SOLUTION: find the area of the shaded polygon http://homework.russianschool.com/resource?key=17351ioj8gsb0

Question 1120647: find the area of the shaded polygon
http://homework.russianschool.com/resource?key=17351ioj8gsb0
Found 2 solutions by solver91311, greenestamps:
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

There are four unshaded right triangles surrounding the shaded quadrilateral. Calculate the areas of the four triangles, calculate their sum, then subtract that sum from the total area of the square figure.

John

My calculator said it, I believe it, that settles it

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

I'm going to try to get you to look at the problem in different ways to find different ways of finding the answer. In other words, I'm encouraging you to get some good mental exercise with the problem -- rather than just getting the answer the easiest way possible and then forgetting it.

Here is a picture of your polygon:

One common way to find the area of the polygon is to enclose it in a rectangle; then the area of the polygon is the area of the rectangle, minus the area of the triangular regions that are inside the rectangle but outside the polygon.

The picture might look like this:

The area of the rectangle is 4*4=16. Starting from lower left and going clockwise, the triangular regions outside the polygon are 1x2 (area 1), 3x2 (area 3), 1x3 (area 1.5), and 3x1 (area 1.5).

The are of the polygon is then

16 - (1+3+1.5+1.5) = 16-7 = 9.

And now here is another way to find the area of the polygon. You may find this method more to your liking than the method described above. (Or not.... Or maybe you don't like either one....)

Add more line segments to the figure so that each side of the polygon is the diagonal of a rectangle, like this:

Each side of the polygon now divides a rectangle into two congruent triangles; so the areas of those rectangles inside and outside the polygon are equal.

That means the area of the polygon is the area of the small enclosed rectangle (2x1, area 2), plus half of the remaining area of the large rectangle:

2 + (16-2)/2 = 2+7 = 9

Of course, you could also use this last figure to calculate the area of the polygon as the sum of the small rectangle and the four triangles that together make up the whole polygon:

2+(1+3+1.5+1.5) = 2+7 = 9

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