mathschallenge.net logo

Quarter Circles

Problem

Four quarter circles are drawn from each vertex in a unit square.


Find the area of the shaded region.


Solution

Consider the diagram below.


By using the unit circle (see diagram below), we can find the area between 0.5 and a by integrating under the curve, x² + y2 = 1.
That is, y = radical(1 minus x2); taking positive root, because we only want the top part of the circle.


Using the Pythagorean Theorem, 12 = a2 + ½2 implies a = radical3/2.

Area under curve =
radical3/2


1/2
radical(1 minus x2) dx

Using the substitution, x = sin(u), dx = cos(u) du, and we get:

Area under curve =
π/3


π/6
cos2(u) du
  =
π/3


π/6
½(1 minus sin(2u)) du
therefore Area under curve = ½[u + ½cos(2u)]
π/3
π/6
= π/12

Area of rectangle = ½(radical3/2 minus 1/2) = (radical3 minus 1)/4.

therefore Area of ¼ shaded region = π/12 minus (radical3 minus 1)/4.

Hence the area of the shaded region is, π/3 minus radical3 + 1.

Problem ID: 142 (Dec 2003)     Difficulty: 4 Star

Only Show Problem