A flower in a hexagon












2












$begingroup$


The area of the flower



So I had a geometry problem in my maths exam test and I couldn't solve it, though it didn't seem that hard. I tried to connect the points from center to end and create an equilateral triangle but still was unable to find the solution.

Would be thankful if you could help me out.

The question is the following:




We have an equal hexagon with sides equal to $1$ and six circular arcs with radius equal to $1$ from each vertices of the regular hexagon in it create a flower-shaped-like object. Find the area of the flower.
enter image description here




Here are the possible answers ↓

1. $pi$

3. $3pi/2$

4. $4sqrt2-pi$

5. $pi/2+sqrt3$

6. $2pi-3sqrt3$










share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    I count six ellipses
    $endgroup$
    – Ross Millikan
    5 hours ago










  • $begingroup$
    Are the ellipses tangent to the sides at the vertices and meeting without overlap at the center? If so, they are not ellipses. Do you mean pairs of circular arcs?
    $endgroup$
    – Oscar Lanzi
    4 hours ago
















2












$begingroup$


The area of the flower



So I had a geometry problem in my maths exam test and I couldn't solve it, though it didn't seem that hard. I tried to connect the points from center to end and create an equilateral triangle but still was unable to find the solution.

Would be thankful if you could help me out.

The question is the following:




We have an equal hexagon with sides equal to $1$ and six circular arcs with radius equal to $1$ from each vertices of the regular hexagon in it create a flower-shaped-like object. Find the area of the flower.
enter image description here




Here are the possible answers ↓

1. $pi$

3. $3pi/2$

4. $4sqrt2-pi$

5. $pi/2+sqrt3$

6. $2pi-3sqrt3$










share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    I count six ellipses
    $endgroup$
    – Ross Millikan
    5 hours ago










  • $begingroup$
    Are the ellipses tangent to the sides at the vertices and meeting without overlap at the center? If so, they are not ellipses. Do you mean pairs of circular arcs?
    $endgroup$
    – Oscar Lanzi
    4 hours ago














2












2








2


1



$begingroup$


The area of the flower



So I had a geometry problem in my maths exam test and I couldn't solve it, though it didn't seem that hard. I tried to connect the points from center to end and create an equilateral triangle but still was unable to find the solution.

Would be thankful if you could help me out.

The question is the following:




We have an equal hexagon with sides equal to $1$ and six circular arcs with radius equal to $1$ from each vertices of the regular hexagon in it create a flower-shaped-like object. Find the area of the flower.
enter image description here




Here are the possible answers ↓

1. $pi$

3. $3pi/2$

4. $4sqrt2-pi$

5. $pi/2+sqrt3$

6. $2pi-3sqrt3$










share|cite|improve this question











$endgroup$




The area of the flower



So I had a geometry problem in my maths exam test and I couldn't solve it, though it didn't seem that hard. I tried to connect the points from center to end and create an equilateral triangle but still was unable to find the solution.

Would be thankful if you could help me out.

The question is the following:




We have an equal hexagon with sides equal to $1$ and six circular arcs with radius equal to $1$ from each vertices of the regular hexagon in it create a flower-shaped-like object. Find the area of the flower.
enter image description here




Here are the possible answers ↓

1. $pi$

3. $3pi/2$

4. $4sqrt2-pi$

5. $pi/2+sqrt3$

6. $2pi-3sqrt3$







geometry logic triangle area






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 1 hour ago









J. W. Tanner

2,6161217




2,6161217










asked 6 hours ago









DAVODAVO

135




135








  • 2




    $begingroup$
    I count six ellipses
    $endgroup$
    – Ross Millikan
    5 hours ago










  • $begingroup$
    Are the ellipses tangent to the sides at the vertices and meeting without overlap at the center? If so, they are not ellipses. Do you mean pairs of circular arcs?
    $endgroup$
    – Oscar Lanzi
    4 hours ago














  • 2




    $begingroup$
    I count six ellipses
    $endgroup$
    – Ross Millikan
    5 hours ago










  • $begingroup$
    Are the ellipses tangent to the sides at the vertices and meeting without overlap at the center? If so, they are not ellipses. Do you mean pairs of circular arcs?
    $endgroup$
    – Oscar Lanzi
    4 hours ago








2




2




$begingroup$
I count six ellipses
$endgroup$
– Ross Millikan
5 hours ago




$begingroup$
I count six ellipses
$endgroup$
– Ross Millikan
5 hours ago












$begingroup$
Are the ellipses tangent to the sides at the vertices and meeting without overlap at the center? If so, they are not ellipses. Do you mean pairs of circular arcs?
$endgroup$
– Oscar Lanzi
4 hours ago




$begingroup$
Are the ellipses tangent to the sides at the vertices and meeting without overlap at the center? If so, they are not ellipses. Do you mean pairs of circular arcs?
$endgroup$
– Oscar Lanzi
4 hours ago










3 Answers
3






active

oldest

votes


















4












$begingroup$

Assuming those are circular arcs and not "ellipses", you can indeed find the area exactly.





The "flower" has six "petals" Each of those petals has axial symmetry and can be divided into two halves. Each of those halves is the segment subtending a central angle of $frac{pi}{3}$ (radian measure) of a circle of radius $1$.



The area of one such segment is $frac 12 r^2(theta - sintheta) = frac 12(frac{pi}{3} - frac{sqrt{3}}{2})$.



There are $12$ such segments, yielding the total area of the "flower" as $2pi - 3sqrt 3$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Since the shape of each petal is claimed to be ellipses, your answer is not valid in assuming that they consist of circular arcs. However, I do reckon that they are intented to be arcs.
    $endgroup$
    – Trebor
    4 hours ago










  • $begingroup$
    @Trebor I had edited my answer (I think before your comment) to clarify that assumption. However, I think it's a valid assumption, and just a very poor question.
    $endgroup$
    – Deepak
    4 hours ago










  • $begingroup$
    @AnirbanNiloy Thank you for the nice diagram. :)
    $endgroup$
    – Deepak
    12 mins ago










  • $begingroup$
    @Deepak Welcome! Anytime!!!! You deserve more than that.
    $endgroup$
    – Anirban Niloy
    10 mins ago





















3












$begingroup$

We can't easily find the area exactly - we just don't have enough information for that. But then, what are the estimates we're looking at? (1) $approx 3.1$. (3) $approx 4.7$. (4) $approx 2.5$. (5) $approx 3.3$. (6) $approx 1.1$.



If the area's close to $3$, we might have difficulty deciding. If it's farther away, it'll be an easy choice.



Now, what's the area of the hexagon? That's something we can calculate exactly; an equilateral triangle of side $1$ has area $frac{sqrt{3}}{4}$, and the hexagon is six of those stuck together for a total area of $frac32sqrt{3}approx 2.6$. The flower is clearly much less than that. Only one of the answers is plausible. It has to be (6), $2pi-3sqrt{3}$.



And, reverse-engineering from the answer... the curves that define the "petals" are the six circles centered at the vertices of the hexagon with radius $1$. That bit about "five ellipses" is just wrong.






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    Here's a solution that doesn't require trigonometry. The hexagon is made up of 6 "petals" and 6 "wedges" (the scooped out triangles between the petals). Call the area of a petal $p$, and the area of a wedge $w$.



    What is the area enclosed by two adjacent sides of the hexagon and the arc drawn from the vertex between them. (For example, the region EFAGE). It's one third of a unit circle, and contains three petals and two wedges:



    $3p + 2w = {frac 1 3} pi r^2 = frac pi 3$



    What is the area of one triangle (one sixth of the hexagon). The altitude of the triangle is $sqrt{1 - {frac 1 4}} = frac{sqrt 3}{2}$, so the area is $frac{sqrt 3}{4}$. The triangle is made up of two half-petals and one wedge:



    $p + w = frac{sqrt 3}{4}$



    Solving the two simultaneous equations:



    $2p + 2w = frac{sqrt 3}{2}$



    $p = frac{pi}{3} - frac{sqrt 3}{2}$



    And the area of the flower is $6p = 2pi - 3sqrt 3$.






    share|cite|improve this answer








    New contributor




    AndyB is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.






    $endgroup$













      Your Answer





      StackExchange.ifUsing("editor", function () {
      return StackExchange.using("mathjaxEditing", function () {
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      });
      });
      }, "mathjax-editing");

      StackExchange.ready(function() {
      var channelOptions = {
      tags: "".split(" "),
      id: "69"
      };
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function() {
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled) {
      StackExchange.using("snippets", function() {
      createEditor();
      });
      }
      else {
      createEditor();
      }
      });

      function createEditor() {
      StackExchange.prepareEditor({
      heartbeatType: 'answer',
      autoActivateHeartbeat: false,
      convertImagesToLinks: true,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      imageUploader: {
      brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
      contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
      allowUrls: true
      },
      noCode: true, onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      });


      }
      });














      draft saved

      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3133044%2fa-flower-in-a-hexagon%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      4












      $begingroup$

      Assuming those are circular arcs and not "ellipses", you can indeed find the area exactly.





      The "flower" has six "petals" Each of those petals has axial symmetry and can be divided into two halves. Each of those halves is the segment subtending a central angle of $frac{pi}{3}$ (radian measure) of a circle of radius $1$.



      The area of one such segment is $frac 12 r^2(theta - sintheta) = frac 12(frac{pi}{3} - frac{sqrt{3}}{2})$.



      There are $12$ such segments, yielding the total area of the "flower" as $2pi - 3sqrt 3$.






      share|cite|improve this answer











      $endgroup$













      • $begingroup$
        Since the shape of each petal is claimed to be ellipses, your answer is not valid in assuming that they consist of circular arcs. However, I do reckon that they are intented to be arcs.
        $endgroup$
        – Trebor
        4 hours ago










      • $begingroup$
        @Trebor I had edited my answer (I think before your comment) to clarify that assumption. However, I think it's a valid assumption, and just a very poor question.
        $endgroup$
        – Deepak
        4 hours ago










      • $begingroup$
        @AnirbanNiloy Thank you for the nice diagram. :)
        $endgroup$
        – Deepak
        12 mins ago










      • $begingroup$
        @Deepak Welcome! Anytime!!!! You deserve more than that.
        $endgroup$
        – Anirban Niloy
        10 mins ago


















      4












      $begingroup$

      Assuming those are circular arcs and not "ellipses", you can indeed find the area exactly.





      The "flower" has six "petals" Each of those petals has axial symmetry and can be divided into two halves. Each of those halves is the segment subtending a central angle of $frac{pi}{3}$ (radian measure) of a circle of radius $1$.



      The area of one such segment is $frac 12 r^2(theta - sintheta) = frac 12(frac{pi}{3} - frac{sqrt{3}}{2})$.



      There are $12$ such segments, yielding the total area of the "flower" as $2pi - 3sqrt 3$.






      share|cite|improve this answer











      $endgroup$













      • $begingroup$
        Since the shape of each petal is claimed to be ellipses, your answer is not valid in assuming that they consist of circular arcs. However, I do reckon that they are intented to be arcs.
        $endgroup$
        – Trebor
        4 hours ago










      • $begingroup$
        @Trebor I had edited my answer (I think before your comment) to clarify that assumption. However, I think it's a valid assumption, and just a very poor question.
        $endgroup$
        – Deepak
        4 hours ago










      • $begingroup$
        @AnirbanNiloy Thank you for the nice diagram. :)
        $endgroup$
        – Deepak
        12 mins ago










      • $begingroup$
        @Deepak Welcome! Anytime!!!! You deserve more than that.
        $endgroup$
        – Anirban Niloy
        10 mins ago
















      4












      4








      4





      $begingroup$

      Assuming those are circular arcs and not "ellipses", you can indeed find the area exactly.





      The "flower" has six "petals" Each of those petals has axial symmetry and can be divided into two halves. Each of those halves is the segment subtending a central angle of $frac{pi}{3}$ (radian measure) of a circle of radius $1$.



      The area of one such segment is $frac 12 r^2(theta - sintheta) = frac 12(frac{pi}{3} - frac{sqrt{3}}{2})$.



      There are $12$ such segments, yielding the total area of the "flower" as $2pi - 3sqrt 3$.






      share|cite|improve this answer











      $endgroup$



      Assuming those are circular arcs and not "ellipses", you can indeed find the area exactly.





      The "flower" has six "petals" Each of those petals has axial symmetry and can be divided into two halves. Each of those halves is the segment subtending a central angle of $frac{pi}{3}$ (radian measure) of a circle of radius $1$.



      The area of one such segment is $frac 12 r^2(theta - sintheta) = frac 12(frac{pi}{3} - frac{sqrt{3}}{2})$.



      There are $12$ such segments, yielding the total area of the "flower" as $2pi - 3sqrt 3$.







      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited 2 hours ago









      Anirban Niloy

      609218




      609218










      answered 4 hours ago









      DeepakDeepak

      17.2k11536




      17.2k11536












      • $begingroup$
        Since the shape of each petal is claimed to be ellipses, your answer is not valid in assuming that they consist of circular arcs. However, I do reckon that they are intented to be arcs.
        $endgroup$
        – Trebor
        4 hours ago










      • $begingroup$
        @Trebor I had edited my answer (I think before your comment) to clarify that assumption. However, I think it's a valid assumption, and just a very poor question.
        $endgroup$
        – Deepak
        4 hours ago










      • $begingroup$
        @AnirbanNiloy Thank you for the nice diagram. :)
        $endgroup$
        – Deepak
        12 mins ago










      • $begingroup$
        @Deepak Welcome! Anytime!!!! You deserve more than that.
        $endgroup$
        – Anirban Niloy
        10 mins ago




















      • $begingroup$
        Since the shape of each petal is claimed to be ellipses, your answer is not valid in assuming that they consist of circular arcs. However, I do reckon that they are intented to be arcs.
        $endgroup$
        – Trebor
        4 hours ago










      • $begingroup$
        @Trebor I had edited my answer (I think before your comment) to clarify that assumption. However, I think it's a valid assumption, and just a very poor question.
        $endgroup$
        – Deepak
        4 hours ago










      • $begingroup$
        @AnirbanNiloy Thank you for the nice diagram. :)
        $endgroup$
        – Deepak
        12 mins ago










      • $begingroup$
        @Deepak Welcome! Anytime!!!! You deserve more than that.
        $endgroup$
        – Anirban Niloy
        10 mins ago


















      $begingroup$
      Since the shape of each petal is claimed to be ellipses, your answer is not valid in assuming that they consist of circular arcs. However, I do reckon that they are intented to be arcs.
      $endgroup$
      – Trebor
      4 hours ago




      $begingroup$
      Since the shape of each petal is claimed to be ellipses, your answer is not valid in assuming that they consist of circular arcs. However, I do reckon that they are intented to be arcs.
      $endgroup$
      – Trebor
      4 hours ago












      $begingroup$
      @Trebor I had edited my answer (I think before your comment) to clarify that assumption. However, I think it's a valid assumption, and just a very poor question.
      $endgroup$
      – Deepak
      4 hours ago




      $begingroup$
      @Trebor I had edited my answer (I think before your comment) to clarify that assumption. However, I think it's a valid assumption, and just a very poor question.
      $endgroup$
      – Deepak
      4 hours ago












      $begingroup$
      @AnirbanNiloy Thank you for the nice diagram. :)
      $endgroup$
      – Deepak
      12 mins ago




      $begingroup$
      @AnirbanNiloy Thank you for the nice diagram. :)
      $endgroup$
      – Deepak
      12 mins ago












      $begingroup$
      @Deepak Welcome! Anytime!!!! You deserve more than that.
      $endgroup$
      – Anirban Niloy
      10 mins ago






      $begingroup$
      @Deepak Welcome! Anytime!!!! You deserve more than that.
      $endgroup$
      – Anirban Niloy
      10 mins ago













      3












      $begingroup$

      We can't easily find the area exactly - we just don't have enough information for that. But then, what are the estimates we're looking at? (1) $approx 3.1$. (3) $approx 4.7$. (4) $approx 2.5$. (5) $approx 3.3$. (6) $approx 1.1$.



      If the area's close to $3$, we might have difficulty deciding. If it's farther away, it'll be an easy choice.



      Now, what's the area of the hexagon? That's something we can calculate exactly; an equilateral triangle of side $1$ has area $frac{sqrt{3}}{4}$, and the hexagon is six of those stuck together for a total area of $frac32sqrt{3}approx 2.6$. The flower is clearly much less than that. Only one of the answers is plausible. It has to be (6), $2pi-3sqrt{3}$.



      And, reverse-engineering from the answer... the curves that define the "petals" are the six circles centered at the vertices of the hexagon with radius $1$. That bit about "five ellipses" is just wrong.






      share|cite|improve this answer









      $endgroup$


















        3












        $begingroup$

        We can't easily find the area exactly - we just don't have enough information for that. But then, what are the estimates we're looking at? (1) $approx 3.1$. (3) $approx 4.7$. (4) $approx 2.5$. (5) $approx 3.3$. (6) $approx 1.1$.



        If the area's close to $3$, we might have difficulty deciding. If it's farther away, it'll be an easy choice.



        Now, what's the area of the hexagon? That's something we can calculate exactly; an equilateral triangle of side $1$ has area $frac{sqrt{3}}{4}$, and the hexagon is six of those stuck together for a total area of $frac32sqrt{3}approx 2.6$. The flower is clearly much less than that. Only one of the answers is plausible. It has to be (6), $2pi-3sqrt{3}$.



        And, reverse-engineering from the answer... the curves that define the "petals" are the six circles centered at the vertices of the hexagon with radius $1$. That bit about "five ellipses" is just wrong.






        share|cite|improve this answer









        $endgroup$
















          3












          3








          3





          $begingroup$

          We can't easily find the area exactly - we just don't have enough information for that. But then, what are the estimates we're looking at? (1) $approx 3.1$. (3) $approx 4.7$. (4) $approx 2.5$. (5) $approx 3.3$. (6) $approx 1.1$.



          If the area's close to $3$, we might have difficulty deciding. If it's farther away, it'll be an easy choice.



          Now, what's the area of the hexagon? That's something we can calculate exactly; an equilateral triangle of side $1$ has area $frac{sqrt{3}}{4}$, and the hexagon is six of those stuck together for a total area of $frac32sqrt{3}approx 2.6$. The flower is clearly much less than that. Only one of the answers is plausible. It has to be (6), $2pi-3sqrt{3}$.



          And, reverse-engineering from the answer... the curves that define the "petals" are the six circles centered at the vertices of the hexagon with radius $1$. That bit about "five ellipses" is just wrong.






          share|cite|improve this answer









          $endgroup$



          We can't easily find the area exactly - we just don't have enough information for that. But then, what are the estimates we're looking at? (1) $approx 3.1$. (3) $approx 4.7$. (4) $approx 2.5$. (5) $approx 3.3$. (6) $approx 1.1$.



          If the area's close to $3$, we might have difficulty deciding. If it's farther away, it'll be an easy choice.



          Now, what's the area of the hexagon? That's something we can calculate exactly; an equilateral triangle of side $1$ has area $frac{sqrt{3}}{4}$, and the hexagon is six of those stuck together for a total area of $frac32sqrt{3}approx 2.6$. The flower is clearly much less than that. Only one of the answers is plausible. It has to be (6), $2pi-3sqrt{3}$.



          And, reverse-engineering from the answer... the curves that define the "petals" are the six circles centered at the vertices of the hexagon with radius $1$. That bit about "five ellipses" is just wrong.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 5 hours ago









          jmerryjmerry

          11.6k1527




          11.6k1527























              0












              $begingroup$

              Here's a solution that doesn't require trigonometry. The hexagon is made up of 6 "petals" and 6 "wedges" (the scooped out triangles between the petals). Call the area of a petal $p$, and the area of a wedge $w$.



              What is the area enclosed by two adjacent sides of the hexagon and the arc drawn from the vertex between them. (For example, the region EFAGE). It's one third of a unit circle, and contains three petals and two wedges:



              $3p + 2w = {frac 1 3} pi r^2 = frac pi 3$



              What is the area of one triangle (one sixth of the hexagon). The altitude of the triangle is $sqrt{1 - {frac 1 4}} = frac{sqrt 3}{2}$, so the area is $frac{sqrt 3}{4}$. The triangle is made up of two half-petals and one wedge:



              $p + w = frac{sqrt 3}{4}$



              Solving the two simultaneous equations:



              $2p + 2w = frac{sqrt 3}{2}$



              $p = frac{pi}{3} - frac{sqrt 3}{2}$



              And the area of the flower is $6p = 2pi - 3sqrt 3$.






              share|cite|improve this answer








              New contributor




              AndyB is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.






              $endgroup$


















                0












                $begingroup$

                Here's a solution that doesn't require trigonometry. The hexagon is made up of 6 "petals" and 6 "wedges" (the scooped out triangles between the petals). Call the area of a petal $p$, and the area of a wedge $w$.



                What is the area enclosed by two adjacent sides of the hexagon and the arc drawn from the vertex between them. (For example, the region EFAGE). It's one third of a unit circle, and contains three petals and two wedges:



                $3p + 2w = {frac 1 3} pi r^2 = frac pi 3$



                What is the area of one triangle (one sixth of the hexagon). The altitude of the triangle is $sqrt{1 - {frac 1 4}} = frac{sqrt 3}{2}$, so the area is $frac{sqrt 3}{4}$. The triangle is made up of two half-petals and one wedge:



                $p + w = frac{sqrt 3}{4}$



                Solving the two simultaneous equations:



                $2p + 2w = frac{sqrt 3}{2}$



                $p = frac{pi}{3} - frac{sqrt 3}{2}$



                And the area of the flower is $6p = 2pi - 3sqrt 3$.






                share|cite|improve this answer








                New contributor




                AndyB is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                Check out our Code of Conduct.






                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  Here's a solution that doesn't require trigonometry. The hexagon is made up of 6 "petals" and 6 "wedges" (the scooped out triangles between the petals). Call the area of a petal $p$, and the area of a wedge $w$.



                  What is the area enclosed by two adjacent sides of the hexagon and the arc drawn from the vertex between them. (For example, the region EFAGE). It's one third of a unit circle, and contains three petals and two wedges:



                  $3p + 2w = {frac 1 3} pi r^2 = frac pi 3$



                  What is the area of one triangle (one sixth of the hexagon). The altitude of the triangle is $sqrt{1 - {frac 1 4}} = frac{sqrt 3}{2}$, so the area is $frac{sqrt 3}{4}$. The triangle is made up of two half-petals and one wedge:



                  $p + w = frac{sqrt 3}{4}$



                  Solving the two simultaneous equations:



                  $2p + 2w = frac{sqrt 3}{2}$



                  $p = frac{pi}{3} - frac{sqrt 3}{2}$



                  And the area of the flower is $6p = 2pi - 3sqrt 3$.






                  share|cite|improve this answer








                  New contributor




                  AndyB is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.






                  $endgroup$



                  Here's a solution that doesn't require trigonometry. The hexagon is made up of 6 "petals" and 6 "wedges" (the scooped out triangles between the petals). Call the area of a petal $p$, and the area of a wedge $w$.



                  What is the area enclosed by two adjacent sides of the hexagon and the arc drawn from the vertex between them. (For example, the region EFAGE). It's one third of a unit circle, and contains three petals and two wedges:



                  $3p + 2w = {frac 1 3} pi r^2 = frac pi 3$



                  What is the area of one triangle (one sixth of the hexagon). The altitude of the triangle is $sqrt{1 - {frac 1 4}} = frac{sqrt 3}{2}$, so the area is $frac{sqrt 3}{4}$. The triangle is made up of two half-petals and one wedge:



                  $p + w = frac{sqrt 3}{4}$



                  Solving the two simultaneous equations:



                  $2p + 2w = frac{sqrt 3}{2}$



                  $p = frac{pi}{3} - frac{sqrt 3}{2}$



                  And the area of the flower is $6p = 2pi - 3sqrt 3$.







                  share|cite|improve this answer








                  New contributor




                  AndyB is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.









                  share|cite|improve this answer



                  share|cite|improve this answer






                  New contributor




                  AndyB is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.









                  answered 25 mins ago









                  AndyBAndyB

                  1




                  1




                  New contributor




                  AndyB is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.





                  New contributor





                  AndyB is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.






                  AndyB is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.






























                      draft saved

                      draft discarded




















































                      Thanks for contributing an answer to Mathematics Stack Exchange!


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      Use MathJax to format equations. MathJax reference.


                      To learn more, see our tips on writing great answers.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3133044%2fa-flower-in-a-hexagon%23new-answer', 'question_page');
                      }
                      );

                      Post as a guest















                      Required, but never shown





















































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown

































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown







                      Popular posts from this blog

                      How to label and detect the document text images

                      Vallis Paradisi

                      Tabula Rosettana