Expert: Abe Mantell - 9/2/2008

Question
My ninth grader came home from his frist day in physics with this question...Samuel will use stamped concrete for a patio.  He designed the parallelogram pattern shown below.  Find the area of the design to the nearest tenth of a square foot.  They show a parallelogram with a 110 degrees written in the bottom right corner, 8 in above the top line and below the bottom line, 5 inch outside the right and left sides.  Please help us.  Thank you for your time.

Answer
Hello Gerri,

There is a nice formula for this...the area of a parallelogram is:
A=a*b*sin(C), where a and b are the length of the sides that form the
angle C.  In this case, a=8, b=5, and C=110 degrees.  So, the area is
8*5*sin(110) which is about 40*0.9396926 or about 37.6 square feet.

If he is not allowed to use that formula, then simple trigonometry
will do.  Construct a right triangle at that corner, draw the height
perpendicular from the top edge to the lower right corner...so we get
a right triangle with hypotenuse 5 and an angle of 20 degrees (110-90).
Now all we need is the height (since the area is base x height).
The height, h, can be found using right triangle trigonometry.
cos(20)=adjacent/hypotenuse=h/5 ==> h=5*cos(20) or about 4.698463...
so the area is about 8 * 4.698463 or about 37.6 square feet (same as
above).

OK?

Abe