Shapes and Their Areas

Shapes and Their Areas

Length: 867 words (2.5 double-spaced pages)

Rating: Excellent

Open Document

Essay Preview

More ↓
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


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


(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

How to Cite this Page

MLA Citation:
"Shapes and Their Areas." 16 Jul 2018

Need Writing Help?

Get feedback on grammar, clarity, concision and logic instantly.

Check your paper »

How Human Geography Shapes Society and Social Processes Essay

- “Human geography is not just about describing the spatial manifestations of economy and society; it is about explaining how space is transformed and shapes societies and social processes” (Daniels, et al., 2005). Discuss, drawing on at least two substantive areas of Economic Geography. Human Geography is a ‘major field of geography that is centrally concerned with the ways in which place, space and environment are both the condition and in part the consequence of human activities’ (Gregory et al....   [tags: economic, consumption, globalization]

Research Papers
1613 words (4.6 pages)

The Nature of Elections in Rural Areas of China Essay

- The article explains the nature of elections in rural areas of China and what are some aspects that affect voter turnout by citizens in the villages. The decision to vote was mainly affected by the newly available elections in Chinese villages. The Chinese peasants were faced with a particular system, elections and voting systems, in which they had no vague understanding of and with little to no experience. The article further explains that the introduction of elections in the villages provoked Chinese peasants to question what exactly was the intentions of the government through elections as well as how could they benefit from the elections....   [tags: voter turnout, decision]

Research Papers
912 words (2.6 pages)

Essay on Computer Technology Shapes the Future of Education

- Computer Technology Shapes the Future of Education The accessibility of computer education is stratified between the classes. The higher socio-economic schools, homes, and businesses are able to receive computer education and have daily contact with computers. However, in the lower socio-economic areas the computer usage is very low. To break down the divide many school districts and corporations have begun computer education programs, in lower-socio-economic areas. These advancements can assist the students to reach higher levels of education, which can lead to higher level careers....   [tags: Teaching Education Essays]

Free Essays
710 words (2 pages)

Vocabulary Shapes Knowledge Essay

- Vocabulary is a tool of communication that is useful and limiting at the same time. Defined as a collection of symbols and words used by a particular person or group, vocabulary assists in giving instructions, expressing feelings, and sharing knowledge. It has the capability of showing what the user both does and does not know. Furthermore, knowledge of vocabulary limits the reader’s or listener’s ability to understand information. First, in evaluating vocabulary’s effects on one’s knowledge, the question of whether knowledge can exist without vocabulary arises....   [tags: Education]

Research Papers
1586 words (4.5 pages)

Family Systems Come in All Shapes and Sizes Essay

- ... Our readings from Development Across the Lifespan outline factors that play a role in development such as a child’s nutrition, their environment, environmental factors such as culture, language development, and their sense of self. All of these factors are things in which a family is expected to provide to a child in order to promote healthy development. If a child is lacking something such as nutrition, then the child could not grow properly. For example, our text states, “Because of the rate of growth during this period is slower then during infancy, preschoolers need less food to maintain their growth” (Feldman, 2014)....   [tags: children, nourishment, development]

Research Papers
847 words (2.4 pages)

The Significance of Forests Essay

- Introduction Forests can shape our landscape; they can provide immeasurable appeal to rural and peri-urban areas, defining the inherent landscape character. Furthermore, the significance of design interpretation can facilitate management objectives and outcomes; particularly in proximity to centers of population. Wherein, the community has an affinity, vested interest, general interest or typically a fear of change. Historic landscape design: The major influence on the forest landscape was the Forestry Act 1919, and creation of the Forestry Commission (FC); there remit was to create a strategic timber reserve demanded by the war effort....   [tags: Landscapes, Rural, Peri-Urban Areas]

Research Papers
974 words (2.8 pages)

Essay on Living in Diverse Areas is Beneficial with Social Interactions

- In American Diversity is one of the biggest issues in America. America is a home to all, such that no one group can call itself more “American” than another. Even the America is very diverse, not everyone has the privilege in living in a cultural/racial diverse area. With this being said every person is not use to interactions with people much different from their own race. After reviewing many articles and books I have gained enough knowledge on my topic: prove why living in diverse area is not only just beneficial with social interactions, but with your self-esteem, and just life in general....   [tags: culture, ethnicity, diversity, america]

Research Papers
965 words (2.8 pages)

Economic Development Because of Rural Areas Migration Essay

- Urban rural migration refers to as the change of people from one geographical area to another, whether permanently or temporarily. The reasons as to why people make migrate usually differ from ones personal level. Migration, therefore, has socio- economic, demographic and educational characteristics that affect a particular individual or family. If development was referred as having access to available resources, we see that whether it is access to clean water, income, health care, good and clean environment, or education then rural areas would not be at par....   [tags: urban areas, rural migration, agriculture]

Research Papers
1277 words (3.6 pages)

Essay about Investigation of Shapes

- Investigation of Shapes I am doing an investigation to look at shapes made up of other shapes (starting with triangles, then going on squares and hexagons. I will try to find the relationship between the perimeter (in cm), dots enclosed and the amount of shapes (i.e. triangles etc.) used to make a shape. From this, I will try to find a formula linking P (perimeter), D (dots enclosed) and T (number of triangles used to make a shape). Later on in this investigation T will be substituted for Q (squares) and H (hexagons) used to make a shape....   [tags: Papers]

Free Essays
3296 words (9.4 pages)

Essay on Shapes Investigation

- Shapes Investigation doing an investigation to look at shapes made up of other shapes (starting with triangles, then going on squares and hexagons. I will try to find the relationship between the perimeter (in cm), dots enclosed and the amount of shapes (i.e. triangles etc.) used to make a shape. From this, I will try to find a formula linking P (perimeter), D (dots enclosed) and T (number of triangles used to make a shape). Later on in this investigation T will be substituted for Q (squares) and H (hexagons) used to make a shape....   [tags: Papers]

Free Essays
3299 words (9.4 pages)

Related Searches

check to be sure. Now I will experiment with some 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

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)))


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

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


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.


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


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.


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.


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.





Graph to show areas of rectangles



Side a

Graph to show area of isosceles





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.)



Return to