shoelace

Coordinate geometry is one of the rare topics from which problems appear in CAT. The questions are pretty simple and formula based but many a time, students do not know shortcuts and direct formulae that make their life a lot easier. One such concept is that of the shoelace formula or the shoelace theorem.

Simply put, if you have three or more coordinates given which form a closed figure, and you would need to find the area, it would look difficult at the first glance, especially for an untrained aspirant. But, if you know the shoelace formula, you can calculate the area in a couple of steps.

shoelace 1

You can find the area of the triangle using Heron’s formula by finding individual lengths or find the perpendicular distance from a vertex to the opposite base and then using ½ * base * height. But these methods would be time consuming and the more numbers you deal with, higher are the chances of making silly mistakes. We can use Gauss’ area formula, commonly known as the shoelace formula to calculate areas in these cases.

  1. You have to simply write down the closed figure as ABCA (it should be closed and so, the starting vertex and the last vertex should be same and the vertices should be in the order in which they appear (clockwise or counterclockwise as you may wish)
  2. Write the coordinates of these points as follows:shoelace 2

3. Starting with the first element, multiply the pairs in a diagonal fashion and add them. Essentially, get the sum of x1y2 + x2y3 + x3y1

4. Starting with the bottom element in the second row, multiply the pairs in a diagonal fashion and add them. Essentially, get the sum of y1x2 + y2x3 + y3x1

5. The value of the determinant will be given by the difference between these two terms. Multiply it by half and you get the answer

So, in this case, the answer will be:

shoelace 3

which will be the area of the closed figure.

You can use this formula for any closed figures. Just remember to ‘close’ the figure when you are putting it in the expression and follow the steps and you should be good to go.

Examples:

What is the area of the polygon formed by the points (3,4), (5,11), (12,8), (9,5), and (5,6)?

Simply write it down as

shoelace 4

The applications of the concept are immense. While direct area based questions can be tackled using this formula, this concept will be particularly useful when you are trying to find out the number of points with integral coordinates inside a closed region which we will be covering up next.

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