# 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:

- 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 = 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

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

This student written piece of work is one of many that can be found in our GCSE Fencing Problem section.

## Found what you're looking for?

- Start learning 29% faster today
- 150,000+ documents available
- Just £6.99 a month