The Shoelace Formula

Hi, today I will introduce you all to the Shoelace Formula, which is also known as Gauss’s Area Formula. This formula would provide a quick and easy way to find the area of a triangle given the coordinates of the triangle. Since this formula would be hard to explain without an appropriate visual, I added in a picture with a diagram of a triangle along with a summary of steps. Make sure you refer to this picture as you read the explanations for each step :)

Step 1: Write down all the x and y coordinates in a manner shown in “Step 1” in the picture. Note that one of the coordinates will have to be written twice. The order in which you list the coordinates do not matter.

Step 2: Starting on the x coordinates, draw an arrow moving down diagonally (depicted with pink lines). Starting on the y coordinates, draw an arrow moving up diagonally (depicted with blue lines).

Step 3: Multiply the pair of values that are tied by a line. For the values that were connected with a pink line, place them in the absolute value without changing their signs and for values that were connected with a blue line, place them in the absolute value with an opposite sign (in other words, add a negative sign).

I apologize if my explanation was a little confusing or vague, but in my next post, I will put up an example and you guys will see how convenient and quick this formula can be in some cases. There are so many different ways to find the area of a triangle. However, when you are only given the coordinates of each vertex, the Shoelace Formula would most likely be the most convenient. For instance, if we tried to use the traditional 1/2 times bases times height, it would be hard to find the height. If we tried to use Heron's formula, we would have to start out by finding the lengths of each sides (which is possible, of course, using the distance formula), which can be very time consuming. In my Calculus class, we approached this same problem using concepts of Calculus; we applied the method of finding area between two curves, and this process to turned out harder and more time consuming. So I hope you all find this method helpful!! Have a great restful day!