The diameter of circle S is equal in length to the side

[hwilliamsvi]

Can you provide help with this question?

"The diameter of circle S is equal in length to the side of a certain square. The diameter of circle T is equal in length to a diagonal of the same square. Th area of circle T is how many times the area of circle S."

A.√2
B.√2+1
C. 2
D. π [= pi]
E.√π [= √pi]

I initially thought the diagonal of the square to be equal to s√2 which I then squared, and I ran into issues moving forward from there. Can you provide any assistance into how the answer is C?

[naresh.kumar82]

Can you provide help with this question?

"The diameter of circle S is equal in length to the side of a certain square. The diameter of circle T is equal in length to a diagonal of the same square. Th area of circle T is how many times the area of circle S."

A.√2
B.√2+1
C. 2
D. π
E.√π

I initially thought the diagonal of the square to be equal to s√2 which I then squared, and I ran into issues moving forward from there. Can you provide any assistance into how the answer is C?

Imagine radii of circles S and T are ...S and T.

Then as per the given relationship among circles and square, you'll get T = √2 S.

Now you know the Area of circle: Pi * radius^2
so, area of circle T = Pi * T^2 => Pi * ( √2 S)^2 => 2 ( Pi*S^2)
Hence Area of circle T = 2 times the Area of Circle S.
That's why C is the Answer.

[RonPurewal]

I initially thought the diagonal of the square to be equal to s√2 which I then squared, and I ran into issues moving forward from there.

that's a legitimate approach, so you must have just been having an off-day with algebra.
as you (correctly) wrote here, the diameter of the larger circle is √2 times the diameter of the smaller circle.
this means that the radius of the larger circle is also √2 times the radius of the smaller circle. so let's call those r and r√2.
in this case, the area of the smaller circle is just (pi)(r^2), while the area of the larger circle is (pi)[(r√2)^2] = pi * 2(r^2). that's exactly twice as big.

--

the real trick here, however, is to realize that there are no definite values in this problem -- and so you can PICK YOUR OWN NUMBERS for the diameters/radii of the two circles.

let the diameters of the smaller and larger circles, respectively, be 2 and 2√2. (according to the relationships for squares, the latter of these must be √2 times the former, but apart from that they can have any values whatsoever.)
then the radii are 1 and √2.
therefore, the areas of the circles are pi(1)^2 = 1pi and pi(√2)^2 = 2pi, so the conclusion is immediate.