• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Perfect Shapes

Extracts from this document...

Introduction

Perfect Shapes A perfect shape is said to be one that has the same area and perimeter. (1) Find the area and perimeter of the rectangle below. (2) Is the rectangle perfect? (3) Investigate perfect rectangles (4) Extend your investigation into other shapes. Perfect Shapes A perfect shape is defined to be one in which its perimeter is equal to its area. Rectangles I will begin by assuming that two of the sides of the rectangle are of length 5: A 5 by 1 rectangle is therefore not perfect and we need to investigate others: A 5 by 2 rectangle is not perfect either. I will tabulate further results in my quest to find a perfect 5 by something rectangle: Rectangle Length Width Area Perimeter Perimeter - Area 5 1 5 12 7 5 2 10 14 4 5 3 15 16 1 5 4 20 18 -2 5 5 25 20 -5 5 6 30 22 -8 5 7 35 24 -11 5 8 40 26 -14 5 9 45 28 -17 5 10 50 30 -20 Table 1 Looking at Table 1 it can be seen that no perfect shape has been found. ...read more.

Middle

7 28 22 -6 4 8 32 24 -8 4 9 36 26 -10 4 10 40 28 -12 This is interesting in that it shows that a 4 by something perfect rectangle is a square. Let us consider a general square to see if any other perfect square exists: A graph to illustrate this relationship might be useful: Graph 1 We can see from Graph 1 that in two places: and . The zero solution makes no sense but this confirms that if a square is 4 by 4 then it is perfect. As there is only one non-zero crossing point it also illustrates that there is only one perfect square. Let us return to the algebra briefly: Which confirms the result from the graph and proves that a 4 by 4 square is the only one that exits. It might be interesting to see if this pattern is true for other regular polygons, is there only one perfect equilateral triangle? Is there only one perfect equilateral pentagon? These questions I will attempt to address later. So far in this investigation the length of a rectangle has been fixed, it will be useful to relax this condition and consider the more general case of a perfect rectangle: Suppose that the length and ...read more.

Conclusion

#NUM! 25 #NUM! B C D E F G 2 y x h A P 3 Length of the equal sides Length of the base Perpendicular height Area of Triangle Perimeter of triangle Perimeter -Area 4 7 1 6.98 3.49 15 11.51 5 7 2 6.93 6.93 16 9.07 6 7 3 6.84 10.26 17 6.74 7 7 4 6.71 13.42 18 4.58 8 7 5 6.54 16.35 19 2.65 9 7 6 6.32 18.97 20 1.03 10 7 7 6.06 21.22 21 -0.22 11 7 8 5.74 22.98 22 -0.98 12 7 9 5.36 24.13 23 -1.13 13 7 10 4.90 24.49 24 -0.49 14 7 11 4.33 23.82 25 1.18 15 7 12 3.61 21.63 26 4.37 16 7 13 2.60 16.89 27 -10.11 Note the lack of sign change in the Area - Perimeter column suggesting that perfect 4, 4, x and 6, 6, x isosceles triangles do not exist. There are two sign changes in the 7, 7, x table and so it seems that two perfect isosceles triangles exist with these dimensions. Let me investigate this further: To be completed... let me know if you would like a copy of the completed version.......... Page 1 of 1 ...read more.

The above preview is unformatted text

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

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Fencing Problem essays

  1. Investigating different shapes of gutters.

    9 You can see that for the multiples of three the angle, which gives the best area, is 30? and the best value for x is 1/3 of the Length (L). I notice that the maximum value on the two-way table is surrounded by symmetry as you can see more clearly on the following sheets.

  2. Impacts of tourism Positive and Negative effects in Castleton

    it is away from the main roads in the village and only very small amounts of traffic passes through the area. There is very little pollution in the are because of the lack of traffic. There is a large tree and soime other smaller trees in the area but apart

  1. GCSE Maths Coursework Growing Shapes

    The pattern for 3D shapes is as follows. I drew the shapes using individual cubes which I found on AutoShapes in Word. Area/Number of Cubes Pattern no.

  2. Investigate different shapes of guttering for newly built houses.

    A = 30x�-2x� Differentiating with respect to x dA dx = 30 - 4x� If the area is to be a maximum then: dA dx = 0 30 - 4x = 0 4x = 30 x

  1. Geography Investigation: Residential Areas

    However, we can see one thing and that is that the highest penalty point is for Sarum Hill which is located in the centre of town and with the lowest score of The Beaches, located in the district of Hatch Warren in Basingstoke.

  2. In my investigation I am going to work on different shapes with a perimeter ...

    After calculating the area of a regular hexagon I can say that the area of a regular 6-sided shape with a perimeter of 1000 meters is greater than the area of a 5-sided shape with the same perimeter. 72169 > 68819 The next step in my investigation is to get the area of an eight-sided shape (Octagon)

  1. I have been asked to find out the isoperimetric quotients of plane shapes using ...

    = 4? x Tan54 x 1.25?2 25?� = 4? x Tan54 20 = ? x Tan54 5 = 0.864806206 I.Q. = 0.86 to 2d.p Therefore, the I.Q of a regular pentagon is always 0.86 to 2d.p. Now I will find the I.Q of a regular 6-sided shape and see if it has

  2. To investigate the areas of different shapes when they are joined together on square ...

    Algebra notations: - a=Area do=dots outside di=dots inside Shapes with 1 dot inside: - Shape No. of dots inside No of dots outside Area (cm�) 1 2 3 4 5 1 1 1 1 1 4 7 6 12 8 2 3.5 3 6 4 From this first table I

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work