semi circle problem: square inside has area 36 cm^2; find radius of semi-circle

apple2357

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Can anyone offer me hints or check the answer i have got?
I got sqrt(45) using a mixture of theorem involving semi-circles and similar triangles.
But i would be interested..

1) if i am correct?
2) Any other approaches?
 

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Beer soaked ramblings follow.
Yes, thou art correct.
Another approach I see is to divide the square in the middle and connect the bottom midpoint with either the left or the right upper corner. Apply the pythagorean theorem to solve the length of the left or right diagonal (which is equal to the radius of the half circle) and you're done.
 
Last edited:
Beer soaked ramblings follow.
Yes, thou art correct.
Another approach I see is to divide the square in the middle and connect the bottom midpoint with either the left or the right upper corner. Apply the pythagorean theorem to solve the length of the left or right diagonal (which is equal to the radius of the half circle) and you're done.

Ah, thanks that seems like a nice approach. So the centre of the circle. That seems easier
 
It's correct, but unnecessarily complex. I used the pythagorean theorem approach mentioned above.


I wonder why i went down such a complicated route? Why is it that some people can just spot the simple way in while others ( like me) look for complications!
 
Easier still: straight line from center of circle to an upper corner = radius,
and creates ye olde 30-60-90 triangle with sides 3,6,r ....
 
I wonder why i went down such a complicated route? Why is it that some people can just spot the simple way in while others ( like me) look for complications!

I would imagine the figure made you think of one you've seen before, so you took the route associated with that other problem. The more problems you've seen, the more possibilities you have to notice.

A good habit, also, is after solving a problem to look back at it and see what other methods you can find (sometimes inspired by something about the answer, such as its simplicity). If you do that enough, you will develop an ability to see alternative methods for new problems, too.
 
I would imagine the figure made you think of one you've seen before, so you took the route associated with that other problem. The more problems you've seen, the more possibilities you have to notice.

A good habit, also, is after solving a problem to look back at it and see what other methods you can find (sometimes inspired by something about the answer, such as its simplicity). If you do that enough, you will develop an ability to see alternative methods for new problems, too.

Thanks. I think i thought i needed more facts to use so i was reminded about the angle in the semi-circle and then similar triangles.
But all it needed was a bit of Pythagoras! In these problems i find you have to know what to add to the diagram and where!
 
Can anyone offer me hints or check the answer i have got?
I got sqrt(45) using a mixture of theorem involving semi-circles and similar triangles.
But i would be interested..
1) if i am correct?
2) Any other approaches?
Un2.jpg
Put the figure onto an axis system.
The center \(\displaystyle \mathcal{C}: (0,0)\) the upper-right-hand vertex \(\displaystyle \mathcal{A}: (3,6)\)
The radial line-segment \(\displaystyle \overline{\mathcal{C}\mathcal{A}}\) has length \(\displaystyle \sqrt{3^2+6^2}=\sqrt{45}~.\)
 
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