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Geometry question

i am revising for the physics aptitude test for oxford and came across this question. it's quite easy and yet i really don't get what i'm doing wrong:

In the figure below, the radius of the larger circle is twice that of thesmaller circle. Find an expression for the fraction of the area of thesquare which is occupied by the two circles.

The figure can be seen on question 12 of the maths section of the paper through this link: https://www2.physics.ox.ac.uk/sites/default/files/PAT-2006-Paper.pdf

What I did was

area of square= (5r)^2=25r^2
area of bigger circle = 4pir^2
area of smaller circle = pir^2

total area that the circles occupy = 5pir^2
5pir^2/25r^2 = pi/5

answer = pi/5

but that's not even close to the answer and idk how
Reply 1
Avatar for TeeEm
TeeEm
19
Original post by tanyapotter
i am revising for the physics aptitude test for oxford and came across this question. it's quite easy and yet i really don't get what i'm doing wrong:

In the figure below, the radius of the larger circle is twice that of thesmaller circle. Find an expression for the fraction of the area of thesquare which is occupied by the two circles.

The figure can be seen on question 12 of the maths section of the paper through this link: https://www2.physics.ox.ac.uk/sites/default/files/PAT-2006-Paper.pdf

What I did was

area of square= (5r)^2=25r^2
area of bigger circle = 4pir^2
area of smaller circle = pir^2

total area that the circles occupy = 5pir^2
5pir^2/25r^2 = pi/5

answer = pi/5

but that's not even close to the answer and idk how


how?
Original post by TeeEm
how?


that is the incorrect part of my answer but i don't know how.
if the radius of the bigger circle is 2r and the smaller circle is r then won't the side of the square be 5r? the bigger circle, from top to bottom, takes up 4r and the smaller circle r,...?
Reply 3
Avatar for TeeEm
TeeEm
19
Original post by tanyapotter
that is the incorrect part of my answer but i don't know how.
if the radius of the bigger circle is 2r and the smaller circle is r then won't the side of the square be 5r?


post a diagram with what you are doing
Original post by tanyapotter
that is the incorrect part of my answer but i don't know how.
if the radius of the bigger circle is 2r and the smaller circle is r then won't the side of the square be 5r? the bigger circle, from top to bottom, takes up 4r and the smaller circle r,...?


What is the solution we are trying to get?
Original post by TeeEm
post a diagram with what you are doing


image.jpeg
Original post by Eux
What is the solution we are trying to get?


the answer is
10pi / 27+18root2
Reply 7
Avatar for TeeEm
TeeEm
19
Original post by tanyapotter
image.jpeg


I disagree with your diagram ... do you have the answers?
I got around 0.5989 ... (pain to type the exact answer)
Original post by tanyapotter
the answer is
10pi / 27+18root2


Okay, thats fine. You are getting the side of the square wrong. You are not working the side of the square out correctly. Try and find the distance from the centre of the big circle to the top left corner, and the distance from the centre of the small circle to the bottom right corner, as you can then work out the diagonal of the square, and with that the sides. Remember that the diagonal of a square is root 2 * its side length
Original post by TeeEm
I disagree with your diagram ... do you have the answers?
I got around 0.5989 ... (pain to type the exact answer)


Yeah thats right
Original post by TeeEm
I disagree with your diagram ... do you have the answers?
I got around 0.5989 ... (pain to type the exact answer)


yep, that's the right answer
could you help me understand why my diagram is wrong? why is the side of the square incorrect?
Original post by Eux
Okay, thats fine. You are getting the side of the square wrong. You are not working the side of the square out correctly. Try and find the distance from the centre of the big circle to the top left corner, and the distance from the centre of the small circle to the bottom right corner, as you can then work out the diagonal of the square, and with that the sides. Remember that the diagonal of a square is root 2 * its side length


i do know that my side of the square is wrong, but i don't know why. and yes that's how you work it out - find the distance from the centres of the circles to the two corners, then add them to the radii to get the full diagonal of the square and use pythagoras to find the side of the square
but why can't it be 2r + 2r + r? sorry if this is a dumb question but i don't understand why i can't use the diameters of both circles to measure the length of a side
Reply 12
Avatar for TeeEm
TeeEm
19
Original post by tanyapotter
yep, that's the right answer
could you help me understand why my diagram is wrong? why is the side of the square incorrect?


It looks that Eux is trying to "evict me" and he looks "extra keen".
I leave you to his "capable" hands.

All the best
Original post by TeeEm
It looks that Eux is trying to "evict me" and he looks "extra keen".
I leave you to his "capable" hands.

All the best


Noooo come back! You're way more qualified than me
Original post by tanyapotter
i do know that my side of the square is wrong, but i don't know why. and yes that's how you work it out - find the distance from the centres of the circles to the two corners, then add them to the radii to get the full diagonal of the square and use pythagoras to find the side of the square
but why can't it be 2r + 2r + r? sorry if this is a dumb question but i don't understand why i can't use the diameters of both circles to measure the length of a side


You are making an assumption that just isn't true- can you give any proof that the side of the square = the diameter of the big circle and the radius of the little circle other than just by inspection of the diagram?
Reply 15
Avatar for TeeEm
TeeEm
19
Original post by Eux
Noooo come back! You're way more qualified than me


I was only joking ...
I just woke up from a nap and need to wake up, plus my "to do tray" on my desk is getting taller by the day.
Original post by Eux
You are making an assumption that just isn't true- can you give any proof that the side of the square = the diameter of the big circle and the radius of the little circle other than just by inspection of the diagram?


just before i read this i understood!! i assumed that the end of the bigger circle is directly in line with the centre of the smaller one, but i literally got my ruler out and checked and there's actually a small gap in between. what a stupid mistake
thank u!!
Original post by TeeEm
I was only joking ...
I just woke up from a nap and need to wake up, plus my "to do tray" on my desk is getting taller by the day.


Um. It's 9 o clock. And its a saturday...
Reply 18
Avatar for TeeEm
TeeEm
19
Original post by Eux
Um. It's 9 o clock. And its a saturday...


older people get tired ...
Original post by tanyapotter
i am revising for the physics aptitude test for oxford and came across this question. it's quite easy and yet i really don't get what i'm doing wrong:

In the figure below, the radius of the larger circle is twice that of thesmaller circle. Find an expression for the fraction of the area of thesquare which is occupied by the two circles.

The figure can be seen on question 12 of the maths section of the paper through this link: https://www2.physics.ox.ac.uk/sites/default/files/PAT-2006-Paper.pdf

What I did was

area of square= (5r)^2=25r^2
area of bigger circle = 4pir^2
area of smaller circle = pir^2

total area that the circles occupy = 5pir^2
5pir^2/25r^2 = pi/5

answer = pi/5

but that's not even close to the answer and idk how


Was doing the same paper :
IMG_3470.jpg500.1KB

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