Math Problem

Math Problem

I like these little math problems i think math at school should be taught this way.

Here is the problem find the Area of the Purple.

This can be done in 5 Steps.

Step 1 – Area of the Quarter Circle

We know the math formula for the area of a Circle which is PI*r^2 we know that the Radius of the large circle is 2×3 = 6 so the area of the quarter circle is 1/4 * PI * 6^2 = 1/4 * PI * 36 or 9*PI.

Step 2 – Area of the Larger Half Circle

Next we can calculate the area of the large half circle which has a radius of 3, using 1/2 * PI * 3^2 = 1/2 * PI * 9 or 4.5*PI.

Step 3 – Calculate the Radius of the Smaller Half Circle

So, we can draw a Line through the center of both smaller circles which defines a right-triangle with the following sides.

• 6-r
• r+3
• 3

We can now use Pythagorean theorem to calculate r, X^2 + Y^2 = Hyp^2 lets substitute.

(6-r)^2 + 3^2 = (r+3)^2, lets expand

(6-r)*(6-r) +9 = (r+3)*(r+3) ,next well multiple out both sides

6*6 -6r -6r + r^2 + 9 = r^2 +3r + 3r + 9 ,next we can simplify

36 – 12r + r^2 + 9 = r^2 + 6r + 9 ,lets simplify further

r^2 – 12r + 45 = r^2 + 6r + 9 , lets bring all of the terms to one side

( r^2 – r^2 ) + ( -12r -6r ) + (45 – 9) = 0, lets simplify

-18r + 36 = 0

finally we can move the -18r back over and divide both side by 18

36 / 18 = 18r/18 or r=2

Step 4 – Area of the Smaller Half Circle

This is easy since we now know the radius is 2, the area of the smaller half circle which has a radius of 2, using 1/2 * PI * 2^2 = 1/2 * PI * 4 or 2*PI.

Step 5 – Final Calculation

Area of the Purple = 9*PI – (4.5*PI+2*PI )

28.2743units – 20.4203units = 7.8539units

There we go