isopometric quotients

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Cara Foley 10i

Isoperimetric Quotients

Isoperimetric Quotients of plane shapes are calculated using the formula:

I.Q. = 4π x Area of shape

                (Perimeter of shape)²

I am going to investigate isoperimetric quotients of plane shapes and interpret my findings.

Firstly, I am going to look at flat shapes. Using the formula, I will calculate the isoperimetric quotients of the shapes.  Starting with the smallest 2D shape- a triangle- I will calculate the I.Q s of right-angled triangles. I will also do this with isosceles and equilateral triangles. I will move on to quadrilaterals and look at the I.Q s. Maybe there will be something about the results that will help me with further plane shapes; pentagon, hexagon, heptagon, nonagon, decagon and possibly a circle. With comparison, the results might show something about the shapes, such as a pattern.

Triangles

I am now going to study right-angled triangles.

Right-angled triangles

I will first look at the 3, 4, 5 right-angled triangles and then enlargements of it.

1.    

Perimeter= 3+4+5= 12 cm

Area= ½ x 4 x 3= 6cm²

I.Q. = 4 x π x 6

                    12²

I.Q. = 0.5236

I will look at similar enlarged right-angled triangles, based on the 3, 4, 5 triangle.

2.  

Perimeter= 6+8+10= 24 cm

Area= ½ x 6 x 8= 24 cm ²

I.Q= 4 x π x 24

                   24²

I.Q. = 0.5236

3.

Perimeter= 16+20+12=48 cm

Area= ½ x 16 x 12= 96 cm²

I.Q. = 4 x π x 96

                  48²

=0.5236

Similar 345 right-angled triangles have the same I.Q. The answers are all the same.  I predict that similar enlarged shapes make the same I.Q. answer.

I am now going to look at other right-angled triangles that are not versions of the 3, 4, 5.

Other right angled triangles

Firstly, in these triangles, I will need to find the hypotenuse length so I will use Pythagoras’ theorem.

4.

Hypotenuse = 4² + 6² = 52

                           =    √52

                           = 7. 2111 cm

Perimeter = 4+6+7.2111

                     = 17.2111 cm

Area= ½ x 4 x 6

           = 12 cm ²

I.Q. = 4 x π x 12

              17.2111²

I.Q. = 0.5090

5.  

H = 8² + 12² = 208

       = 208

       = 14.422 cm

P = 12+8+14.422

     = 34.422

A = ½ x 8 x 12

      = 48

I.Q. = 4 x π x 48

               34.422²

I.Q. = 0.5090

6.

H = 3² + 2² = 13

      = 13

      = 3.6

P = 3+2+3.6

    = 8.6

A= ½ x 2 x 3

      = 3

I.Q. = 4 x π x 3

                  8.6²

I.Q. = 0.5097

7.

H = 6²+4² = 52

      =52

     = 7.211

P = 6+4+7.211

     = 17.211

A= ½ x 4 x 6

     = 12

I.Q. = 4 x π x 12

               17.211²

I.Q. = 0.5091

8.

H = 7²+9² = 130

      = 130

      = 11.4018

P = 7+9+11.4018

     = 27.4018

A = ½ x 7 x 9

      = 31.5

I.Q. = 4 x π x 31.5

               27.4018²

I.Q.= 0.5271

9.  

H = 18²+14²= 520

      = 520

      = 22.8035 cm

P = 18+14+22.8035

     = 54.8035 cm

A= ½ x 14 x 18

     = 126 cm²

I.Q. = 4 x π x 126

               54.8035²

I.Q. = 0.5271

Different right-angled triangles have different I.Q. s but enlarged versions have the same I.Q. s as smaller triangles.

Isosceles Triangles

1.

P = 6+6+4

    = 16 cm

H = c²=a²+b ²

        6² = a² + 2²

         6² - 2² = a²

               32=a²

               32 = a

               5.66 = a

A of isosceles= ½ x 4 x 5.66

                             = 11.32 cm²

I.Q. = 4 x π x 11.32

                        16²

I.Q.= 0.556

I predict that an enlarged version of the isosceles triangle will have the same I.Q.

2.

P = 8+12+12

    = 32

H = c²=a²+b ²

       12² = a² + 4²

         12² - 4² = a²

              128= a²

                128 = a

       11.314 cm = a

A of isosceles= ½ x 8 x 11.314

                             = 45.256

I.Q. = 4 x π x 45.256

                       32²

I.Q. = 0.55537

The answers were the same my prediction was correct. (Do not round for more accurate answers.)

I will now look at a different isosceles and an enlarged version.

3.

P= 8+8+5

    = 21

H = c²=a²+b ²

       8² = a² + 2.5²

        8² - 2.5² = a²

             57.75= a²

            57.75 = a

7.599342077 cm = a

A= ½ x 5 x 7.599342077

     = 18.99835519

I.Q. = 4 x π x 18.99835519

                              21²

I.Q. = 0.541361388

I am now looking at an enlarged version of the above triangle; I predict that the I.Q. will be 0.54136188.

4.

P= 16+16+10

   = 42cm²

               

H = c²=a²+b ²

     16² = a² + 5²

      16² - 5² = a²

             231= a²

            231 = a

15.19868415 cm = a

A= ½ x 10 x 15.19868415

     = 75.99342075 cm ²

I.Q. = 4 x π x 75.99342075

                         42² 

I.Q. = 0.541361388

The I.Q.  s of the isosceles enlarged triangle is equal to the smaller triangle, therefore my prediction was correct.

The I.Q. s of enlarged isosceles triangles are equal. However, when they are not similar or enlarged isosceles triangles they have different I.Q. s

Equilateral Triangles

I think that all equilateral triangles will have the same I.Q. because each one is an enlargement of another; they are all similar.

1.

I am going to us “½absinCto find the area because all the angles in an equilateral triangle are 60°.

A= ½ x absinC

     = ½ x 3 x 3 x sin 60

= 3.897114317

P= 3+3+3 = 9

I.Q. = 4 x π x 3.897114317 

                              9²

I.Q. = 0.604599788

I will now look at other similar enlarged versions.

2.

A= ½ x absinC

     = ½ x 5 x 5 x sin 60

     = 10.82531755

P= 5+5+5 = 15

I.Q. = 4 x π x 10.82531755

                          15²

I.Q. = 0.604599788

3.

A= ½ x absinC

     = ½ x 11 x 11 x sin 60

     = 52.39453693

P= 11+11+11

   = 33

I.Q. = 4 x π x 52.39453693

                                33²

I.Q. = 0.604599788

The I.Q. for each triangle was 0.604599788. They were all equal.

I will now construct a table comparing all of the isoperimetric quotients I have found. I will use this to try and discover a pattern between the I.Q. s.

Join now!

Table of Results for the I.Q. s of Triangles

All the I.Q. s are under 1; they are 0. numbers.

From the table, equilateral triangles have the largest I.Q. s. This could possibly be because it is a regular plane shape with equal sides and angles.

Quadrilaterals

I am now going to study quadrilaterals and their enlargements. I think that enlargements will give the same I.Q. s. I think also that a square will give the largest I.Q. because it is a regular plane shape; this is my prediction. I will look at the following:

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