Using Pythagorean Help

Geometry Level 1

In a circle of diameter 50, can we fit a 30 × 40 30 \times 40 rectangle?

Yes No

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Calvin Lin by Calvin Lin

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4 solutions

Sam Bealing
Apr 2, 2016

3 0 2 + 4 0 2 = 5 0 2 30^2+40^2=50^2 so the length of the diagonal of the rectangle is 50 50 .

Set this to the diameter of the circle and clearly the other two points lie on the circle by Thales' Theorem .

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After reading the solution using Pythagoras' theorem, I feel silly, but I used the formula chord length=2 radius sin(theta/2) where theta is the subtending angle and checked if the angles added to 2pi.

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Finn C
Apr 10, 2016

This is not as accurate or speedy as Sam's method, but I found the the area of the circle: 3.14 x 25 x 25= 1962.5 Then I found the area of the rectangle: 30 x 40= 1200 Therefore, the circle has bigger area than the rectangle. So the answer is: Yes

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Moderator note:

That claim is not true. Just because the area is larger doesn't mean that we can put the rectangle in the circle.

That claim is not true. Just because the area is larger doesn't mean that we can put the rectangle in the circle.

Can you think of an easy counterexample to your claim?

Calvin Lin Staff - 5 years, 2 months ago

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Yes it does, the question doesn't precise we can't cut or modify the forms. In a circle diameter 50 we can fit a rectangle 45*40.

damien G - 5 years, 2 months ago

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Obviously you can't cut or modify the forms. Otherwise, you might as way cut it up and stack it over itself, making this inquiry pointless.

Calvin Lin Staff - 5 years, 2 months ago

You are saying that because the rectangle has an area of 1200 then it must fit inside a circle with an area of 1962.5, this is not true because a rectangle that was 2 x 600 would also have an area of 1200 but clearly would not fit inside the same circle.

John Lawrie - 5 years, 1 month ago
DarkMind S.
Jan 6, 2017

No equations are needed. The diameter of the circle is 50. The maximum side of the rectangle is 40. So of course. !

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Can we fit a square of side length 40 into a circle of diameter 50?
If yes, how?
If no, why not?

Calvin Lin Staff - 4 years, 5 months ago

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