Question: Radius vs. Diameter

Comment on Radius vs. Diameter

When you divide both sides by b in the last step to get root 2 and 2, How you make sure that b is not negative?
greenlight-admin's picture

Good question.
b represents the radius of Circle B, and the radius of a circle will always be positive.

Hi Brent,
I solved it this way.
circle A area = Pie *(r square)
circle B area = 2* pie * r square
Pie *(r square) = 2* pie * r square (as per shared info)

comparing these both areas. (used matching operations)

divide by pie to both quantities to get

r square (qty A)

2 r square (qty B)

divided by 2 to both quantities:
r square/2 (qty A)
r square(qty B)
take sq. root of both quantities:
sq root(r square/2) (qty A)
r (qty B)
r/sq. root 2 (qty A) can be rewritten as r*sq root 2/2

Now multiplied both sides by 2 to get:
r*sq root 2 (qty A)
2*r (qty B) which is nothing but diameter of Qty B.
Since 2*r is greater than sq root 2* r , so qty B is greater than qty A. Is this approach correct?
greenlight-admin's picture

I'm not sure I follow your solution.

What does r represent?
I ask because your equation πr² = 2πr² boils down to saying 1 = 2.

I think it would be useful to assign a different variable to the radius of each circle.

Yes right,(with due respect) using 2 different variables will avoid unwanted confusions.

I meant , πr² = 2πr² because area of circle A is twice the area of circle B so I wrote it as , πr² = 2πr² and solved further, but yes, it boils down to saying, 1=2 which is not a perfectly sound equation. So how will I know in such QC questions whether to use one variable or 2? Because using one variable I still got the answer.
greenlight-admin's picture

Your question: So how will I know in such QC questions whether to use one variable or 2?

We can use one variable to create more than one variable expressions as long as the variable represents the SAME VALUE for all expressions.

Here's what I mean:

If we let x = Max's age...
Then x+5 = Joon's age

In both cases, x represents Max's age. So, it's okay to use one variable.

In your example, you used r to represent two DIFFERENT VALUES (the radius of the small circle AND the radius of the large circle.

Does that help?


Makes perfect sense. thanks a ton Brent :) You are genius.
Abdul Hannan's picture

Hi Mr Hanneson,

I plugged in some nice value for the area of circle B and I found the values of both the radii and then I figured out the answer by multiplying the radius of circle B by 2.

Did I do it right?
greenlight-admin's picture

That's a great approach, Abdul.


I tried plugging in and I got a strange result.
I said area A = 4π
I said area B = 2π

So radius A = 2
radius B = √2

Then when I multiplied radius B by 2 to get diameter B:
radius A = 2
diameter B = 2√2

Shouldn't I end up with 2 vs √2?

I must have made a mistake somewhere, but I don't see it. Could you please take a look.
greenlight-admin's picture

Your answer is also good.

Regardless of what values you use, Quantity B will be √2 times the value of Quantity A.
In your case, 2√2 is √2 times 2.
In my case, 2 is √2 times √2.


Ok, I guess that makes sense when I think about it. Given that the area of circle A is greater than circle B, it makes sense that it's radius must be larger, in this case 1.4 times larger or √2 larger. Thank you.

Have a question about this video?

Post your question in the Comment section below, and a GRE expert will answer it as fast as humanly possible.

Change Playback Speed

You have the option of watching the videos at various speeds (25% faster, 50% faster, etc). To change the playback speed, click the settings icon on the right side of the video status bar.

Let me Know

Have a suggestion to make the course even better? Email us today!

Free “Question of the Day” emails!