1. Question: Consider two planets with uniform mass distributions. The mass density and the radius of planet 1 are p1 and R1, respectively, and those of planet 2 are p2 and R2.

Question

What is the ratio of their masses?

Ans: M1/M2 = (p1/p2)(R1/R2)^3...I got this answer correct.

I need help in the next, as below, affiliated question.

2. What is the ratio of the circular areas defined by the two equators? Can you please tell me which one you think is correct?

1. A1/A2 = (R2/R1)^3

2. A1/A2 = (R2/R1)^2

3. A1/A2 = pi (R1/R2)^3

4. A1/A2 = (R2/R1)

5. A1/A2 = (R1/R2)^3

6. A1/A2 = pi (R2/R1)^2

7. A1/A2 = (R1/R2)

8. A1/A2 = (R1/R2)^2

9. A1/A2 = pi (R2/R1)^3

10. A1/A2 = pi (R1/R2)^2

What I am thinking is that in the top question, R refers to the radius, so R should also stand for the radius in the below question since they are both connected. But the bottom question talks about "equators" which I take to be diameters. That is why I am confused; if the bottom question's R's are radii, then only those with an exponent of 2 and pi should be right because the area of a circle is pi*r^2. So it should only be either 6 or 10. But if the bottom question has R as the diameter, then I don't know which one it is because the formula would be (pi*d^2)/4 (according to what I found online). Could you please tell me what you think it is, because I don't understand it? I also don't get how to tell whether R1 or R2 should be on the top.

Answer 1

The answer is
A1/A2=(R1/R2)^2

Not exactly sure how to explain but i'll give it a shot.
Since the area is already first over the 2nd, the radius should also be the same. So pluggin in the radius, it would also be r1/r2 to match the a1/a2 part. Not sure but somehow the pis just cancel out. I originally picked A1/a2=(r1/r2) but got that incorrect and chose the ^2 one. Got it right. Can't really explain it. I sort of did half math and half instinct.