# A math question: fitting a square into a circle **URL:** \ **Category:** Factual Questions\ **Created:** [March 9, 2005, 7:52am UTC](https://boards.straightdope.com/t/a-math-question-fitting-a-square-into-a-circle/293650 "2005-03-09T07:52:19Z")\ **Posts on this page:** 8\ **Page:** 1
**Author:** ![bienville](https://avatars.discourse-cdn.com/v4/letter/b/4bbf92/32.png) [@bienville](https://boards.straightdope.com/u/bienville)\ **Post date:** [March 9, 2005, 7:52am UTC](https://boards.straightdope.com/t/a-math-question-fitting-a-square-into-a-circle/293650/1 "2005-03-09T07:52:19Z")
I swear to Cecil I have not been in school for 8 years, so this isn’t a homework question, I just have a personal little project that I’m too dumb for. How do you figure out the measurement for the side of a square if you know the measurement for the diagonal? (The square’s gotta fit inside a circle, the points at the corners of the square rest on the circle, so I figure the diagonal of the square is equal to the diameter of the circle- right?) ---
**Author:** ![Triskadecamus](https://avatars.discourse-cdn.com/v4/letter/t/b19c9b/32.png) [@Triskadecamus](https://boards.straightdope.com/u/Triskadecamus)\ **Post date:** [March 9, 2005, 7:55am UTC](https://boards.straightdope.com/t/a-math-question-fitting-a-square-into-a-circle/293650/2 "2005-03-09T07:55:58Z")
It’s the square root of two times the radius of the circle. ---
**Author:** ![Smeghead](https://avatars.discourse-cdn.com/v4/letter/s/f1d935/32.png) [@Smeghead](https://boards.straightdope.com/u/Smeghead)\ **Post date:** [March 9, 2005, 7:56am UTC](https://boards.straightdope.com/t/a-math-question-fitting-a-square-into-a-circle/293650/3 "2005-03-09T07:56:52Z")
When you have a square divided by a diagonal, you have two equilateral right triangles. Remember the Pythagorean Theorem? a[sup]2[/sup] + b[sup]2[/sup] = c[sup]2[/sup]. In this case, a=b, and you know c (the length of the diagonal). So, 2a[sup]2[/sup] = c[sup]2[/sup], where a=the length of the side, and c=the length of the diagonal. ---
**Author:** ![Triskadecamus](https://avatars.discourse-cdn.com/v4/letter/t/b19c9b/32.png) [@Triskadecamus](https://boards.straightdope.com/u/Triskadecamus)\ **Post date:** [March 9, 2005, 7:57am UTC](https://boards.straightdope.com/t/a-math-question-fitting-a-square-into-a-circle/293650/4 "2005-03-09T07:57:56Z")
Oops. ---
**Author:** ![nivlac](https://avatars.discourse-cdn.com/v4/letter/n/3bc359/32.png) [@nivlac](https://boards.straightdope.com/u/nivlac)\ **Post date:** [March 9, 2005, 8:31am UTC](https://boards.straightdope.com/t/a-math-question-fitting-a-square-into-a-circle/293650/5 "2005-03-09T08:31:47Z")
> [@bienville](#): > > I swear to Cecil I have not been in school for 8 years, so this isn’t a homework question, I just have a personal little project that I’m too dumb for. > > How do you figure out the measurement for the side of a square if you know the measurement for the diagonal? > > (The square’s gotta fit inside a circle, the points at the corners of the square rest on the circle, so I figure the diagonal of the square is equal to the diameter of the circle- right?) Don’t know what a circle has to do with it. Side = diagonal/(square root of 2), in case you can’t solve the equation in the prior post. ---
**Author:** ![bienville](https://avatars.discourse-cdn.com/v4/letter/b/4bbf92/32.png) [@bienville](https://boards.straightdope.com/u/bienville)\ **Post date:** [March 9, 2005, 8:41am UTC](https://boards.straightdope.com/t/a-math-question-fitting-a-square-into-a-circle/293650/6 "2005-03-09T08:41:32Z")
> [@Smeghead](#): > > When you have a square divided by a diagonal, you have two equilateral right triangles. > > Remember the Pythagorean Theorem? a[sup]2[/sup] + b[sup]2[/sup] = c[sup]2[/sup]. > > In this case, a=b, and you know c (the length of the diagonal). So, > > 2a[sup]2[/sup] = c[sup]2[/sup], where a=the length of the side, and c=the length of the diagonal. O.K. so the diameter of the circle / diagonal of the square is **4.75"** squared is: **22.5625"** half of which is equal to a[sup]2[/sup], right? So a[sup]2[/sup] is **11.28125**" and then a = **3.358757yadda yadda yadda** (for my purposes 3 1/3 is o.k.) Am I correct? ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [March 9, 2005, 9:12am UTC](https://boards.straightdope.com/t/a-math-question-fitting-a-square-into-a-circle/293650/7 "2005-03-09T09:12:44Z")
> [@Triskadecamus](#): > > Oops. Actually, you’re correct (as are the others). According to **Smeghead** (correctly), 2a[sup]2[/sup]=c[sup]2[/sup] (a=side of square, c=diameter of circle). So a=c / sqrt(2). Since c=2r (r=radius), a=2r / sqrt(2) = sqrt(2)r, which was your answer. > [@bienville](#): > > O.K. so > > the diameter of the circle / diagonal of the square is **4.75"** > > squared is: **22.5625"** > > half of which is equal to a[sup]2[/sup], right? So a[sup]2[/sup] is **11.28125**" > > and then a = **3.358757yadda yadda yadda** (for my purposes 3 1/3 is o.k.) > > Am I correct? Yep. ---
**Author:** ![bienville](https://avatars.discourse-cdn.com/v4/letter/b/4bbf92/32.png) [@bienville](https://boards.straightdope.com/u/bienville)\ **Post date:** [March 9, 2005, 9:23am UTC](https://boards.straightdope.com/t/a-math-question-fitting-a-square-into-a-circle/293650/8 "2005-03-09T09:23:30Z")
Thanks, all!