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As I am using trigonometry to find the height I have to find out which one to choose out of: SOH
CAH TOA. It looks like the greatest area will be when all the sides are exactly equal just like it was
for the four-sided shapes. I then tried to find the area of polygons with more sides using my formula
to see if the area was bigger than my other shapes. To work out the maximum area I am going to take
out a triangle from the octagon, work out its height from doing trigonometry and work out the area
for the triangle. This means that only if I was to use a circle that has no side’s will I get the maximum
value for 1000m of fencing. I am only going to use isosceles triangles because if I know the base I
can work out the other 2 lengths because they are the same. I shall plot my results onto a graph to see
the correlation between them. I have gone down by taking 10m off the base (x) every time. Now I
am going to look at other four-sided shapes to work out that the square has the biggest area in the
four-sided shape family. This proved that I was right as the decagon had a bigger area than the
equalateral triangle. This is the same pattern that occurred with the rectangles occurred with the
triangles as well. I want to investigate these shapes because I think that after going through the
rectangles and triangles and investigating them and their sides I have decided to do theses shapes as
all of them have different number of sides. However, there are many gaps left between the shape and
the circle. I have used this formula to work out the area when the base is different heights. This
means the bigger the length and the bigger the height, the bigger the area. And if we cut these in half,
clearly the equilateral triangle will have a bigger area (32cm 2 ) than the isosceles triangle (32cm 2 ). I
shall divide the shape into triangles, which shall give me nine triangles. This is in contrast to a,
where the equal sides are. This represents the extra are the circle has and answers why a circle has
the maximum area. If the base is 200m long then I can subtract that from 1000 and divide it by two.
Tan 180 10 50 50 Next is to work out the two brackets. 0.324919696 50 153.884177 N ext work OR
50 x 153.884177 this brackets. 7694.208848 However, this is not the final bit of the formula, because
there was 10 triangles in the decagon so we have to times our answer (7694.208848) by 10 to get the
area of the whole decagon. According to the table and the graph, the rectangle with a base of 250m
has the greatest area. GnJr27sHS from GnJr27sHS student GnJr27sHS central GnJr27sHS co
GnJr27sHS uk. The 'n' number of sides stands for 'any' number of sides. (Not drawn to scale) Here is
a heptagon that I am going to use as an example. 1000m n 'n' being the number of sides. These
polygons that have been discovered are all regular. As I have found the radius I can work out the
area of the circle. This will require me to test all shapes and see which shape gives the biggest area. In
this investigation I am going to using trigonometry to find the area of the whole shape. For the
Hexagon the maximum area I got was 72192m. I also know that each side is 200m long, so the base
of the triangle will be 100m.
This means I can work out both the other angles by subtracting 72 from 180 and dividing the answer
by 2 (as it is an isosceles triangle, two of the angles are equal). So by considering using regular
polygons it would produce the maximum area. It shows that when the sides of the polygon decrease,
the area decreases. To do this I had to find the formula for an n sided polygon. Below are 2
rectangles showing how different shapes with the same perimeter can have different areas. I shall
calculate the area of one triangle then multiply the figure that will be obtained by the number of
triangles present within the hexagon. The area of one of these triangles is found and then multiplied
by 5 to get the overall area of the pentagon. Why does an equilateral triangle have a larger area than
an icosoles triangle. This is the shape that I shall be examining during this segment of my
coursework. To prove this I need to experiment on more time with the next polygon. A hexagon.
After doing my investigation I went over the maximum areas for each family of shapes and
compared them. I need it to be a right-angle triangle because I can then use trigonometry on it to find
out what the height is. I have chosen to increase the length and decrease the width by 50m.
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. This means The closer the sides are
in the ratio of 1:1, the bigger the area. I have worked out a formula to find the area of an any sided
polygon. This can be found by excluding 180 0 from the exterior angle. To work out the area for the
triangle I am going to half the base and multiply it with the height. The following is the shape that I
shall be investigating. If the triangle were folded on any axes of symmetry, it would produce two
identical right-angled triangles. This can be found by excluding 180 0 from the exterior angle. In the
diagram, the 10 x 6 rectangle is being multiplied step by step as well as the 8 x 8 square. The only
shape that has infinite lines of symmetry is the circle. Because every isosceles triangle can be split
into 2 right-angled triangles, I can work out the area of the triangle, using trigonometry. I shall half
the interior angle to find this angle. As I have found the radius I can work out the area of the circle.
But to prove that the above triangle is a right-angled triangle I shall use the Pythagoras’s theorem.
The trigonometry function TAN shall be used to find the height, which can also be known as length
of opposite. I went up by 0.1 for the width and I went down 0.1 for the length so that width and
length add up to 1000 meters. I have constructed a formula linking all three sides in and icosoles
triangle.
The perimeter of the pentagon will once again be 1000m. This means I can work out both the other
angles by subtracting 72 from 180 and dividing the answer by 2 (as it is an isosceles triangle, two of
the angles are equal). However, there are many gaps left between the shape and the circle. This is the
combination of two shapes, a rectangle and a trapezium. To calculate the area of this shape I shall
firstly need to find the radius of the shape. I shall look at different types of sizes and shapes of
fences to obtain the biggest area. I went about it in the same way as the triangle and assembled a
table. This proves that equilateral triangles have a bigger area but why is this. I have shown the
results below in a tabulated form. To find the length of the base segment I would divide 1000 by the
number of sides. To make the height bigger you have to sacrifice the length. Because I am increasing
the sides by large amounts and they are not changing I am going to see what the result is for a circle.
Therefore I shall have to half the triangle that has been shown above, to what has been shown below.
This is the same pattern that occurred with the rectangles occurred with the triangles as well.
Because I am increasing the sides by large amounts and they are not changing I am going to see
what the result is for a circle. This shape is also called a square or a regular quadrilateral. To work out
the maximum area I am going to take out a triangle from the polygon, work out its height from doing
trigonometry and work out the area for the triangle. The following are the steps that have been taken
into consideration to find the height of the trapezium above. A shape with more sided will have a
larger area and conclusively the shape with the most area under the restriction of thousand meter
perimeter is the circle. So to find out the maximum area for a farmer with perimeter of 1000m I will
need to look at the regular polygon with as many sides as possible. 100 sides would begin to look
like a circle and as the number of sides increased it would get more and more like a circle so I will
test the area of a circle to see if it has a bigger area than I have already found. The area of a
parallelogram can be found by using the formula (Base multiplied by Height equals area). I have
shown the shape that I shall be investigating during this part of the coursework below. However,
when you try and tilt the height of the square to fit the parallelogram the height will go down so it
would still be the biggest area. The radium shall then be substitutes into the formula of finding out
the area. I shall be taking the same procedures that I took before. Also the equation that I used is a
quadratic equation, and all quadratic equations form parabolas. As the number of sides of the
polygons goes up to infinite, it will looks like a circle, so I assume that circle will have the largest area
with 1000m perimeter, that's why I will then investigate on a circle. I will divide the 12 Sided Shape
into 3 rectangles. This proved that I was right as the decagon had a bigger area than the equalateral
triangle. I shall be taking the same procedures that I took before.
The regular triangle seems to have the largest area out of all the areas but to make sure I am going to
find out the area for values just around 333. The three lengths have to equal 1000 metres so you
therefore have to divide 1000 by 3. I have shown the results below in a tabulated form. This means
The closer the sides are in the ratio of 1:1, the bigger the area. From looking at these quadrilaterals it
shows that the square has the biggest area so far with the perimeter 1000m. Usually, however, the
means the length of the circle, and the interior of the circle is called a. I have shown the formula to
find the area of a trapezium below. The other two sides were also done in two separate columns,
which went up in 5. I shall divide the shape into triangles, which shall give me nine triangles. I
decided to produce a table, and then a graph, showing varying lengths and breadths in a logical
manner, decreasing one by 50, and increasing the other by 50. I shall half the interior angle to find
this angle. To find the length of the base segment I would divide 1000 by the number of sides. This
has 12 sides. I have shown the illustration of the shape that I shall be investigating below. I shall now
find the height of the triangle shown below. I also found that the heptagon gives a higher maximum
area than the pentagon and hexagon. After I had finished the regular polygons I guessed that because
it was the only shape I had not done the shape with the most maximum area will be a circle. This
proves that equilateral triangles have a bigger area but why is this. This shows that you do not have to
investigate Kites and Rhombuses as they all fall under the categories of the square. An isosceles
triangle also has two equal internal angles (namely, the angles where each of the equal sides meets
the third side). I shall not be using the area of a triangle formula during this part of the investigation.
To work out the area for the triangle I am going to half the base and multiply it with the height. The
biggest rectangle was with length 450cm with an area of 202500cm?, the biggest triangle was of base
600cm with an area of 155884.5727cm?. Now that I have found the area of a 4 sided shape and a 3
sided shape, I will find out the area of other regular polygons, with up to 50 sides. To make sure that
my results are as accurate as they can be, I’m going to draw another table but in decimals and I will
be looking between 249.8 and 250.2. When this number is plugged into the formula we see that this
is actually an equilateral triangle. Therefore in a rectangle of 100m x 400m, there are two other that
are 100m long and 2 sides next tot hem that are opposite each other that are 400m long. I will refine
my search even further to find the exact maximum area for the triangles. I shall find the length of
side of the triangle by dividing the total perimeter by the number of sides that the shape consists of.
This shape is also called a square, or a regular quadrilateral. To calculate the area of this shape I shall
firstly need to find the radius of the shape. I will also have to keep in mind that the perimeter has to
be 1000m.
Also when I reached a 40-sided shape the area change in a very small amount. An equilateral triangle
has three axes of symmetry. I shall use the same method as I did with the previous shape. After a
couple of workings out of some triangles I came to discover another faster and easer way to calculate
a triangle and this formula is called the semi perimeter. If the base is 200m long then I can subtract
that. To work out the area for the triangle I am going to half the base and multiply it with the height.
As you can see from my graph (on the previous page) the line goes up then curves round and
becomes a constant line until we reach the area of a circle, where it stops as I found out because a
circle has the maximum area. I shall multiply this figure by the number of triangles present within the
pentagon. The results of the area shall also be observed and noted down on a table. So therefore, a
circle must have the maximum area because it has the most number of sides from any other shape.
She is not concerned about the shape of the plot but it must have a perimeter of 1000 m. Because
every isosceles triangle can be split into 2 equal right-angled triangles, I can work out the area of the
triangle, using trigonometry. I shall multiply this figure by the number of triangles present within the
hexagon. When I narrow my search down to rises and falls of 12.5, I obtained these results. To find
the height, as before, a perpendicular must be drawn so the new triangle is a right-angled triangle
with two unknown lengths and one known angle. I have gone down by taking 10m off the base (x)
every time. I shall divide the shape into triangles, which shall give me ten triangles. The following
are the steps that have been taken into consideration to find the height of the trapezium above. Using
a spreadsheet and formula I have created a table that shows my prediction is right. I have gone down
by taking 10m off the base every time. This will decrease the height, which means the area will be
smaller. I shall calculate the area of one triangle then multiply the figure that will be obtained by the
number of triangles present within the hexagon. I shall divide the shape into triangles, which shall
give me nine triangles. Then I can work out the area of one of the right-angled triangles using
trigonometry and then multiply it by 2 to get the area of the triangle above. Firstly I will find the area
of the triangle the followed by the rectangle that it is places on. The graph below shows that the
circle has the most area. The graph shows that the area is nearing its limit, the limit is also the area of
the circle and you cannot get any higher value then it. When you split the shape into triangle, the
more sides the shape has the smaller the angle gets in between the two equal sides but the perimeter
of these triangles increase as the shape has more sides. This means that you can work out the area if
you only have the length of one side. So if a 1000m fence is used these dimensions are the maximum
for a rectangle which is really a square.
I have learnt that a one sided shape can hold the maximum area with a perimeter of 1000cm. The
formula to find the area of a triangle has been shown below. So if a 1000m fence is used these
dimensions are the maximum for a rectangle which is really a square. The area for a quadrilateral was
found to be considerably larger than the area for a triangle. These triangles shall include equilateral
triangle, an isosceles triangle, a right-hand triangle and a scalene triangle. The following are the four
triangles that shall be investigated in part 1. From what I have written I now can show you how the 2
formulas are the same. I shall find the length of side of the triangle by dividing the total perimeter by
the number of sides that the shape consists of. I have shown the steps that shall be taken into
consideration alongside with how the stage shall be considered in calculating and besides this I have
shown the formula. I shall multiply this figure by the number of triangles present within the decagon.
I shall half the interior angle to find this angle. The following shape is an octagon and I shall be
investigating the shape during this segment of the coursework. To find the length of the base
segment I would divide 1000 by the number of sides. These will include: triangles, quadrilaterals, a
pentagon, a hexagon, a heptagon, and octagon, a nonagon and a decagon. Also when I reached a 40-
sided shape the area change in a very small amount. This means that the farmer should use a circle
for her plot of land so that she can gain the maximum area. As I am using trigonometry to find the
height I have to find out which one to choose out of: SOH CAH TOA. Out of all triangles the
equilateral triangle holds the maximum area. Regular shapes have numerous properties; they can be
split up into icosoles triangles. Below are 2 rectangles (not drawn to scale) showing how different
shapes with the same perimeter can have different areas. First I shall look at rectangles as they are
easiest to calculate. However not much space is wasted from the octagon to the cricle, also when you
have a triacontagon(30-sided shape) and you fit it in a circle there is hardly any wasted spce so this is
tell us that, as the sided of the polgyen increases there is less wasted spcae. This is because with each
area rise, the amount it rises by decreases each time. I have discovered the area of five different
quadrilaterals. I shall construct the method as I have done before. The term would be entered into a
spreadsheet program such as Microsoft Excel. I also know that each side is 200m long, so the base of
the triangle will be 100m. To calculate the area of this shape I shall firstly need to find the radius of
the shape. We can put this in a fromula and to prove that no shape has the area bigger than a circle as
we could replace n with 1000 and still the area would be smaller than the circle.