SOLUTION: (49) In the diagram, ABCD is a unit square. Arc AC has center D and arc BD has center C. Find the shaded area in square units. Link to diagram: https://ibb.co/zhCbbmhz

Algebra ->  Circles -> SOLUTION: (49) In the diagram, ABCD is a unit square. Arc AC has center D and arc BD has center C. Find the shaded area in square units. Link to diagram: https://ibb.co/zhCbbmhz      Log On


   

Question 1209525: (49) In the diagram, ABCD is a unit square. Arc AC has center D and arc BD has center C. Find the shaded area in square units.
Link to diagram: https://ibb.co/zhCbbmhz
Answer by ikleyn(52921) About Me  (Show Source):
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(49) In the diagram, ABCD is a unit square. Arc AC has center D and arc BD has center C.
Find the shaded area in square units.
Link to diagram: https://ibb.co/zhCbbmhz
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The shaded area is the sum of areas of two 60°-degrees sectors of a circle of the radius of 1 unit

minus the area of the equilateral triangle with the side length of 1.


The area of one 60°-sector of a circle of the radius 1 is 1/6 of the area of the circle, i.e.

    %281%2F6%29pi%2A1%5E2 = pi%2F6.


The area of the equilateral triangle with the side length 1 is  sqrt%283%29%2F4.


Therefore, the shaded area is  

    pi%2F6 + pi%2F6 - sqrt%283%29%2F4 = %282pi%29%2F6 - sqrt%283%29%2F4 = pi%2F3 - sqrt%283%29%2F4 = 3.14159265%2F3 - sqrt%283%29%2F4 = 0.614184848  square units (approximately).


ANSWER.  The shaded area is  pi%2F3+-+sqrt%283%29%2F4 = 0.6142  square units (approximately).

Solved.