Geometry Question : Geometry Question By, Kousha Talebian
Slide2 : Given the square above with side ‘r’, and that each curve is a quarter of a circle, find:
Area red
Area purple
Slide3 : Please refer to this question, entitled ‘Geometry Question 1’, by Kousha Talebian, for a solution of the area of red and purple combined.
You should be able to complete ‘Geometry Question 1’ before continuing.
Slide4 : The question becomes relatively simple if we notice couple of facts:
The circle will touch the four curved sides exactly in the middle
The diameters of the square also pass through the four points above (and hence the diameter of the circle)
By noticing this, we can construct the image above. We can now define the two curves:
Now, we can find the intersection of the diameter (line y = x ), with the two curves (Points P&Q).
Slide5 : By equating A = y & B = y, we can solve for the x-co-ordinate of P and & Q:
We can now find L by subtracting P-Q. Further note that theta is 45 here. So we can now easily compute d as:
Slide6 : Using simple formula for area of the circle, we can find area purple. Then subtracting the area determined from ‘Geometry Question 1’, gives us area red. We can then find the ratio between the two.
Copyrighted – Kousha Talebian : Copyrighted – Kousha Talebian