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mathsmarts
a square is inscribed in a circle, each side of sq measures 4^2 in. Find an expression for the exact area of the shaded region... That being outside the square within the circle...I know how to get the shaded area.. I need the expression which is (16 pi - 32) but I dont understand the answer. can you help with that???
area of circle - area of square
I know how to get the shaded area.. I need the expression which is (16 pi - 32) but I dont understand the answer. can you help with that???
is this correct? |dw:1385249213040:dw|
|dw:1385250369163:dw|
|dw:1385250006166:dw| The area of the shaded region of the problem you described is \[A= 128 \pi - 256\]
|dw:1385250495005:dw|
on the right, it should say \[r=c/2\]
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To understand the answer, we can look at another example. Say we have two rectangles, one that's 8x5 and another 2x5 - the smaller one is inset in the other. |dw:1385273283416:dw|
If we want to find the area of the "shaded region" (like in the circle problem) |dw:1385273501118:dw| we can subtract the area of the little rectangle from the bigger rectangle \[8*5-2*5 = 40-10=30\] In this instance, it's easy to see that the shaded area is the larger area minus the smaller area - we check it knowing that the rectangular shaded region has dimensions 6x5 |dw:1385273671303:dw| \[6x5=30\] So the area of the shaded rectangle is proved to be equivalent to the area of the large rectangle minus the area of the small rectangle \[6*5 = 8*5-2*5\] \[30=40-10\] \[30=30 \ \Huge \color{green} \checkmark\] Similarly, the answer in the back of the book gives the area of the larger circle |dw:1385273946213:dw| minus the area of the inscribed square |dw:1385274015235:dw| And the radius of the circle was found using the Pythagorean Thm. to find the diameter of the circle (and thus the radius) |dw:1385274176101:dw|