Geometry Question : Geometry Question By, Kousha Talebian
Slide2 : Find the shaded area if each curve is a quarter of a circle, and the square has side ‘r’.
Slide3 : Like all hard questions, we shall use the method of divide and conquer; the question given has a relatively complicated shape and cannot be analyzed directly.
Continue for step-by-step solution.
One common ground note:
The solution uses a co-ordinate system method. The origin in this question is set as the lower left hand corner of the square.
Slide4 : By removing two of the lines, we get the above image. Our very step is to determine the co-ordinate of point P.
To do this, we first need to define the two curve A, B. Both are part of a circle. Recall the equation of a circle of general form:
Further note that the base of the square is the radius of the circle. Thus, the two curves can be written as:
Equating A=B, and solving for x, yields the x-co-ordinate of P. Substituting this in either curve will give the y-co-ordinate:
P(x,y)=(r/2,r/2.sqrt(3))
Slide5 : Now, drawing the following lines, we can easily compute the two shaded area as outlined below.
First note that ‘h’ is the y-co-ordinate of P, and ‘b’ is x-co-ordinate.
Let us first compute the area of the purple, or the right angle triangle as simply base x height / 2:
Next step is to fine theta. This value is simply 90-alpha. Alpha is simply arctan of h/b (here I shall use radians). Note that the value of alpha is constant regardless of value of r.
Now, to calculate the blue area, we shall use the following ratio method,
saying that the total area of the circle has an angle of 2pi, and thus an angle of theta would have area blue
Slide6 : We can now compute the yellow area. From the previous image, if we add the blue and pink area, multiplied it by 2, and then have the area of the square subtracted from it, we get the yellow area:
Next step is to find the green area.
Slide7 : By looking at only one curve, it can be seen that the brown area is simply one quarter of the area of the circle. The purple area is then the area of the square minus the brown area:
Slide8 : The question is basically now done. The purple area if 2 yellow area plus one green area. Thus we can compute the green area as:
The shaded area is then the area of the square minus four times the addition of the green and yellow areas. Answer is on the next slide.
Slide9 : Copyrighted - Kousha Talebian