200 225
Area= 200 X 300 Area= 225 X 275
= 60000 = 84375
250
250
Area= 250 X 250
= 62500
From these results, I have realised that the maximum area for a rectangle is square.
This made me realise that a square is a rectangle which is something I have learnt in this section which I didn’t know before.
Parallelograms
I have now done squares and rectangles; I have to move on to quadrilaterals. I have chosen to do Parallelograms.
To get a wide range of results the base will have a 200m length.
(1)I will have to find out the (2)The side is 300 m because
Perpendicular height and (1000- (200 X2)) divided by
Find the length of the other 2 is 300cm.
Sides which is 300m.
(3)To find the height I have
To use sin and then times by
Base to get height.
Here are some examples and the angles go in 10º:
1. SIN 10º= x
200
200 SIN 10º = x
x= 34.72963553 x 200
Area=6945.927107
2. SIN 20º= x
200
200 SIN 20º = x
x= 68.40402867 x 200
Area=13680.80573 m2
3. SIN 30º= x
200
200 SIN 30º = x
x= 100 x 200
Area=20000m2
4. SIN 40º= x
200
200 SIN 40º = x
x= 128.5575219 x 200
Area=25711.50439m2
5. SIN 50º= x
200
200 SIN 50º = x
x= 153.2088886 x 200
Area= 30641.77772
6. SIN 60º= x
200
200 SIN 60º = x
x= 173.2050808 x 200
Area=34641.01615
7. SIN 70º= x
200
200 SIN 70º = x
x= 187.9385242 x 200
Area=37587.70483
8. SIN 80º= x
200
200 SIN 80º = x
x= 196.9615506 x 200
Area=39392.31012 m2
9. SIN 90º= x
200
200 SIN 90º = x
x= 200 x 200
Area=40000 m2
10. SIN 100º= x
200
200 SIN 100º = x
x= 196.9615506 x 200
Area=39392.31012 m2
11. SIN 110º= x
200
200 SIN 110º = x
x= 187.9385242 x 200
Area=37587.70483
12. SIN 120º= x
200
200 SIN 120º = x
x= 173.2050808 x 200
Area=34641.01615
13. SIN 130º= x
200
200 SIN 130º = x
x= 153.2088886 x 200
Area= 30641.77772
14. SIN 140º= x
200
200 SIN 140º = x
x= 128.5575219 x 200
Area=25711.50439m2
15. SIN 150º= x
200
200 SIN 150º = x
x= 100 x 200
Area=20000m2
16. SIN 160º= x
200
200 SIN 160º = x
x= 68.40402867 x 200
Area=13680.80573 m2
17. SIN 170º= x
200
200 SIN 170º = x
x= 34.72963553 x 200
Area=6945.927107
18. SIN 180º= x
200
200 SIN 180º = x
x= 0 x 200
Area=0 m2
If I do this working out it is zero because 180 º would be a straight line.
The maximum area is with 90 º angle, it has 1000m perimeter and it has 40,000 m² area .
90 º
This parallelogram is similar to a trapezium because it has two right angle triangle and one is turned over.
I have learnt that when parallelogram angles increases the height increases which makes the area increase up to the value 90° when the highest height is achieved. After 90° the height starts to decrease.
TRIANGLES
Different triangles have same features so there will be same ways to work them out, these are some examples:
- Scalene Triangles: it is easier to work out with Hero’s rule.
- Isosceles triangle: it is easier to work out on with Hero’s rule.
- Equilateral triangle : it is easier to work out with Hero’s rule..
Here are some Scalene Triangles examples.
Side a: 301
Side b: 200
Side c: 499
√(500*(500-301)* (500-200)* (500-498)) =5463.515352
Side a: 201
Side b: 300
Side c: 499
√(500*(500-201)* (500-300)* (500-499)) =5468.089246
Side a: 211
Side b: 290
Side c: 499
√(500*(500-211)* (500-290)* (500-499)) =5508.629594
Side a: 251
Side b: 250
Side c: 499
√(500*(500-251)* (500-250)* (500-499)) =5578.978401
Side a: 250.5
Side b: 250.5
Side c: 499
√(500*(500-250.5)* (500-250.5)* (500-499)) =5578.989604
Here are some Isosceles Triangles examples:
Side a: 300
Side b: 300
Side c: 400
√(500*(500-300)* (500-300)* (500- 400)) =44721.35955
Side a: 400
Side b: 400
Side c: 200
√(500*(500-400)* (500-400)* (500- 200)) =38729.83346
Side a: 450
Side b: 450
Side c: 100
√(500*(500-450)* (500-450)* (500- 100)) =22360.67977
Side a: 250.5
Side b: 250.5
Side c: 499
√(500*(500-250.5)* (500-250.5)* (500-499)) =5578.989604
Here is a equilateral Triangle example:
Side a: 333.333
Side b: 333.333
Side c: 333.333
√(500*(500-333.333)* (500-333.333)* (500-333.333)) =48112.285 cm ²
TRIANGLES
I have collected information about triangles that show me what type triangle has the maximum area. I have organised all my data on a spreadsheet.
To work out the area I use Hero’s rule, Pythagoras or another formula.
Scalene
I have used Heros rule to work out the area of this Scalene triangle to ensure that the spreadsheet data gets the same results.
S=500
Area= √(500*(500-3)* (500-498)* (500-499)) =704.9822693
These are the spreadsheet data results in 100’s.
As you can see these results are not very accurate.
I now have to go in 10’s for more accurate results.
These results show me that the maximum area for a scalene triangle may be the area of an isosceles triangle.
I will go in 0.5’s just to see if my prediction is right.
My prediction was right that the maximum area for a scalene triangle is an isosceles triangle.
Isosceles a²+b²=c²
I have used Pythagoras to work out the area of the Isosceles triangle.
Example:
0.5² + b² = 499.5²
0.25 + b = 249500.25
249500.25 - 0.25 =249500
b² =249500
b = √249500
b/ height= 499.4997497
=499.49 cm
Area = 499.49 x 1
2
Area= 249.745
I have input data on a spreadsheet to find out the maximum area for an isosceles triangle.
I put my data in 100’s to begin with this would show me between which numbers have the maximum area.
The highlighted row has the biggest area of an isosceles out of all these results.
To have a more accurate result I went in 10’s.
These results show me there is a more accurate answer possible to get.
I now am going to go in 1’s
These results are starting to show me that maybe the equilateral triangle has the maximum value.
I have now decided to go in 0.111’s to see if it really is an equilateral triangle.
This shows that the equilateral triangle has the maximum area.
To see if these results are true that the equilateral triangle has the maximum area I will work it out without using spreadsheet but with a calculator.
Equilateral
There is one triangle area possible to get for a equilateral triangle with a perimeter of 1000cm².
a² + 166.6665² = 333.333²
a²+ 27777.72222= 111110.8889
_______________________
a=√111110.8889 - 27777.72222
__________
a=√3333.16667
a= 288.6748459
Area= base X height
2
Area= 333.333 X 288.674 =96224.570
Area= 96224.570
2
Area= 48112.285 cm ²
This shows the area from the spreadsheet is right even though this is a less accurate calculation.
Overall, from these triangles I have learnt that that the maximum area for a scalene triangle is an isosceles triangle and the maximum area of a isosceles triangle is the area of an equilateral triangle. Out of all triangles the equilateral triangle holds the maximum area.
Polygons
I have done some calculations of polygons with a perimeter of 1000cm this data will enable me to find out which polygon has the maximum area and it will help me to create a formula which will possibly work out the maximum area of nearly any 2d shape.
1000 = 100 cm 1000 = 83.333
10 12
360 = 30º
360 =36º 12
10
x tan (36) = 100 x tan(30) = 83.333
Tan (36) = 100 Tan(30)= 83.333
x x
x= 83.333
Tan(30)
x= 100
t an (36) x= 144.2798323
x=137.638192 x= 144.279
½ x bx h ½ x b x h
100 x 137.638= 13763.8 83.333 x 144.279= 12023.201
2 = 6881.9 2
=6011.600
Area= 6881.9 x 10 Area= 6011.6 x 12
Area= 68819 Area= 72139.2
x = 148.3236624
1000 =71.428
14 x = 148.323
360 =25.714 ½ x b x h
14
71.428 x 148.323= 10594.415
x tan 25.714 = 71.428 2
= 5297.207
tan 25.714 = 71.428
x 5297.207 x 14 = 74160.898
x = 71.428 Area = 74160.898
tan 25.714
I have worked out the area of three different polygons, but to have more accurate anwers at a faster process I have to find/create a formula.
N= number of sides
The angle of a shape is: 360
2N
The length of one side is: 1000
2N
Height of a triangle is length of a side divideded by the angle times tan.
1000 tan 360
2N 2N
= 1000
2N tan 180
N
I have found what the height is so I have to times the height with the base and divide by 2.
BASE HEIGHT
1000 X 1000 X 1 I EIGHTER DIVED BY TWO OR TIMES BY 1/2
N 2N tan 180 2
N
= 1,000,000 X N 1
2N² tan 180 X 2
N
= 1,000,000 X N = 250,000 X N
4N² tan 180 N² tan 180
N N
From the results I have got I can see that the more sides the polygon has the higher the area, So I have decided to find out the area of a circle.
2пr= 1000 cm r = radius
пr=1000 п= 3.141592654…..
2
r = 500
п
п x (500 ) ² = 79577.47155
П
This shows me that the circle has the maximum area with a perimeter of 1000cm.
My prediction was wrong that the polygon with the most sides would have the highest area
But actually the shape with only one side (circle) has the maximum area this is something I have just learnt after doing the calculations.
To conclude I have examined many shapes such as triangles (equilateral, scalene & isosceles), quadrilaterals and many polygons. I have learnt that a one sided shape can hold the maximum area with a perimeter of 1000cm.
