Introduction: Activities - Pick's Theorem
Okay, good, now let's do it another way. There's a method of calculating the area of any polygon on a geoboard quickly and easily using what's called Pick's theorem. It's almost too good to be true, but Pick's theorem computes the area of an arbitrary polygon simply by counting pegs. Note that the polygon in Figure 7 touches ten pegs and surrounds six pegs. Now if we take the number of boundary pegs (which is 10 in this case) and divide it in half, add the number of interior pegs (which is 6) and subtract one, we get
which should agree with what you obtained earlier by summing the areas of the polygon's constituent squares and triangles.
Let's check to see that Pick's theorem holds for two of the simple examples in Figure 6. The rectangle in Figure 6a, for instance, has ten boundary pegs and two interior pegs. By Pick's theorem, the area is
which checks with our earlier calculation via the area formula
A = b x h.Similarly, for the parallelogram in Figure 6b, we have
We leave it to you to use Pick's theorem to verify the areas of the triangle and trapezoid in Figures 6c and 6d.
Pick's theorem, then, can be stated as follows. Let b be the number of boundary pegs and i be the number of interior pegs of any polygon on the geoboard. Then the area of the polygon is given by
With a little practice, you'll be able to simply look at a polygon and compute its area in your head.
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