Can you solve this tricky interview question?

A square is inscribed in a circle. A smaller square is drawn; one side is shared with the inscribed square, and the other two corners touch the circle. What is the ratio of the large square's area to the small square's area? This problem was asked during a job interview. I could not solve it; can you? Watch the video for a solution.

*Small correction: at 1:35 I said "small circle" but I meant "small square."

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[Your-BestNightmare]

McDonalds is really focusing on hiring the best of the best

[vexrav]

I came up with an alternative solution: To find the side length of the small circle I used the following equations: x^2 + y^2 = 1 (formula for a unit circle) and y = 2x + root(2)/2 because the circle should bisect the square sitting on the line y = root(2)/2 (the height of the large square). Using this method I arrived at the same result.

[alexhaynes]

Most of your puzzles are WAY over my head. A few of the Pythagorean type puzzles are a delight because they boost my ego. Sunday is 'Daisy Day' the one day I can devote to my wife, my daughter and Little Daisy, my grandaughter.
Your video comes around that 'my cup runneth over' moment.
Thank you. You are a true friend.

[Mar]

A fun follow-up question on this is to look at the subsequent, even smaller squares you get between the circle and the small (and following) square(s). It's actually possible to obtain a recursive formula that describes the side length (or area) of the i-th square constructed this way. d_0 = r*√2 is our starting value, the biggest inner square. I'll leave the way to solve this open as a challenge. If you want to check your final result, here's what I got:

d_i = 2/5 * ( √(4*r^2 + 1/4 * d_{i-1}^2) - √(4*r^2 - d_{i-1}^2) )

Or the whole thing squared for the area. Taking the ratio to the previous term if you want to compare ratios. As you'd expect intuitively, the incremental ratios get bigger and bigger (because the circle segment approaches a flat line better and better the further you "zoom in").

A hint to get started, I did this by looking at the coordinates of an outer corner of each following square. There is an obvious recurrence relation between subsequent y-coordinates, d_i determines the x-coordinates in a straightforward way, and the (x|y) couples must satisfy a specific condition so that they lie on the circle.

[backyard]

That feeling when you can't solve it but then you see the solution and realize how simple it is

[RabblesTheBinx]

Is it weird that I got it right just by looking at it and thinking, "Yeah, that small square looks like it could fit 2 more on each side, so the small square is 1/25th the size of the big one."? Like, I know the math to do it the right way, but it just seemed so _obvious._

[RolandsSh]

I had a very similar approach, but I think mine was a little bit more elegant.
I denoted half the large square side length as "a" and half the small square side length as "b", which also means that radius of the circle is equal to a*sqrt(2). In that same triangle using Pythagorean theorem I got this equation:
b^2 + (a+2b)^2 = 2a^2
b^2+2^2+4a*b+4b^2=2a^2
a^2-4ab-5b^2=0 - I divided this equation by b^2
(a/b)^2-4(a/b)-5=0

a/b = 5 or a/b = -1
Ratio of area would be square of side length ratio, which gives me 25.

[Pining_for_the_fjords]

I honestly thought it would be some irrational number related to pi. It's not often you get nice whole numbers when talking about circles.

[garymitchell]

I have to say that x25 seemed obvious from a visual point of view- the lower side of the small square looked like a fifth of the side of the larger square. The good thing about geometry- circles, squares etc- is that proportion remains the same throughout, regardless of size, so proportional and initial calculations can be made. Square and circle examples are interesting- inscribed or circumscribed- fascinating stuff but one needs to start from basic principles and work NOT what I did when I worked backwards in this example! Haha The ongoing thing with any circle problem is good old because it's irrational and r is the only quantity available. Thanks again Presh for an interesting example.

[tomdekler]

There's something magical about forms like this having perfect integer solutions. And out of all integers, I wouldn't expect 5 to show up in a question about squares and circles.

[treesarecool]

I guessed it based on the appearance of the diagram. Full marks for me

[SuperStargazer]

Ok. It’s like this. You cut out a template of the smaller square, then you use it to make lots of small squares. Then you lay the small squares like tiles down onto the larger square. Finally you simply count how many smaller squares it took to completely cover the larger square. My method, while being less mathematical, allows me to demonstrate my scissor skills. Top marks to me:-)

[charbeleid]

That's the first problem I actually figured out. I'm so happy

[stevekerp]

I eye-balled it and guessed 25, but I like your solution better.

[silversurfer]

I just took a ruler, measured the small top square side, roughly 4 cm at my screen resolution at full window. Then measured the large square at 20cm - thereby 20/4=5, then 5*5 =25

[unlincecosmico]

I think the easiest way to do it is: observe that the side that the small square shares with the big square can fit 2 times in each side, so this is a total if 5 small square in a side of the the big square, do the maffs, and you can get 25 small squares in the big square

[benjaminbrady]

"That stumped me" wait!? You get the answer to most of the questions on your channel?

[jagmarz]

That's an excellent drawing, so I started by estimating the answer, then I went and figured out that you find the smaller square by taking a 60 degree line off the center of a big square edge, then I found that when you solved for the intersection of that line and the circle, you get a nice simple linear ratio of r, and from there it was easy to calculate the real ratio, which was surprisingly exactly the same as the estimate. Wow, what a neat problem!

[robertheal]

You can get the same conclusion, by considering the angle of the small pointy triangle, and then determining the sin and cosine of that angle and using the sin**2 + cos**2 = 1 identity,

[BeyondPatmos]

The question ask for ration, not precise numbers and scale won't matter either as long as the circle is... a circle. You can easily solve this by vision alone