# Can you square pi (3.1415965...)? If so, what is pi squared? **URL:** \ **Category:** Factual Questions\ **Created:** [April 30, 2012, 11:04pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454 "2012-04-30T23:04:28Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Sheriru](https://avatars.discourse-cdn.com/v4/letter/s/5f8ce5/32.png) [@Sheriru](https://boards.straightdope.com/u/Sheriru)\ **Post date:** [April 30, 2012, 11:04pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/1 "2012-04-30T23:04:28Z")
Random question šŸ˜ƒ please help me find the answer! ---
**Author:** ![beowulff](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/beowulff/32/542_2.png) [@beowulff](https://boards.straightdope.com/u/beowulff)\ **Post date:** [April 30, 2012, 11:05pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/2 "2012-04-30T23:05:27Z")
9.8696044ā€¦ ---
**Author:** ![zoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zoid/32/336_2.png) [@zoid](https://boards.straightdope.com/u/zoid)\ **Post date:** [April 30, 2012, 11:05pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/3 "2012-04-30T23:05:40Z")
Fess up - is this homework? ---
**Author:** ![Shagnasty](https://avatars.discourse-cdn.com/v4/letter/s/9dc877/32.png) [@Shagnasty](https://boards.straightdope.com/u/Shagnasty)\ **Post date:** [April 30, 2012, 11:09pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/4 "2012-04-30T23:09:47Z")
Of course not, all good pi is round. ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [April 30, 2012, 11:10pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/5 "2012-04-30T23:10:51Z")
> [@Sheriru](#): > > Random question šŸ˜ƒ please help me find the answer! Of course you can square pi. You can square any real number and many others of course. You canā€™t get an exact answer, but then you canā€™t get an exact answer to pi itself either, but you can get as accurate approximation as you want. You know that pi ~ 3.14159 so 3.1415 \< pi \< 3.1416 making 3.1415[sup]2[/sup] = 9.8690225 \< pi[sup]2[/sup] \< 3.1416[sup]2[/sup] = 9.8696506. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [April 30, 2012, 11:17pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/6 "2012-04-30T23:17:59Z")
You can square any number. Why should it be surprising that this includes pi? As for what itā€™s used for, it shows up in the formula for the hypervolume of a sphere in 4 or 5 dimensions (pi^3 is in 6 or 7 dimensions, pi^4 in 8 or 9, etc.), and pi^2/6 is the sum of the reciprocals of the perfect squares (that is, 1 + 1/4 + 1/9 + 1/16 + ā€¦). For something more trivial, if you had a measuring-wheel of diameter 1 meter, and used it to measure out the sides of a square, itā€™d have an area of pi^2 square meters. Of course, even if it werenā€™t used for anything, itā€™d still be perfectly valid to square it. By comparison, the number e+pi (5.859874482ā€¦) doesnā€™t show up in any interesting formula, but itā€™s still a perfectly valid number. ---
**Author:** ![Xema](https://avatars.discourse-cdn.com/v4/letter/x/9de053/32.png) [@Xema](https://boards.straightdope.com/u/Xema)\ **Post date:** [April 30, 2012, 11:28pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/7 "2012-04-30T23:28:36Z")
> [@Sheriru](#): > > Can you square pi (3.1415965ā€¦) Note that this is not the right value - should be 3.14159265ā€¦ ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [April 30, 2012, 11:36pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/8 "2012-04-30T23:36:58Z")
Is there a proof that the square of pi is irrational? Is there a general proof that the square of any irrational number is irrational? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 30, 2012, 11:53pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/9 "2012-04-30T23:53:02Z")
> [@Little\_Nemo](#): > > Is there a proof that the square of pi is irrational? Is there a general proof that the square of any irrational number is irrational? The square root of two is irrational, so no, there isnā€™t a general proof. ---
**Author:** ![Scarlett67](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/scarlett67/32/7387_2.png) [@Scarlett67](https://boards.straightdope.com/u/Scarlett67)\ **Post date:** [April 30, 2012, 11:58pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/10 "2012-04-30T23:58:17Z")
Pi are not square. Pi are round. Cornbread are square. (Iā€™m sorry, I couldnā€™t resist the straight line.) ---
**Author:** ![merrick](https://avatars.discourse-cdn.com/v4/letter/m/58956e/32.png) [@merrick](https://boards.straightdope.com/u/merrick)\ **Post date:** [April 30, 2012, 11:59pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/11 "2012-04-30T23:59:53Z")
> [@Little\_Nemo](#): > > Is there a proof that the square of pi is irrational? Is there a general proof that the square of any irrational number is irrational? No, because the square root of 2 (for example) is irrational. Root 2 squared is, of course, 2. Pi, on the other hand, is trancendental (it canā€™t be expressed as the root of any polynomial equation with rational coefficients). That (I think) means its square has to be irrational, because if pi^2 was some rational number a/b the equation bx^2 - a = 0 would have root pi. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [May 1, 2012, 12:00am UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/12 "2012-05-01T00:00:26Z")
> [@ultrafilter](#): > > The square root of two is irrational, so no, there isnā€™t a general proof. But there is a proof that the square of pi is irrational; hereā€™s a sketch of the proof: pi has been proved to be transcendental, meaning that it is not the root of any polynomic equation with rational coefficients. All transcendental numbers are irrational. If pi^2 were not transcendental, it would be the root of some polynomic equation with rational coefficients - and it would be simple to convert that equation to one which had pi as a root; therefore pi^2 is transcendental (and thus irrational) ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [May 1, 2012, 12:22am UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/13 "2012-05-01T00:22:34Z")
> [@Andy\_L](#): > > But there is a proof that the square of pi is irrational; hereā€™s a sketch of the proof: pi has been proved to be transcendental, meaning that it is not the root of any polynomic equation with rational coefficients. All transcendental numbers are irrational. If pi^2 were not transcendental, it would be the root of some polynomic equation with rational coefficients - and it would be simple to convert that equation to one which had pi as a root; therefore pi^2 is transcendental (and thus irrational) I was thinking along those lines, but Iā€™m having trouble seeing exactly how that transformation works. I think if the polynomial that has pi^2 as a root has any odd degree terms that you could be in trouble. > [@merrick](#): > > No, because the square root of 2 (for example) is irrational. Root 2 squared is, of course, 2. > > Pi, on the other hand, is trancendental (it canā€™t be expressed as the root of any polynomial equation with rational coefficients). That (I think) means its square has to be irrational, because if pi^2 was some rational number a/b the equation bx^2 - a = 0 would have root pi. This is beautiful. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [May 1, 2012, 12:28am UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/14 "2012-05-01T00:28:58Z")
> [@ultrafilter](#): > > I was thinking along those lines, but Iā€™m having trouble seeing exactly how that transformation works. I think if the polynomial that has pi^2 as a root has any odd degree terms that you could be in trouble. You could be right about that. > [@ultrafilter](#): > > This is beautiful. Yes it is. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [May 1, 2012, 1:13am UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/15 "2012-05-01T01:13:03Z")
> [@Sheriru](#): > > Can you square pi (3.1415965ā€¦)? If so, what is pi squared? [Let me WolframAlpha that for you.](http://www.wolframalpha.com/input/?i=pi%5E2) Ah yes, now I remember that pi[sup]2[/sup]/6 is the sum of the infinite series 1/1 + 1/4 + 1/9 + 1/16 + ā€¦ + 1/n[sup]2[/sup] + ā€¦ (so pi[sup]2[/sup] = 6/1 + 6/4 + 6/9 + 6/16 + ā€¦) ---
**Author:** ![Sheriru](https://avatars.discourse-cdn.com/v4/letter/s/5f8ce5/32.png) [@Sheriru](https://boards.straightdope.com/u/Sheriru)\ **Post date:** [May 1, 2012, 1:45am UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/16 "2012-05-01T01:45:20Z")
No, itā€™s not for homework šŸ™‚ Iā€™m one to ask unusual, random and/or factual questions either regarding theory, opinion or fact. Sorry that I missed the two. I never really memorized pi. 3.14159265ā€¦ Since pi goes on forever and ever amen, you can never multiply itself times itself, correct? You can only round to the nearest thousandth. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [May 1, 2012, 1:50am UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/17 "2012-05-01T01:50:00Z")
> [@Sheriru](#): > > Since pi goes on forever and ever amen, you can never multiply itself times itself, correct? You can only round to the nearest thousandth. I think **OldGuy** already answered this in Post #5. You can compute it to whatever (finite) degree of accuracy you need. ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [May 1, 2012, 1:53am UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/18 "2012-05-01T01:53:29Z")
> [@ultrafilter](#): > > The square root of two is irrational, so no, there isnā€™t a general proof. Okay, itā€™s obvious when you put it like that. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [May 1, 2012, 2:01am UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/19 "2012-05-01T02:01:53Z")
> [@ultrafilter](#): > > I was thinking along those lines, but Iā€™m having trouble seeing exactly how that transformation works. I think if the polynomial that has pi^2 as a root has any odd degree terms that you could be in trouble. If the polynomial x |-\> p(x) has pi^2 as a root, then the polynomial x |-\> p(x^2) has pi as a root. Itā€™s only the other direction thatā€™s tricky (although itā€™s also true that if x is algebraic, so is x^2; itā€™s just not as immediate to see). The concern about odd degree terms is only for going in the other direction. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [May 1, 2012, 2:18am UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/20 "2012-05-01T02:18:38Z")
> [@Sheriru](#): > > No, itā€™s not for homework šŸ™‚ Iā€™m one to ask unusual, random and/or factual questions either regarding theory, opinion or fact. > > Sorry that I missed the two. I never really memorized pi. 3.14159265.. > Since pi goes on forever and ever amen, you can never multiply itself times itself, correct? You can only round to the nearest thousandth. Infinite series are exact answers. Exact as in exact. Itā€™s true that they can be computed only to a finite degree, but thatā€™s a practical limitation, not a theoretical one. The infinite series that Thudlow Boink gave is the exact expression of pi[sup]2[/sup]/6. Itā€™s as exact as the diagonal of a unit one square is exactly the square root of 2. BTW, if you donā€™t already know, you can enter in any mathematical relationship into the Google box and get an answer. You donā€™t even need symbols. Just put in ā€œpi squaredā€ and youā€™ll get an answer to seven decimal places. [Next page](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454.md?page=2)