I will use the cosine rule to work out one of the angles:
Cos A = (b2 + c2 – a2) / (2bc)
Cos A = (3402 + 326.62 – 333.32) / (2 x 340 x 326.6)
Cos A = 0.5006 so A = 59.96o
Now that I have one angle, I can work out the area:
Area = ½ x b x c x Sin A
Area = ½ x 340 x 326.6 x Sin (59.96) = 48,074.02cm2
This shows that shapes with all sides the same length have the highest formula.
Why do regular shapes have the biggest area?
All regular sided shapes have an n (n being the number of sides in that shape) number of equilateral triangles. Therefore, it is important to realise why an equilateral triangle has the biggest area compared to any other type of triangle. However, it is easier to see why a square has a higher area than a rectangle because there are no fractions involved in finding its area and squares and rectangles are related to triangles anyway.
Let us look at two 4-sided shapes – a rectangle with dimensions 10 x 6 and a square with dimensions 8 x 8 (note that if we cut the square in half, it becomes an equilateral triangle and the rectangle becomes an isosceles triangle). The area of the rectangle is 60cm2 and the area of the square is 64cm2. And if we cut these in half, clearly the equilateral triangle will have a bigger area (32cm2) than the isosceles triangle (32cm2). This proves that equilateral triangles have a bigger area but why is this? A simple diagram can explain this. In the diagram, the 10 x 6 rectangle is being multiplied step by step as well as the 8 x 8 square.
10 8
20 16
30 24
40 32
50 40
60 48
- 56
- 64
As you can clearly see, when both numbers have been multiplied by 6, the 10 x 6 rectangle has a bigger area. At this point, the difference in the areas is 60 – 48 = 12. It has become 12 because with each step, the square is losing 2cm2 as it started with 2cm less. However, the rectangle cannot be multiplied anymore but the square can be multiplied twice more giving it and extra 8 x 2 = 16cm2 area. If we add this to the 48cm2 that it already has, it becomes 64cm2, making its area bigger than the rectangle by 4cm2.
If we apply this to the area of a triangle (½ x base x height), the base of the isosceles triangle is bigger than the base of the equilateral triangle, but the height is smaller which means that the area of the equilateral triangle will overtake it as it did with the square. Here is the diagram for the triangles:
5 4
10 8
15 12
20 16
25 20
30 24
- 28
- 32
Thus, this proves why equilateral triangles have a bigger area than any other triangle. If we refer this back to our regular polygons, we see that each polygon is made up of n (n being the number of sides in that shape) number of equilateral triangles. An irregular one will not have equilateral triangles which will mean that when we finally multiply the area of each triangle by the amount of triangles in that polygon to get the area, it will be less than the regular polygons because the regular ones will be multiplied by bigger areas of triangles as they will be equilateral ones.
Pentagons
I will only look at a regular pentagon because I have already worked out that it will give me the maximum area for a pentagon.
This pentagon can be split into 5 equilateral triangles. Here is one of them:
The top angle is 72o because this area for each of the triangles is 360 / n and the base length is 200cm because the base for each of the triangles is 1000 / n. To find out the height of the triangle (so I can ultimately calculate the area), I will split the triangle into two right-angles triangles and will find the height of one of them using trigonometry. Because I will split it into two, I will need to halve the top angle and the base length. Here is the triangle:
The equation to find the height will be Tan 54 = O / 100, so the height can be worked out with the formula (Tan 54) x 100 = 137.638, which means that the area of the triangle is ½ x 50 x 137.638 = 6881.9cm2. This means that the area for the whole pentagon is 6881.9 x 10. It is multiplied by 10 because the right-angled triangle is 1/10 of the whole pentagon. This means that the area of the pentagon is 68,819cm2.
Hexagons
A hexagon can be split into six equilateral triangles. Here is one of them:
I worked out the top angle using 360 / n and the base using 1000 / 6.
Here is the triangle split into two to make a right-angled triangle so I can work out the height. All of the values have been divided by two.
The equation to find the height will be Tan 60 = O / 83.3, so the height can be worked out with the formula (Tan 60) x 83.3 = 144.3, which means that the area of the triangle is ½ x 83.3 x 144.3 = 6,014.1cm2. This means that the area for the whole pentagon is 6,014.1 x 12. It is multiplied by 12 because the right-angled triangle is 1/12 of the whole hexagon. This means that the area of the hexagon is 72,168.8cm2.
Beyond hexagons
Using a spreadsheet, and entering in the formulas that I used to calculate the other shapes, I have been able to obtain this information about the area of shapes from triangles to decagons.
Table
Graph
By looking at this graph, I can conclude that as the number of sides in a regular polygon increases, the area also increases. However, it increases at a decreasing rate. This is because with each area rise, the amount it rises by decreases each time.
General formula
Now that I know that as the amount of sides increases, the area increases, I can write a general formula on how to calculate any regular polygon with an n number of sides.
Each shape has an n number of equilateral triangles. This is what a general triangle looks like for each shape:
To find the area of this triangle, I need to split it into two to make two right-angles 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. This is what a general right-angled triangle looks like for each shape:
So to find out the height, I will need to do the equation Tan (180 – ((180/n) + 90)) x (500/n), which can be simplified to 500 x Tan ((90 x (n – 2)) / n)
Then to find the area, I need to multiply the height by 500 / n and then divide it by 2, which makes the overall equation (125000 x Tan (90 – (180 / n))) / n2.
I then need to multiply this by 2n to get the overall area of the whole polygon. Then the formula would be 2n x ((125000 x Tan (90 – (180 / n))) / n2). This can be simplified to: (250000 x n x Tan ((90 x (n – 2)) / n)) / n2, which can again be simplified to (250000 x Tan ((90 x (n – 2)) / n)) / n. We can simplify the fraction and it becomes:
(250000 x Tan (90 – (180/n))) / n
To check the equation, I will test it on the area of a square. As I already know that the area is 62,500cm2, if the equation is right, it will prove that it is true.
Area = (250000 x Tan (90 – (180/4))) / 4 = 62,500
Just to double check, I will do the formula on a decagon, which has an area of 76942cm2.
Area = (250000 x Tan (90 – (180/10))) / 10 = 76,942
I am now definitely sure that my formula is right.
Finding the shape with the biggest area
Now that I have a general formula, I can now check which shape has the biggest are. The appendix shows how the area increases until 35000 sides. It is increasing but the rate at which it is increasing is decreasing. However, I will never find a shape with the maximum area unless I look at a shape with an unlimited amount of sides. This shape is a circle and I will try to work out the area of a circle with a circumference of 1000m. However, I will need to know the radius of the circle before I can calculate its area. I can calculate the radius by rearranging the formula for the circumference:
Circumference = 2πr
1000 = 2πr
1000 / 2π = r
159.1549431 = r
Now I can calculate the area:
Area = πr2
Area = 79,577.4715459477cm2.
This shows that the shape with the maximum is with a perimeter (or circumference) of 1000m is a circle. I firstly worked out that as the number of sides increased, the area increased as well but at a decreasing rate. The reason that I realised that a circle would give the maximum area was because as the number of sides increased, it would at one point reach an infinite number, which would be a circle. So therefore, a circle must have the maximum area because it has the most number of sides from any other shape.
Why a circle has the maximum area
As we can see from the two pictures above, the triangle and octagon fit into the circle easily. However, there are many gaps left between the shape and the circle. This represents the extra are the circle has and answers why a circle has the maximum area.
Conclusion
Ultimately, I would recommend the farmer build a fence in the shape of a circle, with a circumference of 1000m and a maximum area of 79577.4715459477cm2. This will give her the maximum area with the length of fence that she has.
