A general formula for the area of a polygon

• In this advanced topic, we show that the area of a polygon in the plane is given by a simple formula involving the coordinates of its vertices.

• Click on any vertex near the top of the polygon. You will see a yellow trapezoid. The line segment at the top of the trapezoid joins two vertices, P(x,y) on the left and Q(X,Y) on the right. Notice that y and Y are the lengths of the trapezoid's vertical sides, which are its bases. Since the bases are vertical, and the height h is measured perpendicular to the bases, the height is equal to the horizontal distance X - x. Therefore the yellow trapezoid's area is h(b1 + b2) = (X - x)(Y + y)/2 .

• To understand our method for finding the polygon's area, click 'Under top.' The area T of the shaded red region is obtained by adding all the trapezoid areas (X-x)(Y+y)/2 for edges PQ above the polygon, with P to the left of Q.

Similarly, click 'Under bottom.' The area B of the shaded red region is obtained by adding all the trapezoid areas (X-x)(Y+y)/2 for edges PQ below the polygon, with P to the left of Q.

• The polygon's area is the 'under top' area minus the 'under bottom' area, which is T + -B, since subtracting B means adding -B. Notice that -B is obtained by adding all expressions (X-x)(Y+y)/2 for edges PQ below the polygon, with P to the right of Q.

• Move clockwise along the edges of the polygon. For each edge, let P(x,y) be the current vertex and let Q(X,Y) be the next vertex as you move clockwise around the polygon. Along the top of the polygon, P is to the left of Q. Along the bottom of the polygon, P is to the right of Q.

Conclusion: To find the polygon's area, add up all expressions(X-x)(Y+y)/2 as you move clockwise around the polygon.

• Drag the green slider to get polygons with more vertices. Also, clicking 'no trapezoids' will allow you to move any vertex when you click on it.

• This area formula is really a special case of an advanced calculus formula called Green's Theorem.