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  • Level: GCSE
  • Subject: Maths
  • Word count: 2211

In my investigation I am going to work on different shapes with a perimeter of 1000 meters. I will try to find the shape with the largest area.

Extracts from this document...

Introduction

Mohamed El Sherif        Y11A        Math Coursework

Math Coursework

In my investigation I am going to work on different shapes with a perimeter of 1000 meters. I will try to find the shape with the largest area.

First I have decided to try the simplest type of shapes, which are triangles (only 3 sides).

3 Types of triangles:

  1. Isosceles: This triangle has only two sides equal while the third is not.

                                Bimage00.pngimage01.png

image07.pngimage16.png

                          A                  Cimage07.png

                                 M

In triangle ABC:

image02.pngimage02.png

AB = BC, BM is perpendicular to AC and divides into two equal halves (AM & CM).

If AB = 400m , BC = 400m  , AC = 100m

Therefore AM = MC = 250m

By Pythagoras theorem:

BM2 = BC2 – MC2

BM2 =  4002 – 1002

BM2 = 160000 – 10000

         = 150000

BM = 387.2983346m

Area of triangle BMC = ½ x base x perpendicular height

                            = ½ x 100 x 387.2983346

                        = 19364.91673

Area of triangle ABC = 19364.91673 x 2 = 38729.83346m2

Formula for any isosceles triangle:

Base                                =         b

One of the two equal sides =  simage03.png

Height = √ (s2 – (b/2) 2)

Area of small triangle (BMC)  = ½ x (b/2) x √ (s2 – (b/2) 2)

Area of big triangle = 2 (½ x (b/2) x √ (s2 – (b/2) 2))

By simplifying this formula I get: b/2 x √ (s2 – (b/2) 2)

I will try the formula in the above example:

100/2 x √ (4002 – (100/2) 2) = 19364.91673m2

This answer is exactly what I got when using the other method. Using this formula I can now put down the area of different isosceles triangles in a table.

Table 1: Different areas of isosceles triangles:

One of the equal sides (s)

Base of triangle (b)

Area Using the formula

(b/2 x √ (s2 – (b/2) 2)

350

300

47434.1649

400

200

38729.83346

450

100

22360.67978

...read more.

Middle

By looking at the graph it is now clear that the area keeps increasing until the shape is a square and then decreases at the same rate that it has increased by.

A square: all sides are equal

  A                   Bimage11.png

  D                   C  

Area of a square= length2

                   = 2502

  = 62500

Comparing the area of a square with that of an equilateral triangle, the square has a bigger area. Therefore 4-sided regular shapes with a perimeter of 1000 meters have a bigger area than 3-sided regular shapes with the same perimeter.

62500 > 48113

The next step in my investigation is to look at the area of five-sided regular shapes (regular pentagons) with a perimeter of 1000 meters.

Pentagons: five sides

                                                                                                         A        image13.pngimage12.png

         E                                 B   image14.pngimage15.png

image19.pngimage18.pngimage17.png

image05.pngimage05.pngimage06.pngimage06.png

            D       N       C

AB = BC = CD = DE = 200meters

By dividing the pentagon ABCDE into five equal triangles I can get its area.

In triangle DMC:

MN is the perpendicular bisector of DC.

DN = CN = 200 / 2

             = 100m

Angle DMC = 360 / 5

                     = 72°

Therefore angle NMC = 72 / 2

                        = 36°

Since angle MNC = 90°

Therefore angle NCM = 180 – (90 + 36)

                         = 54°

tan = opp / adj

tan 36 = 100 / adj

adjacent = 100 / tan 36

          = 137.6387192

MN = 137.62387192 meters

Area of triangle MCN = ½ x base x height

= ½ x 100 x 137.6387192

= 6881.909602 m2

Thereforearea of triangle DMC = area of triangle MCN x 2

                                  = 6881.909602 x 2

                                  = 13763.8192 m2

Area of ABCDE = area of triangle DMCx 5

                   = 68819.096 m2

                   = 68819 (nearest d.p.)

Comparing the area of a regular 5-sided shape

...read more.

Conclusion

A Circle:

                              B

AB is the radius of the circle with perimeter 1000 meters.

Circumference  =     d

Therefore d = Circumference

               =  1000

               = 318.3098862 m

Therefore radius = d / 2

                   = 159.1549431 m

Area of a circle =    r2

=         X (159.1549431) 2

                     = 79577.47155

                 = 79577 (nearest d.p)

The area of the circle with perimeter 1000 meters is greater than the area of all the other shapes that I have tried. Even when there is 100000 sides, still the area of the circle is greater.

Using the formula to get the area of a shape with 100000 sides:

       250000

100000tan (180)

                100000

= 79577.47152

The area of a circle is equal to the area of a shape with 1000000 sides.

Using the formula to get the area of a shape with 1000000 sides:

        250000

1000000tan (180)

                1000000

= 79577.47155

A shape with sides more than 1000000 will still have the same area as a circle. It is not possible to establish a shape with this number of sides.

Using the formula to get the area of a shape with 10000000 sides:

        250000

10000000tan (180)

                10000000

= 79577.47155

This means that no other shape will have a greater area than a circle if they both have the same perimeter.

In conclusion, according to my investigation I can say that no other shape will have an area greater than that of a circle if they both have the same perimeter. Therefore the circle is the best choice for fencing the maximum area with a perimeter of 1000 meters.

Mrs. Claire Woffendon        Math        El Alsson School

...read more.

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