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Calculate the area of the shape from problem #16 by using Pick's Law

Calculate the area of the shape from problem #16 by using Pick's Law-example-1

1 Answer

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Answer:

14 square units

Explanation:

You want to use Pick's theorem to find the area of the figure shown.

Pick's theorem

The area of a simple polygon can be found using the formula ...

A = i +b/2 -1

where i is the number of interior points, b is the number of boundary points.

We can divide the figure into three triangles and sum their areas.

Left triangle

The left triangle has 4 interior points and 12 boundary points. Its area is ...

A1 = 4 +12/2 -1 = 9 . . . . square units

Center triangle

The center triangle has 1 interior point and 8 boundary points. Its area is ...

A2 = 1 +8/2 -1 = 4 . . . . square units

Right triangle

The right triangle has 0 interior points and 4 boundary points. Its area is ...

A3 = 0 +4/2 -1 = 1 . . . . square unit

Total

The total shaded area is ...

A = A1 +A2 +A3 = 9 + 4 + 1 = 14 . . . . square units

The area of the shape is 14 square units.

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Additional comment

Here, we have treated the figure as 3 separate triangles. One can consider the figure as having one boundary containing 24 points (2 points are counted twice). Then the "-1" in the formula needs to be adjusted to "-3" to account for the fact that there are 3 separate enclosed areas:

A = 5 +24/2 -3 = 5 +12 -3 = 14 . . . . square units

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