Lattice Polygons Part 1

A lattice polygon:

polygon 1

It has straight sides, and each point lies on a lattice (grid). Lattice polygons are really fun because we can practice area calculations without getting messy numbers. For example: consider the following lattice polygon:

polygon 2

Since it is a rectangle, we can use the formula arearectangle = base * height which in this case is 2*3 = 6. We could also have counted the 6 squares inside the shape. Ok, that was too easy; we need to move on to a more complicated shape. How about the following lattice polygon:

polygon 3

If you use the square counting method, then the area is clearly 8. How could we apply our formula to get the same result? The shape is not a rectangle.

If you take a shape and cut it into pieces, those pieces will have the same area as the original shape. Therefore, if we are clever enough with our cuts, we can turn the shape into a bunch of smaller rectangles.

polygon 4

Now we have:

  • a rectangle with base 1 and height 1 (area 1)
  • a rectangle with a base of 3 and a height of 1 (area 3)
  • a rectangle with a base of 2 and a height of 2 (area 4)

Putting that all together, we have the following area: 1 + 3 + 4 = 8

Time to ramp it up a notch. We need one more formula and things will get crazy. The area of a triangle is areatriangle = 1/2 * base * height

Consider the original lattice polygon. Here is how we could split it up:

Polygon 5

We have the following:

  • a triangle of base 3 and height 1 (area 1.5)
  • a triangle of base 1 and height 1 (area 0.5)
  • a triangle of base 2 and height 2 (area 2)

Putting that all together, we have the following area: 1.5 + 0.5 + 2 = 4

One last, massive, lattice polygon:

Polygon 6

Here is one way to split up the giant shape:

Polygon 7

If you do all of the relevant calculations, you will find that the area is 18. Stay tuned for the next post.

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