How to Find the Area of a Scalene Triangle

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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 area. It's also possible to calculate 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.

TL;DR (Too Long; Didn't Read)

The area of a scalene triangle with base b and height h is given by 1/2 bh. If you know the lengths of all three sides, you can calculate area using Heron's Formula without having to find the height. If you know the value of an angle and the lengths of the two sides that form it, you can find the length of the third side using the Law of Cosines and then use Heron's Formula to calculate area.

General Formula for Finding Area

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} \text{ base} × \text{height}

Heron's Formula

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 half-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 half-perimeter

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)}

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 calculate area using Heron's Formula.

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