Find the area of the shaded polygons: PLZZ HELPP

Step-by-step explanation:

Green

It can work well to divide the figure into a triangle of base 1 and height 3, and a trapezoid with bases 1 and 4 and height 1. The formulas to use will be ...

  A = (1/2)bh . . . . . triangle

  A = (1/2)(b1 +b2)h . . . . . trapezoid

The triangle area is ...

  A = (1/2)(1)(3) = 3/2

The trapezoid area is ...

  A = (1/2)(1+4)(1) = 5/2

So, the total green area is 3/2 +5/2 = 8/2 = 4.

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Note that the 4×4 grid has an area of 16 square units. The upper right (white) triangle has an area of (1/2)(4)(3) = 6, and the one on the left has an area of (1/2)(4)(3) = 6. Then the shaded area can be found by subtracting those white areas from the total: 16 -6 -6 = 4. This matches the above result.

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Yellow

The attached figure shows a dividing line 1 unit down from the top to separate the top triangle from the remaining figure. The triangle area is found using the above formula with a base of 3 and a height of 1.

The odd remaining figure has an area that may be best found by subtracting the areas that are not part of the figure. The three (white) triangles around the outside are shown in the attachment outlined with blue lines. Their areas are shown, figured using the above formula for triangle area.

The yellow area is ...

  (3/2) +(12 -1 -2 -9/2) = 3/2 +9/2 = 6

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Blue

The blue area can be divided into a square and a number of triangles, but it is just as easy to subtract the areas of the triangles around the outside:

The blue area is ...

  16 -1 -1 -1 -2 -1 = 10 . . . . . . (CCW from upper left)

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The last attachment was made using a graphics program. It figures the area of an enclosed polygon. The numbers it shows match our calculations above.