Shapes and Their Areas

Shapes and Their Areas

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Shapes and Their Areas

The objective of this coursework is to find out which shapes have the
biggest area. The perimeter must be 1000m, and the shapes can be
regular or irregular.

First of all I will experiment with different rectangles, the
different triangles, then pentagons. Then I will experiment with more
regular shapes (or whatever type of shape has the largest area) to see
the effect on area changing the number of sides has. I predict that
the largest shape will be a regular circle, and the more sides a shape
has and the more regular it is, the larger its area. (Taking a circle
as having infinite straight sides, not one side).

After I have experimented I will try to prove everything using
algebra. I will try and develop a formula to work out the area of any
polygon.

Rectangles

When I looked at the spreadsheet of rectangle areas I could instantly
see that the more regular the shape the larger the area.

However I also noticed that if you turned the graph of for this
spreadsheet upside down you would have a y=xsquared graph, with the
250x250 value being where the y- axis would be.

This means that the area of the values on either side of the square
have a square difference from the area of the square. This is because
if you "move" some of the perimeter (d) from length to with, (i.e.
decrease one dimension and increase the other) the perimeter has not
changed, but the equation for working out the area has.

It changes from

(250)(250) =250 squared

to

(250-d)(250+d) =250 squared - d squared.

So the area difference between a rectangle and a square of the same
perimeter is the difference from one of the squares sides and one of
the rectangles sides, squared.

Because all "real" square numbers are positive, the square will always
have the larger area.

It is very likely that this rule is the same for any shape but I must

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check to be sure. Now I will experiment with some triangles.

Triangles

Have written two formulae for working out the area of triangles; one
for working out any triangle, the other for working out the area of
any Isosceles.

I will explain how I worked them out later, but for the moment, I am
going to use them for this coursework.

First I will experiment with Isosceles, then scalene. I will make 2
tables.

I predict that with the Isosceles the one with the biggest are will be
base 333 because as that is (roughly) one third of 1,000 that will
make the shape equilateral.

On Excel; =0.25*sqrt(((1000*b)^2)-(2000*(b^3)))

Triangles

As you can see from my chart, the Equilateral triangle had the largest
area, and the more equal the sides, the greater the area. As with the
square, I will prove why this is so using algebra and it will be
included at the end of my coursework.

I also noticed the area was considerably less than that of the square,
so my prediction of number of sides and area is so far correct. Now I
will experiment with scalene triangles. I will use my other triangle
formula;

On Excel; =sqrt(500(500-a2)*(500-b2)*(500-c2))

So I don't get more values than I need, I will go up in 50's, and

a ≤ b ≤ c

Triangles

From the spreadsheet we can see that the larger side a and the more
equal sides b and c (i.e. the further down the table) the larger the
area. (There are some overlaps where side a changes and the area gets
less; this is because side b and side c are less equal than the
previous value.)

Now that I have shown that triangles have are larger area when they
are equilateral. It is obvious that the same is true for any shape.
Now I will try and develop a formula for working out the area of any
shape, then see if I was right in saying that the area increases as
the number of sides is increased.

After I have done that, I will show how I worked out some of my
formulae, and why equilateral triangles are the best.

Pentagons

Formular for any shape

My formula for working out the area of any shape is

On Excel; =(B2^2)/(4*A2*(TAN(RADIANS(180/A2))))

A2= Sides, B2= Perimeter

Formula for any Shape

To prove that as the number of sides increases so does the area, I
must prove that the denominator of pxp goes down as the number of
sides is increased.

The denominator is;

As 4 is a constant, I must show that

Decreases as S is increased. I will try different values of S and see
if the value that pxp is divided by goes down.

(I predict that it will)

Yes it does go down.

Developing my Isosceles Formula

Developing my General Formula

Proving Equilateral triangles have the largest area

Circles

Because a circle has an infinite number of straight sides making up
its curve, if my formula is true then the circle will have the largest
area. As the number of sides goes the gap between the shape and a
circle of the same perimeter will get smaller, but no-matter how much
the number of sides is increased it will never equal or surpass the
area of the circle.

The circles area can be used as a boundary; the largest area possible
from a straight-sided shape: Circle - m where "m" is infinitely small.

I will now work out the area of the circle and make another
spreadsheet comparing it with the other values.

Radians

When I was working in excel I noticed that the angles were in radians,
so I decided to include some information about them in my coursework
because I had finished and was bored.

Graphs

Now that I have finished the main part of the coursework I have two
more things to do; some tests, which will be done by drawing out
shapes, and some graphs. I will do 3 graphs including the one on this
page, one showing different polygons area their area, one showing
different rectangles and their area, and one showing different
isosceles triangles and their area.

Graph to show area relationships

between different polygons and a circle.

(p=1,000)

(Circle)

Area

Sides

Graph to show areas of rectangles

(p=1,000)

Area

Side a

Graph to show area of isosceles

P=1,000

Area

Base

Testing

Now will do some tests on the following shapes; hexagon, heptagon, and
octagon (Triangle, square and pentagon already covered). I will
firstly work out the area to see if it matches the area generated by
my general formula, then I will do a few "house" style shapes with the
same number of sides and compare area. (I predict that the regular
shape will be smaller.)

Hexagon

Heptagon

Octagon
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