How To Find The Area Of A Scalene Triangle

Unlike an equilateral triangle with its three equal sides and angles, an isosceles one with its two equal sides, or a right triangle with its 90-degree angle, a scalene triangle has three sides of random lengths and three random angles. If you want to know its area, you need to make a couple of measurements. If you can measure the length of one side and the perpendicular distance of that side to the opposing angle, you have enough information to calculate the area. It's also possible to calculate the area if you know the lengths of all three sides. Determining the value of one of the angles as well as the lengths of the two sides that form it also allows you to calculate area.

In trigonometry, if the sides of a triangle are all different lengths (i.e. unequal sides), we can determine it to be scalene. This also means that the interior angles will all be distinct.

Consider a random triangle. It's possible to scribe a rectangle around it that uses one of the sides as its base (it doesn't matter which one) and just touches the apex of the third angle. The length of this rectangle equals the length of the side of the triangle that forms it, which is called the base (‌b‌). Its width is equal to the perpendicular distance from the base to the apex, which is called height (‌h‌) of the triangle.

The area of the rectangle you just drew equals ‌b‌ × ‌h‌. However, if you examine the lines of the triangle, you'll see they divide the pair of rectangles created by the perpendicular line from the base to the apex exactly in half. Thus, the area inside the triangle is exactly half that outside it, or 1/2 ‌bh‌. For any triangle:

\(\text{Area} = \frac{1}{2}\times \text{base} \times \text{height}\)

A scalene triangle's area also follows this relationship, and by multiplying the height of a scalene triangle by its base we the area. However sometimes it can be very difficult to find the height, especially if there is an obtuse triangle which can complicate computational sine and cosine calculations. In these cases we can use the following scalene triangle formulas to find the area.

Mathematicians have known how to calculate the area of a triangle with three known sides for millennia. They use Heron's Formula, named after Heron of Alexandria. To use this formula, you first have to find the semi-perimeter (‌s‌) of the triangle, which you do by adding all three sides and dividing the result by two. For a triangle with sides ‌a‌, ‌b‌ and ‌c‌, the semi-perimeter is half the perimeter of the triangle:

\(s = \frac{1}{2}(a + b + c)\)

Once you know ‌s‌, you calculate area using this formula:

\(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\)

This works to find the area of the scalene triangle defined by sides ‌a, b,‌ and ‌c‌. A scalene triangle is the general case to find the area of any triangle, so this area formula works on an type of triangle (equilateral and isosceles triangles included)!

Using the Law of Cosines

Consider a triangle with three angles ‌A‌, ‌B,‌ and ‌C‌. The lengths of the three sides are ‌a‌, ‌b‌ and ‌c‌. Side a is opposite angle ‌A‌, side ‌b‌ is opposite angle ‌B‌, and side ‌c‌ is opposite angle ‌C‌. If you know one of the angles – for example, angle ‌C‌ – and the two sides that form it – in this case, ‌a‌ and ‌b‌ – you can calculate the length of the third side using this formula:

\(c^2 = a^2 + b^2 – 2ab\cos C\)

Once you know the value of ‌c‌, you can find the perimeter of a scalene triangle and calculate the area using Heron's Formula. The law of cosines may look very similar to the Pythagorean theorem where ‌c‌ would be the hypotenuse. It is in fact closely related to this iconic relationship, and the differentiating last term acts as a correction factor. Most scalene triangles are not right-angled triangles, but the sides of the scalene triangle are still linked together by this relationship in the law of cosines. A scalene triangle is also still in agreement with the law of sines.