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Solve question 3 on the attached worksheet. Explain your working.

There are two questions here. I am answering the first of the questions (number 3 on the worksheet) as the policy is one question per post.


The length AB is equal to the radius of the circles (which have the same radius), as are the lengths AC and AD, BC and BD, so we have


AB = AC = AD = BC = BD = 1


This implies that both the triangles ABC and ABD are equilateral triangles with equal lengths of sides 1 and equal angles of 180/3 = 60 degrees.


Angle CAD is equal to angles CAB + BAD = 120 degrees. The area of the circle segment from C to D is then a third of the area of the circle centered at A, since CAD = 120 = 360/3 (a third of the full 360 degrees of the circle).


The area of the circle centered at A is given by the formula for the area of a circle


Area =  `pi r^2 ` 


where r is the radius. Here we have `r=1 `, so that the area is equal to `pi ` . The area of the circle segment from C to D (call this area S) is hence a third of this. So we have that the area `S = pi/3 `.


The shaded area we are required to calculate is the area of the circle segment (S) minus the area of the triangle ACD (call this T). Note that the area of the triangle ACD is equal to that of the triangles ABC and ABD since it is made up of half of each of those and they have the same area. Those triangles are equilateral with sides equal to 1 so that their area is half base x height = `0.5 times sqrt(1^2 - 0.5^2) = 0.5 times sqrt(3)/2 = sqrt(3)/4 `


Therefore the area of the triangle ACD is  `T = sqrt(3)/4 ` and the shaded area we are required to calculate is given by


`S - T = pi/3 - sqrt(3)/4 `


Finally the area common to both circles is equal to twice the shaded area, so is equal to 2(S - T). If this is indeed 39% of the area of either circle, this being 0.39pi, it will be approximately equal to 1.23. This is the case since


` `` 2(S-T) = 2 (pi/3 - sqrt(3)/4) = 2 times 0.6141848 approx 1.23 approx 0.39pi `


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