The point of this was to see which rectangle has the biggest area and whether I could see any relationships. The following table shows the area of all the rectangles in order of smallest to largest. I have also calculated the difference in metres between the length and width.
In this table I have found that as the area increases the difference between the width and the length decreases until it is zero. When the difference becomes zero the maximum area of the shape is found. In this case the width is 250 metres and the length is 250 metres, therefore this shape is a square.
PREDICTION: If I increase the length by 1 and decrease the width by 1 the area will be less than 62500m² (250m x 250m) but more than 62400m² (240m x 260m).
area = length x width
a = l x w
a = 251 x 249
a = 62499m²
My theory was correct because I predicted that the area would be between 62400 and 62500 and the calculation shows that it was.
Investigating Triangles
For my investigation into triangles I will repeat the procedure that I used for the rectangles. I will try isosceles, scalene, equilateral and right-angled triangles first and find the area I will use Heros Theorem. Here is his theorem.
I will need to use to find the area for a right-angled triangle.
PREDICTION: I think that the triangle with the biggest area will be an equilateral triangle because the square had the largest area and its sides are equal.
Now I will test the various triangles to find which will have the biggest area: I will put the answers to 2 decimal places if it is necessary.
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Isosceles
area = 38729 m²
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Scalene
area = 33541 m²
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Equilateral
area = 48160 m²
4. Right-angled
area = 41625 m²
Again the shape with the largest area is the one where all the sides are the same length. Therefore my prediction was correct.
Investigating the relationship between the number of sides and the area
I have now proved that regular shapes have the largest area. Now I will try to find a relationship between regular shapes with up to ten sides.
Pentagons
I will start by drawing and finding the area of a regular pentagon and giving the answer to two decimal places. I will find the area by using the SIN rule.
This is the SIN Rule
I will first split the shape into ten right-angled triangles, then find the angles, then height. By then I will I will have enough information to use the formula:
All I need to do then is multiply by ten then I will have the area for the pentagon.
c = 137.64m This is the height of the right-angled triangle in the regular pentagon.
The area of the right-angled triangle can now be calculated.
= = 6882.5m²
Therefore the area of the regular pentagon is 6882.5 x 10 = 68825m²
Hexagons
I will split the shape into twelve right-angled triangles, then find the angles, then height. By then I will I will have enough information to use the formula:
All I need to do then is multiply by twelve then I will have the area for the hexagon.
c = 144.33m This is the height of the right-angled triangle in the regular hexagon.
The area of the right-angled triangle can now be calculated.
= = 6013.51m²
Therefore the area of the regular hexagon is 6013.51 x 12 = 72162.11m²
PREDICTION: I think that as the number of sides on the shape increases then the area of the shape will increase.
Heptagons
I will split the shape into fourteen right-angled triangles, then find the angles, then height. By then I will I will have enough information to use the formula:
All I need to do then is multiply by fourteen then I will have the area for the heptagon.
c = 148.33m This is the height of the right-angled triangle in the regular heptagon.
The area of the right-angled triangle can now be calculated.
= = 5296.86m²
Therefore the area of the regular heptagon is 5296.86 x 14 = 74156.04m²
Octagons
I will split the shape into sixteen right-angled triangles, then find the angles, then height. By then I will I will have enough information to use the formula:
All I need to do then is multiply by sixteen then I will have the area for the octagon.
c = 150.88m This is the height of the right-angled triangle in the regular octagon.
The area of the right-angled triangle can now be calculated.
= = 4715m²
Therefore the area of the regular octagon is 4715 x 16 = 75440m²
Nonagons
I will split the shape into eighteen right-angled triangles, then find the angles, then height. By then I will I will have enough information to use the formula:
All I need to do then is multiply by eighteen then I will have the area for the nonagon.
c = 152.62m This is the height of the right-angled triangle in the regular nonagon.
The area of the right-angled triangle can now be calculated.
= = 4239.02m²
Therefore the area of the regular nonagon is 4239.02 x 18 = 76302.36m²
Decagons
I will split the shape into twenty right-angled triangles, then find the angles, then height. By then I will I will have enough information to use the formula:
All I need to do then is multiply by twenty then I will have the area for the decagon.
c = 153.88m This is the height of the right-angled triangle in the regular decagon.
The area of the right-angled triangle can now be calculated.
= = 3847²
Therefore the area of the regular decagon is 3847 x 20 = 76940m²
Results Table
Conclusion
I have found out from my studies that as the number of sides increases the area will increase. This means that the shape with the biggest area will be a circle as this has so many sides that it becomes round.
I have found that the following formula will calculate the area of a shape with any number of sides with a total length of 1000 metres.
number of sides = n
total length = 1000
length of sides =
I had to split the shape up into right-angled triangles and found that there was a relationship between the number of sides on the shape and the number of right-angles in the shape.
number of right-angles = 2n
The area of a right-angled triangle is
The base (b) is half of the length of the sides
There for the base (b) = x =
The height (h) is calculated by
Where c = height (h)
Angle B is calculated by dividing the number of degrees in a circle (360) by the number of right-angled triangles in the shape.
Angle B =
Angle C = 180 – A - B
Angle C = 180 – 90 -
I can now substitute the information I have gained into the formula to calculate the height (h).
I now have enough information to use the formula to calculate the area of a right-angled triangle.
area =
I now have a formula that will calculate the area the right-angled triangle. To find the area of the shape all I need to do is multiply this by the number of right-angled triangles in the shape.
Therefore the area of a shape with n sides can be calculated using the formula
I have produce a graph using microsoft excel which shows the results of the formula for shapes with between 20 and 1000 sides.
