Problem Solving

1) An inner circle is inscribed in a square, which is inscribed in another circle (outer circle).

The question states "Find out the ratio of the area of the two circles".
Only available information- "The radius of the circle is 'r'."


2) An inner square is inscribed in a circle, which is inscribed in another square(outer square).

The question states "Find out the ratio of the area of the two squares".

wtf?...anyone?

if you are confused about what the question is saying.
see attached

if not here goes:
[spoiler]
1) as you can see on the pic, the area of the smaller circle is
πr0^2 or pie*r_0^2 (_for subscript)

larger circle is
pie*r_1^2

when you look at the pic you should notice that
r_1^2=r_0^2+r_0^2

so ratio of smaller circle's area to larger one is
pie*r_0^2:pie*(r_0^2+r_0^2)
or
pie*r_0^2:2*pie*r_0^2
or
1:2

similar type of thing for question 2[/spoiler]

1) Circum Circle ( i.e. Circle outside Square )

If you clearly observe a circumcircle over a square it is very clear that the diameter of the circle is the same as the diagonal of the square.

Hence the diameter of the circle is sqrt(2) * a where a is the side of square.

2) In Circle. ( i.e. Circle inside square ).

If you clearly observe the in circle it is very clear that the diameter of that circle is the same as side of the square.

Hence diameter of the circle is (a), where a is the side of square.

Hence the ratio of areas is [( sqrt (2) *a ) ^ 2 ] / [ (a) ^2)] = 2

So, the ratio of areas of circum circle to in circle is 2.

Thanks for ur help Zenit, but im still a bit foggy..
The formula for area of a circle is phi multiplied by 'r' squared. or phir^2.

Can u please explain what is πr0^2 or pie*r_0^2 .
i would appreciate it if u could explain what is n?, how is 0 raised to the power of 2?, what is 0 doing there....in other words a word break-up because im not able to understand the formula u posted.

This just happens to be part of my homework...
thanks!

Just to clear your doubt... area of circle is phi*r*r;

but when you compare the same quantity... phi cances out on both Numerator and Denominator.. and in the answer I posted above.. I used diamter as dia= 2*r and 2 of 2*r cancles out on both numerator and denominator...


I hope my explanation is clear... what do u say?

ok, my bad I knew it was bad Idea to try and explain here with subscripts and hyper scripts and such.

See attached,

btw, π (look at it carefully its not n) suppose to be pi.

apparently real name for 3.14 number is "pi" not pie or phi.

hey man,
thanks i understood the part of the diameter of circle equal to the diagonal of square.
My doubt is how did u get sqrt(2)*a ??

isnt diameter of a circle = 2 x radius of circle?

p.s. i understood the method the guy on top used hes made use of the area formulas and obtained the answer....but i am keen on learning it using the diameter.

Wonderful!...i understand now...thanks guys!

Let us take "a" as the side of square.

If you apply phythagoras theorem and take square,

(diagonal) ^2 = (length) ^2 + (breadth) ^2.

(d) ^2 = (a)^2 + (a) ^2 [ Length = Breadh ].


(d) ^2 = 2* [(a)^2].
Hence for square,

d= sqrt (2)*a

Now if you see the big circle or circum circle,

diameter of circle = diagonal of square.

Hence dia = sqrt (2) *a.

For small circle,

diameter of circle = side of square.

therefore dia = a.

And,

Area of Circle = 3.14 * r * r or 3.14 * ( ( dia )/2) ^2) * ( ( dia )/2) ^2)

so when you compare the squares of the diameter of two circles you will get the answer.

Hi Zentih,

How can you say

(r1) ^2 = (r0)2 + (r0)^2+

I strongly feel that the observation is wrong, if not can you give me the supporting proof.....

Apply similar analysis,

Let the diameter of square be "d".


For Small square,

diagonal = diameter

therefore diagonal = d.

Hence side of square is [ (d) / sqrt (2) ].

For Big square,

Side = diameter

Hence side is "d".

Hence the ratio of big square to small square is 2:1.

yup i was wondering about that too...it looks like it has been assumed...but i dont think that is the case..the question does not mention it though...

struggling_guy2001 wrote:Hi Zentih,

How can you say

(r1) ^2 = (r0)2 + (r0)^2+

I strongly feel that the observation is wrong, if not can you give me the supporting proof.....

..? do you mean the part where I use Pythagorean theorem?

I've made a pic please see

i think i understand...the hypotenuse is the radius of bigger circle...and the other two sides of the triangle are the radii of smaller circle.

hold on zenit, one of the radii belongs to a third concentric circle uve drawn in light blue ,,is that allowed?