2x=b-a
2x=450-50 x x
h 250
200
Apply Pythagoras theorem to find height
Apply trapezium formulae to find area:
It is clear that a table I necessary as it will allow me to work out the best trapezium in the quickest manner:
The table shows that the trapezium with the maximum area, 62500m2 is achieved when a=250 and b=250. Therefore this isosceles trapezium is a square.
Shown geometrically:
Transition
Giving bigger area
Parallelograms
A parallelogram is a four-sided plane figure that has two sets of opposite parallel sides. Every parallelogram is a , and more specifically a .
The area of a parallelogram can be seen as twice the area of a triangle created by one of its diagonals. The area of a parallelogram can be found by using the formula (Base multiplied by Height equals area).
My prediction of the greatest area here, is that the closer the shape gets to a square the greater its area will be. This means that when the angles inside the shape are 90° and the length and width is equal.
I first started by drawing a parallelogram and then tried to find out its area:
50m
450m
To find height
50m
h
Use trigonometry
=h
4
Apply parallelogram formula for area
It is clear that a table I necessary as it will allow me to work out the best parallelogram in the quickest manner. Because I am dealing with angles here, I will need to create many tables to test out all possible combinations of lengths, widths and angles.
The highlighted cells again show the shape that gives the highest. This shape is a square thus again proving my prediction that the square gives the biggest area so far.
Shown Geometrically
Transition
Giving biggest area
Kites
In , a kite, or deltoid, is a with two pairs of equal adjacent sides. Technically, the pairs of sides are and . This is in contrast to a , where the equal sides are .
I predict that the closer the shape gets to a square, the bigger area it will provide.
The general formula for calculating the area for a kite is:
-
If d1 and d2 are the lengths of the diagonals, then the area is
However because we don’t have the lengths of the two diagonals. We must spit the kite into two triangles as shown below:
The area of a triangle is calculated using this formula:
Since we are eventually going to double the area we get so we can derive this final formula to calculate the area of a kite.
Worked example:
a=400
b=100
C=150
It is clear that a table I necessary as it will allow me to work out the best kite in the quickest manner.
The highlighted cells again show the shape that gave the biggest area. This shape again is a square. Thus proving yet again that the square, so far, is the best shape to use.
Proving Geometrically:
Transaction
Giving bigger area
Triangles
A triangle is one of the basic of : a with three and three sides which are .
Triangles can be classified according to the relative lengths of their sides:
-
In an equilateral triangle all sides are of equal length. An equilateral triangle is also equiangular, i.e. all its internal are equal—namely, 60°; it is a
-
In an isosceles triangle at least two sides are of equal length. An isosceles triangle also has two equal internal angles (namely, the angles where each of the equal sides meets the third side). An equilateral triangle is actually also an isosceles triangle, but not all isosceles triangles are equilateral triangles
-
In a scalene triangle all sides have different lengths. The internal angles in a scalene triangle are all different.
The general formula for calculating an isosceles or equilateral triangle is half base times height or,
The formula for calculating the area of a scalene triangle is
Below a scalene triangle has been drawn and the angle measured in order to calculate the area. This triangle is not drawn to scale.
Here we realise that scalene triangles (irregular shapes) yield a smaller area than isosceles and equilateral (regular) ones. So I decided to just use regular shapes from now on.
As only one equilateral triangle can be produced with perimeter 1000, the isosceles and equilateral triangles have been grouped below in a table. The area has been given to 2 decimal places.
As the triangle gets closer to an equilateral triangle the area gets bigger. The graph, shown below, is the same shape as it was for the rectangles. However the graph is not quadratic but merely represents a similar trend line.
My prediction was correct; the equilateral triangle does have the maximum area. This is proven by the fact that sides either side of produced an area less than that of the equilateral.
Polygons
Polygons bigger than 4 sides
A polygon is a plane that is bounded by paths composed of a finite number of sequential . The straight line segments that make up the boundary of the polygon are called its edges or sides and the points where the edges meet are the polygon's vertices.
By drawing a line from every one of vertices to the centre of the pentagon, one can draw isosceles triangles. . The number of triangles formed, will always be equivalent to the number of sides of the polygon. An example is shown below in a regular pentagon with sides of 200m:
The 5 isosceles triangles formed in the in the pentagon are equal to the number of sides, 5. All 5 isosceles triangles must be identical, therefore the area of the pentagon can be found quite easily with the use of trigonometry.
Angle x is
Angle x=
Angle x=72°
The isosceles triangles can now be split down the middle which creates two identical right angles triangles. A cross section is shown below, x represents the height of the triangle:
36°
h=137.638
Now that we know the height, h, the area of the triangle can be calculated using the formula:
Area of Triangle =
Area Of Triangle = 13700m2
The area of the pentagon can now be calculated as the area of one triangle is known.
Area Of Pentagon= 13700x 5 triangles
Area Of Pentagon= 68500
Now I had to find out the area of other polygons for example hexagon and heptagon. So I decided to investigate a general formula for all polygon shapes after pentagon. This will allow me to calculate large numbers of polygons very quickly and accurately. As doing it step by step, as shown above, will raise the chances of doing mistakes in my calculations. Calculating the area of all these polygons means that I can compare and maybe spot a pattern that perhaps gives me an idea of which shape will be the best to use.
General Formula
First find the interior angle:
Then find each triangle’s base length
Apply trigonometry to find height
Adapt to full polygon
Simplify
Make h the subject
Simplify to find height of any polygon’s triangle
Apply area of triangle formula
Substitute
Simplify
Simplify further to get area of triangle in any polygon:
Area of polygon= Area of triangle n
Substitute
Simplify to get area of any polygon:
Using the general formula above I calculated the area of al polygons up to a 51 sided shape.
I then chose t plot a graph which showed me the trend between the areas and if there was a pattern that would help me decide the best shape to use.
The graph is shown on the next page:
Displayed above, is a graph comparing the areas of all regular polygons with a perimeter of 1000m. As the graph shows that as the number of sides increases, it tends to a particular value and the gradient is gradually straightening. The reason for this is that it is impossible to notice a difference in such a large number of sides.
Circle
A circle is the of all in a plane at a fixed , called the , from a fixed point, the centre. Circles are , dividing the plane into an interior and exterior. Sometimes the word circle is used to mean the interior, with the circle itself called the circumference(C). Usually, however, the means the length of the circle, and the interior of the circle is called a . An arc is any continuous portion of a circle.
A circle is also a polygon with infinite number of sides. If this is applied to the graph on the previous page, the area of a circle will be the biggest as it has infinite number of sides. Therefore I predict that the circle will have the biggest area.
The area of a circle is calculated using this formula:
The circumference of a circle is calculated using this formula:
Since there is only 1 circumference possible we will only be able to get one area.
We don’t know the radius but because the perimeter,
The formula can be rearranged to give us the radius:
Plug r into area of a circle formula:
Simplify:
The area of a circle with a circumference of a 1000m is approximately, 79577.47155m². So far the circle gave us the largest area.
This is because the circle has infinite number of sides and comparing the two formulas shows that if n was to be infinity it would have the biggest area.
Proving Geometrically
Transaction
Giving us a bigger area
The proof above shows that as number of sides tends to infinity the area of the polygon will tend to that of a circle’s. Therefore, when n is an infinite number of sides, its area will be equivalent to that of a circles and this the largest possible area for the given perimeter of a 1000m.
In conclusion; if the farmer wanted the shape with the maximum area and a perimeter of 1000m, he ought to fence his land in the shape of a circle.