There are many ways to find the area of an object, with the dimensions of its sides, with angles or even with the location of its vertices. Finding the area of a polygon with the use of its vertices takes a fair amount of manual calculation, especially for larger polygons, but is relatively easy. By finding the product of a point's x coordinate times the next point's y coordinate, then subtracting the y coordinate of the first point times the x coordinate of the second coordinate and dividing by two, you will find the area of the polygon.
Multiply the x coordinate of the first point with the y coordinate of the second point. For example, the first point is at 2,3 and the second is 4,5, so you would multiply 2 by 5, getting a product of 10.
Multiply the y coordinate of the first point by the x coordinate of the second point. For example, the product of the two points (the first at 2,3 and the second at 4,5) would be 12.
Subtract the second number from the first. For example, the products would be subtracted (10-12) leaving a difference of -2.
Multiply each of the points with its corresponding counterparts. For example, the second point's coordinates would be multiplied by the coordinates from the third point. When you reach the final point, you will simple multiply it with the first number.
Add the final numbers left from the differences together, to get a single number. Divide this number by 2 and the quotient will be your polygon's area.
TL;DR (Too Long; Didn't Read)
The formula for determining the area of a polygon based on its coordinates is: ( (X1Y2 - Y1X2) + (X2Y3 - Y2X3)+...(XxY1-YyX1) ) / 2 = polygon's area.