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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.

B

A                  C

M

In triangle ABC:

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 =  s

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

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                   B

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

E                                 B

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 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

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

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