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What is the largest area of the semi-circle that can be inscribed in a square of edge length 1 unit?

That is a question posted by harpreet in the topic " maxima & minima" on a Mathematics forum on on 12th July 2009.

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here is the simplest solution of all, since we know that the square has all its sides of unit length. now, the semicircle just inscribed has its diameter of one unit too. and from that we are now able to calculate the are of that semicircle, here we are: area of semicircle = pi *(d/2)^2*1/2 = pi*1/8 square units!
Shahzad, thank you for solving "Largest Semicircle Inscribable in a Unit Square". You did find a large semicircle, and calculate its area. That was good work. There happens to be a larger one, but to find that was a significant project involving several people, looking at the problem from different perspectives. One perspective that I took at first was to draw a unit circle, and cut it with a 45 degree diagonal making a semicircle. Then it is easy to draw a square around it, two sides touching the corners of the semicircle and the opposite two sides being tangents to the circle. You can work out the area of that square using Pythagoras theorem. The ratio of the area of the square to the semicircle is the inverse of the area of the semicircle to the square. It is also possible to draw the inscribed semicircle using compass and ruler, as described in my slideshow.
Thanks, Colin
I got the same answer as Colin.

Tools for this solution:
Pythagorean Theorem
45-45-90 right triangle relations
Area of a circle
Area of a square (half product of the diagonals, and side squared)

There are 3 squares:
The large one... let the side = 4a
The medium one with a side = 1 unit
The small one

I looked at the area of these squares and their relationship to the areas of the circles.
Notice that there is an unknown length of side x
Notice the fact that 4a = 1 + x

The answer is

Math Chique, Your diagram of the semicircle inscribed in a square was well presented, and I think it will help people understand the problem.

It's not obvious at first glance, but inscribing a semicircle in a square is the same problem as inscribing a circle in a right isosceles triangle. You can see this from slides 21 and 22 of my slide show "Inscribe Semicircle In Square by Geometric Construction". That slide show and my diagrams in it are public domain. You can reuse them, but it is quite easy to draw them yourself with compasses and a ruler.


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