even when there's a missing side length. There are different ways to find the areas of an obtuse triangle vs. an acute triangle.
Finding the Area of an Acute Triangle
When you need to find the area of an acute triangle, you must use the law of sines in place of a missing side length. Depending on which are the known sides or known angles, one of the following formulas can be used to find the area of an acute non-right triangle:
See the below acute triangle ΔABC. You can see that we do not know the length of all the sides of the triangle, but we do know that the acute angle measures 56° and that the two adjacent side lengths are 18 and 12. We would use the sine function on a calculator to plug this into the formula , where ab represents the known length sides of a triangle. The area of the triangle is 88.47.
When using a trigonometric formula for finding the area of an acute non-right triangle, a capital "C" is used to represent the known angle that is across from the opposite side length represented by lowercase "c".
Finding the Area of an Obtuse Triangle
The functions of sine, cosine, and tangents, can only be used to find the area of a triangle with an acute angle. So you must use a different method to find the area of a triangle with an obtuse angle. In order to find the area of the below triangle, you must draw a straight line from point C and point A, creating a right triangle where the two lines intersect.
The angle of CAE is supplementary to angle CAB, meaning that the two angles add up to 180 degrees. We can now assume that ∠CAE = 180 -∠A and from the area of ΔCAE, we can tell that sin∠CAE =sin (180-∠A). We can now substitute this formula to get and . Plug this into the standard formula of a triangle to get the following formula:
Formula for Success
Whether you need to find the area of an oblique triangle, obtuse triangle, or if you have two missing angles, if we know two sides and the included angle, we can find the area of a non-right triangle. If you’re stuck on this problem, feel free to bookmark this page as a guide.