About sum of right-angled triangles'area and area of circle.












0












$begingroup$


the image shows right-angled triangles in semi-circle



In Definite Integration, we know that area can be found by adding up the total area of each small divided parts.



So, base on the Definite Integration, we may say the area of circle is equal to
$sum^{n}_{k=1}(h_k)$



And we also know that $A_k=2r(h_k)(1/2)=rh_k$ while $A$ is refer to the right-angled triangle's area.



so that, $A_k(1/r)=h_k$



As a result this equation comes out with fixed position of diameter:



$sum^{n}_{k=1}(h_k)=pi r^2$



$sum^{n}_{k=1}(A_k)(1/r)=pi r^2$



$sum^{n}_{k=1}(A_k)=pi r^3$



I wish to know whether I am correct or not, thanks.






definite-integrals triangle circle







share|cite|improve this question









$endgroup$












  • $begingroup$
    A little correction, sum $h_k$ = $pi r^2/2$. I do not know if this is correct of not though
    $endgroup$
    – Anvit
    Dec 17 '18 at 10:34










  • $begingroup$
    Thanks for your respond, but I am trying to use sum $h_k=π r^2$ for considering the heights of triangles in both sides.
    $endgroup$
    – Students
    Dec 17 '18 at 11:11
















0












$begingroup$


the image shows right-angled triangles in semi-circle



In Definite Integration, we know that area can be found by adding up the total area of each small divided parts.



So, base on the Definite Integration, we may say the area of circle is equal to
$sum^{n}_{k=1}(h_k)$



And we also know that $A_k=2r(h_k)(1/2)=rh_k$ while $A$ is refer to the right-angled triangle's area.



so that, $A_k(1/r)=h_k$



As a result this equation comes out with fixed position of diameter:



$sum^{n}_{k=1}(h_k)=pi r^2$



$sum^{n}_{k=1}(A_k)(1/r)=pi r^2$



$sum^{n}_{k=1}(A_k)=pi r^3$



I wish to know whether I am correct or not, thanks.






definite-integrals triangle circle







share|cite|improve this question









$endgroup$












  • $begingroup$
    A little correction, sum $h_k$ = $pi r^2/2$. I do not know if this is correct of not though
    $endgroup$
    – Anvit
    Dec 17 '18 at 10:34










  • $begingroup$
    Thanks for your respond, but I am trying to use sum $h_k=π r^2$ for considering the heights of triangles in both sides.
    $endgroup$
    – Students
    Dec 17 '18 at 11:11














0












0








0





$begingroup$


the image shows right-angled triangles in semi-circle



In Definite Integration, we know that area can be found by adding up the total area of each small divided parts.



So, base on the Definite Integration, we may say the area of circle is equal to
$sum^{n}_{k=1}(h_k)$



And we also know that $A_k=2r(h_k)(1/2)=rh_k$ while $A$ is refer to the right-angled triangle's area.



so that, $A_k(1/r)=h_k$



As a result this equation comes out with fixed position of diameter:



$sum^{n}_{k=1}(h_k)=pi r^2$



$sum^{n}_{k=1}(A_k)(1/r)=pi r^2$



$sum^{n}_{k=1}(A_k)=pi r^3$



I wish to know whether I am correct or not, thanks.






definite-integrals triangle circle







share|cite|improve this question









$endgroup$




the image shows right-angled triangles in semi-circle



In Definite Integration, we know that area can be found by adding up the total area of each small divided parts.



So, base on the Definite Integration, we may say the area of circle is equal to
$sum^{n}_{k=1}(h_k)$



And we also know that $A_k=2r(h_k)(1/2)=rh_k$ while $A$ is refer to the right-angled triangle's area.



so that, $A_k(1/r)=h_k$



As a result this equation comes out with fixed position of diameter:



$sum^{n}_{k=1}(h_k)=pi r^2$



$sum^{n}_{k=1}(A_k)(1/r)=pi r^2$



$sum^{n}_{k=1}(A_k)=pi r^3$



I wish to know whether I am correct or not, thanks.






definite-integrals triangle circle




definite-integrals triangle circle






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 17 '18 at 10:28









StudentsStudents

153




153












  • $begingroup$
    A little correction, sum $h_k$ = $pi r^2/2$. I do not know if this is correct of not though
    $endgroup$
    – Anvit
    Dec 17 '18 at 10:34










  • $begingroup$
    Thanks for your respond, but I am trying to use sum $h_k=π r^2$ for considering the heights of triangles in both sides.
    $endgroup$
    – Students
    Dec 17 '18 at 11:11


















  • $begingroup$
    A little correction, sum $h_k$ = $pi r^2/2$. I do not know if this is correct of not though
    $endgroup$
    – Anvit
    Dec 17 '18 at 10:34










  • $begingroup$
    Thanks for your respond, but I am trying to use sum $h_k=π r^2$ for considering the heights of triangles in both sides.
    $endgroup$
    – Students
    Dec 17 '18 at 11:11
















$begingroup$
A little correction, sum $h_k$ = $pi r^2/2$. I do not know if this is correct of not though
$endgroup$
– Anvit
Dec 17 '18 at 10:34




$begingroup$
A little correction, sum $h_k$ = $pi r^2/2$. I do not know if this is correct of not though
$endgroup$
– Anvit
Dec 17 '18 at 10:34












$begingroup$
Thanks for your respond, but I am trying to use sum $h_k=π r^2$ for considering the heights of triangles in both sides.
$endgroup$
– Students
Dec 17 '18 at 11:11




$begingroup$
Thanks for your respond, but I am trying to use sum $h_k=π r^2$ for considering the heights of triangles in both sides.
$endgroup$
– Students
Dec 17 '18 at 11:11










2 Answers
2






active

oldest

votes


















0












$begingroup$

You have wrongly assumed that the sum of the heights $h_k$ is equal to the area of the circle. Area has dimensions $[L^2]$, so the area of the circle is actually the sum of the areas of the rectangular strips of height $h_k$ and infinitesimal base length $dx$.



$h_k=A_k/r\displaystylesum_1^infty h_kdx=sum_1^inftyfrac{A_kdx}r=pi r^2$






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    No, you are all wrong.



    If you want to make a sum of areas of some triangles that is approximate to the area of the circle, these triangles must not overlap. On your linked picture the two triangles obviously overlap (have some common area), so the further discussion is pointless.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Thanks for the ans first. But what I want is to make a sum of heights of triangles that is approximate to the area of the circle (not the areas).
      $endgroup$
      – Students
      Dec 17 '18 at 11:05










    • $begingroup$
      @ShubhamJohri I can see your point, so what you say is $lim_{x to 0}sum^{n}_{k=1}(h_k)x=πr^2$?
      $endgroup$
      – Students
      Dec 17 '18 at 11:35










    • $begingroup$
      Yes, $xto0,ntoinfty$
      $endgroup$
      – Shubham Johri
      Dec 17 '18 at 11:36










    • $begingroup$
      It seems that @Students doesn't understand the concept of the differential $dx$ in the integral notation. Of course, it's not for a decorative purpose. It's meaning is that the value is infinitesimally small, i.e. $dx → 0$ but the sum of all these small values is constant – the full length of variable $x$ where integration is done (in case of a circle that sum is the diameter $2r$)
      $endgroup$
      – Mucko
      Dec 18 '18 at 13:21











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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    0












    $begingroup$

    You have wrongly assumed that the sum of the heights $h_k$ is equal to the area of the circle. Area has dimensions $[L^2]$, so the area of the circle is actually the sum of the areas of the rectangular strips of height $h_k$ and infinitesimal base length $dx$.



    $h_k=A_k/r\displaystylesum_1^infty h_kdx=sum_1^inftyfrac{A_kdx}r=pi r^2$






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      You have wrongly assumed that the sum of the heights $h_k$ is equal to the area of the circle. Area has dimensions $[L^2]$, so the area of the circle is actually the sum of the areas of the rectangular strips of height $h_k$ and infinitesimal base length $dx$.



      $h_k=A_k/r\displaystylesum_1^infty h_kdx=sum_1^inftyfrac{A_kdx}r=pi r^2$






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        You have wrongly assumed that the sum of the heights $h_k$ is equal to the area of the circle. Area has dimensions $[L^2]$, so the area of the circle is actually the sum of the areas of the rectangular strips of height $h_k$ and infinitesimal base length $dx$.



        $h_k=A_k/r\displaystylesum_1^infty h_kdx=sum_1^inftyfrac{A_kdx}r=pi r^2$






        share|cite|improve this answer









        $endgroup$



        You have wrongly assumed that the sum of the heights $h_k$ is equal to the area of the circle. Area has dimensions $[L^2]$, so the area of the circle is actually the sum of the areas of the rectangular strips of height $h_k$ and infinitesimal base length $dx$.



        $h_k=A_k/r\displaystylesum_1^infty h_kdx=sum_1^inftyfrac{A_kdx}r=pi r^2$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 17 '18 at 11:35









        Shubham JohriShubham Johri

        5,262718




        5,262718























            0












            $begingroup$

            No, you are all wrong.



            If you want to make a sum of areas of some triangles that is approximate to the area of the circle, these triangles must not overlap. On your linked picture the two triangles obviously overlap (have some common area), so the further discussion is pointless.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Thanks for the ans first. But what I want is to make a sum of heights of triangles that is approximate to the area of the circle (not the areas).
              $endgroup$
              – Students
              Dec 17 '18 at 11:05










            • $begingroup$
              @ShubhamJohri I can see your point, so what you say is $lim_{x to 0}sum^{n}_{k=1}(h_k)x=πr^2$?
              $endgroup$
              – Students
              Dec 17 '18 at 11:35










            • $begingroup$
              Yes, $xto0,ntoinfty$
              $endgroup$
              – Shubham Johri
              Dec 17 '18 at 11:36










            • $begingroup$
              It seems that @Students doesn't understand the concept of the differential $dx$ in the integral notation. Of course, it's not for a decorative purpose. It's meaning is that the value is infinitesimally small, i.e. $dx → 0$ but the sum of all these small values is constant – the full length of variable $x$ where integration is done (in case of a circle that sum is the diameter $2r$)
              $endgroup$
              – Mucko
              Dec 18 '18 at 13:21
















            0












            $begingroup$

            No, you are all wrong.



            If you want to make a sum of areas of some triangles that is approximate to the area of the circle, these triangles must not overlap. On your linked picture the two triangles obviously overlap (have some common area), so the further discussion is pointless.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Thanks for the ans first. But what I want is to make a sum of heights of triangles that is approximate to the area of the circle (not the areas).
              $endgroup$
              – Students
              Dec 17 '18 at 11:05










            • $begingroup$
              @ShubhamJohri I can see your point, so what you say is $lim_{x to 0}sum^{n}_{k=1}(h_k)x=πr^2$?
              $endgroup$
              – Students
              Dec 17 '18 at 11:35










            • $begingroup$
              Yes, $xto0,ntoinfty$
              $endgroup$
              – Shubham Johri
              Dec 17 '18 at 11:36










            • $begingroup$
              It seems that @Students doesn't understand the concept of the differential $dx$ in the integral notation. Of course, it's not for a decorative purpose. It's meaning is that the value is infinitesimally small, i.e. $dx → 0$ but the sum of all these small values is constant – the full length of variable $x$ where integration is done (in case of a circle that sum is the diameter $2r$)
              $endgroup$
              – Mucko
              Dec 18 '18 at 13:21














            0












            0








            0





            $begingroup$

            No, you are all wrong.



            If you want to make a sum of areas of some triangles that is approximate to the area of the circle, these triangles must not overlap. On your linked picture the two triangles obviously overlap (have some common area), so the further discussion is pointless.






            share|cite|improve this answer









            $endgroup$



            No, you are all wrong.



            If you want to make a sum of areas of some triangles that is approximate to the area of the circle, these triangles must not overlap. On your linked picture the two triangles obviously overlap (have some common area), so the further discussion is pointless.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 17 '18 at 10:42









            MuckoMucko

            11




            11












            • $begingroup$
              Thanks for the ans first. But what I want is to make a sum of heights of triangles that is approximate to the area of the circle (not the areas).
              $endgroup$
              – Students
              Dec 17 '18 at 11:05










            • $begingroup$
              @ShubhamJohri I can see your point, so what you say is $lim_{x to 0}sum^{n}_{k=1}(h_k)x=πr^2$?
              $endgroup$
              – Students
              Dec 17 '18 at 11:35










            • $begingroup$
              Yes, $xto0,ntoinfty$
              $endgroup$
              – Shubham Johri
              Dec 17 '18 at 11:36










            • $begingroup$
              It seems that @Students doesn't understand the concept of the differential $dx$ in the integral notation. Of course, it's not for a decorative purpose. It's meaning is that the value is infinitesimally small, i.e. $dx → 0$ but the sum of all these small values is constant – the full length of variable $x$ where integration is done (in case of a circle that sum is the diameter $2r$)
              $endgroup$
              – Mucko
              Dec 18 '18 at 13:21


















            • $begingroup$
              Thanks for the ans first. But what I want is to make a sum of heights of triangles that is approximate to the area of the circle (not the areas).
              $endgroup$
              – Students
              Dec 17 '18 at 11:05










            • $begingroup$
              @ShubhamJohri I can see your point, so what you say is $lim_{x to 0}sum^{n}_{k=1}(h_k)x=πr^2$?
              $endgroup$
              – Students
              Dec 17 '18 at 11:35










            • $begingroup$
              Yes, $xto0,ntoinfty$
              $endgroup$
              – Shubham Johri
              Dec 17 '18 at 11:36










            • $begingroup$
              It seems that @Students doesn't understand the concept of the differential $dx$ in the integral notation. Of course, it's not for a decorative purpose. It's meaning is that the value is infinitesimally small, i.e. $dx → 0$ but the sum of all these small values is constant – the full length of variable $x$ where integration is done (in case of a circle that sum is the diameter $2r$)
              $endgroup$
              – Mucko
              Dec 18 '18 at 13:21
















            $begingroup$
            Thanks for the ans first. But what I want is to make a sum of heights of triangles that is approximate to the area of the circle (not the areas).
            $endgroup$
            – Students
            Dec 17 '18 at 11:05




            $begingroup$
            Thanks for the ans first. But what I want is to make a sum of heights of triangles that is approximate to the area of the circle (not the areas).
            $endgroup$
            – Students
            Dec 17 '18 at 11:05












            $begingroup$
            @ShubhamJohri I can see your point, so what you say is $lim_{x to 0}sum^{n}_{k=1}(h_k)x=πr^2$?
            $endgroup$
            – Students
            Dec 17 '18 at 11:35




            $begingroup$
            @ShubhamJohri I can see your point, so what you say is $lim_{x to 0}sum^{n}_{k=1}(h_k)x=πr^2$?
            $endgroup$
            – Students
            Dec 17 '18 at 11:35












            $begingroup$
            Yes, $xto0,ntoinfty$
            $endgroup$
            – Shubham Johri
            Dec 17 '18 at 11:36




            $begingroup$
            Yes, $xto0,ntoinfty$
            $endgroup$
            – Shubham Johri
            Dec 17 '18 at 11:36












            $begingroup$
            It seems that @Students doesn't understand the concept of the differential $dx$ in the integral notation. Of course, it's not for a decorative purpose. It's meaning is that the value is infinitesimally small, i.e. $dx → 0$ but the sum of all these small values is constant – the full length of variable $x$ where integration is done (in case of a circle that sum is the diameter $2r$)
            $endgroup$
            – Mucko
            Dec 18 '18 at 13:21




            $begingroup$
            It seems that @Students doesn't understand the concept of the differential $dx$ in the integral notation. Of course, it's not for a decorative purpose. It's meaning is that the value is infinitesimally small, i.e. $dx → 0$ but the sum of all these small values is constant – the full length of variable $x$ where integration is done (in case of a circle that sum is the diameter $2r$)
            $endgroup$
            – Mucko
            Dec 18 '18 at 13:21


















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