Circle A and B have the same area i.e. same radius.
However the option B is stated as = \(2 \times(\pi r +2r)\). Twice the perimeter of semicirle B is not Equal to perimeter of circle B since there is an additional term of the diameter in the perimeter.

Here is a picture that illustrates that perimeter of two semi-circles is equal to one circle plus two diameters.

GIVEN: Semicircle B has area \(a/2\)
GIVEN: Circle A has area \(a\)



Since a semicircle is HALF of a circle, we can conclude that, if we combine TWO of semicircle B, the combined area will be \(a\)

In other words, the red and blue circles above must have the same area.

If the red and blue circles have the same area, their radii must also be equal.
So let's let r = the radius of the red circle
This means r = the radius of the blue semicircle


At this point we can compare the two quantities.

QUANTITY A: The circumference of Circle A = 2πr

QUANTITY B: TWICE the perimeter of semicircle B
The perimeter of ONE semicircle B = (length of curved portion) + (length of straight diameter)
= (0.5)(2πr) + (2r)
= πr + 2r

So, TWICE the perimeter of semicircle B = 2(πr + 2r)
= 2πr + 4r

Since 2πr + 4r > 2πr, the correct answer is B.

Cheers,
Brent

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