Question

Each square in the grid below has area 1. Find the area of the irregular quadrilateral below.

Answer 1

Answer:

31.275 units²

Step-by-step explanation:

We'll find the total area by breaking up this given area into two separate parts, finding the area of each part separately, and then adding these two sub-areas together.

First, draw a horizontal line through the rightmost vertex, which creates a triangle of base 7.5 and height 5.  

The formula for the area of a triangle of base b and height h is A = (1/2)(b)(h).  Thus, the area of the triangle we have created is

A = (1/2)(7.5)(5) = 18.75 units²

The bottom half of the given irregular quadrilateral is a trapezoid of width 3 units.  We must use the Pythagorean Theorem twice to find the lengths of the two legs.  On the far left we have a triangle with longer side 3 and shorter side 1.5; the hypotenuse of this triangle is

√(1.5² + 3²), or √11.25.  This is the measure of the shorter leg of the trapezoid.

On the far right we have another triangle with loger leg 4 and shorter leg 3.  The length of the hypotenuse is also the length of the longer leg of the trapezoid.  It is √(3°2 + 4°) = 5.

Now we'll use the formula for the area of a trapezoid:

A = (average length of legs of trapezoid)(width of trapezoid)

        √11.25 + 5                8.35              25.06

   = ------------------- * 3  = ----------- * 3 = ------------- = 12.53 units²

                 2                          2                     2

Our last step is to add together the triangle area found earlier and the trapezoid area just found here:

Total area = 18.75 units² + 12.53 units² =  31.275 units²

The total area of the irregular quadrilateral is 31.275 units²

Answer 2

Answer:

37.5

Step-by-step explanation:

altavistard that is incorrect... you were very close..

Btw: I know this is right because its in AOPS! credit to AOPS

STRATEGY:

I cut the blue shape into 4 different shapes  (i'm using color coding for this)

The green triangle has base 4 and height 8, so its area is 1/2*4*8=16

The yellow rectangle has width 1 and height 3, so its area is 1*3=3 ( I didn't need to multiply 1/2 on this because it wasn't a triangle..)

The orange triangle has base 5 and height 5, so its area is 1/2*5*5=12.5

The red triangle has base 4 and height 3, so its area is 1/2*4*3=6

Adding these together, we see that the blue quadrilateral has area 16+3+12.5+6= 37.5!

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