Pick's Theorem
Pick's Theorem
Pick's Theorem
Area of a polygon = Sum of Interior points + 0.5 (Sum of Edge points including the vertices) - 1
Area of a polygon = Sum of Interior points + 0.5 (Sum of Edge points including the vertices) - 1
In the applet below, you can move the vertices of the triangle to change its shape and see how Pick's Theorem is applied to calculate the area.
In the applet below, you can move the vertices of the triangle to change its shape and see how Pick's Theorem is applied to calculate the area.
Proof
Proof
Step 1: Show that when 2 triangles areas are added together, Pick's Theorem still holds.
Step 1: Show that when 2 triangles areas are added together, Pick's Theorem still holds.
Step 2: Prove for the area of a rectangle with coordinates (0, 0), (a, 0), (a, b) and (0, b) where a and b are integers.
Step 2: Prove for the area of a rectangle with coordinates (0, 0), (a, 0), (a, b) and (0, b) where a and b are integers.
Step 3: Prove for any triangle with vertices on lattice points.
Step 3: Prove for any triangle with vertices on lattice points.