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# Ratio of perimeter to area for inscribed circles watch

1. Hi, can anyone help on the below? I'm quite sure that the answer is the same, but having trouble explaining why:

A quadrilateral contains an inscribed circle. The ratio of the perimeter of the quadrilateral to the circumference of the circle is 3:2. What is the ratio of the area of the quadrilateral to the area of the circle?

Thanks
2. (Original post by EffyB)
Hi, can anyone help on the below? I'm quite sure that the answer is the same, but having trouble explaining why:

A quadrilateral contains an inscribed circle. The ratio of the perimeter of the quadrilateral to the circumference of the circle is 3:2. What is the ratio of the area of the quadrilateral to the area of the circle?

Thanks
Have you actually drawn this out? Draw a circle inside a square and label the circle's diameter. The answer should become clear.

Spoiler:
Show
also: if two lines are in ratio 1:2 then the area of their squares are in ratio 1:4.
3. [QUOTE=lerjj;53026683]Have you actually drawn this out? Draw a circle inside a square and label the circle's diameter. The answer shall become clear.

Well a circle inside a square results in the area being in exactly the same ratio as the perimeter/circumference. What you have said about the lines would suggest that the ratio will be different?
4. [QUOTE=EffyB;53026845]
(Original post by lerjj)
Have you actually drawn this out? Draw a circle inside a square and label the circle's diameter. The answer shall become clear.

Well a circle inside a square results in the area being in exactly the same ratio as the perimeter/circumference. What you have said about the lines would suggest that the ratio will be different?
Yeah, sorry. The stuff in the spoiler was stupid wasn't it?
5. But just because it is true for a circle inside a square, why is it true for all circles inscribed in quadrilaterals? How do I know the circle inside a square is not a special case?
6. (Original post by EffyB)

Well a circle inside a square results in the area being in exactly the same ratio as the perimeter/circumference. What you have said about the lines would suggest that the ratio will be different?
hmm... not sure anymore. The most straightforward (brute force) approach would be to draw some rectangle which is in the correct ratio (a square isn't, and from the looks of it different rectangle produce different ratios) and the calculate the ratio of the areas.

I'm not sure it is the same, because as you make a rectangle wider, the inscribed circle stays the same, but the rectangle's area and perimeter don't grow at the same rate.
7. I don't think a circle inside a rectangle counts as being 'inscribed' because it does not touch every edge. I think that each side of the quad must touch the circle. A circle in a square therefore being the simplest representation. The question is from a book on ratio as opposed to area, so feel like I have missed a connection - drawing more complex quads and trying to work out their areas seems like the wrong way forward...
8. (Original post by EffyB)
I don't think a circle inside a rectangle counts as being 'inscribed' because it does not touch every edge. I think that each side of the quad must touch the circle. A circle in a square therefore being the simplest representation. The question is from a book on ratio as opposed to area, so feel like I have missed a connection - drawing more complex quads and trying to work out their areas seems like the wrong way forward...
I think the square is the only quadrilateral that can contain an inscribed circle
9. I don't think that is the case, see http://en.wikipedia.org/wiki/Tangential_quadrilateral
10. (Original post by EffyB)
I don't think that is the case, see http://en.wikipedia.org/wiki/Tangential_quadrilateral

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