200m
I divide the Isosceles triangle
400 m2 - 100 m2 = Height m2
160,000 – 10,000 = 150,000
Height = square root of 150,000
Height = 387.29 to two decimal places
To calculate the area of this Isosceles triangle we calculate:
Area = ½ x base x height
Area = ½ x 200 x 387.29
Area = 38729 m2 (meter squared)
Triangle 4
I chose another size isosceles triangle to continue my investigation
375m 375m
250m
I divide the Isosceles triangle
375 m2 - 250 m2 = Height m2
140,625 – 15,625 = 125,000
Height = square root of 125,000
Height = 353.55 to two decimal places.
To calculate the area of this Isosceles triangle we calculate:
Area = ½ x base x height
Area = ½ x 250 x 353.55
Area = 44,194m2 (meter squared) to the nearest whole number.
I will now investigate the right – angle triangle. The formula needed for this part of the investigation is
- Trial and improvement.
- Area = ½ x base x height
Triangle 5
This triangle requires trial and improvement to find out the values of the perimeter of the following triangle:
284 squared + 284 squared = 4012 969m – too low
287 squared + 287 squared = 405.92. 979m – too low
290 squared + 290 squared = 4102. 990m – too low
295 squared + 295 squared = 417.22 1007m – too high
292.5 squared + 292.5 squared = 413.72 998.7m – too low
293 squared + 293 squared = 4142 1000m – perfect
We will now put the values into the triangle.
293 414
293
To calculate the area of this Right – Angle Triangle we calculate:
Area = ½ x base x height
Area = ½ x 293 x 293
Area = 42,924.5 m2 (meter squared)
I will now investigate the equilateral triangle. The formula needed for this part of the investigation is
Area = ½ x base x height
All lengths of an equilateral triangle are equal. Therefore each side will be 1000 meters divided by, which is 333.33 meters to 2 decimal places
333.33m 333.33m
333.33m
I divide the equilateral triangle to calculate the height
333.3 m2 - 166.65 m2 = Height m2
111108.89m – 27 772.23m = 83336.66 m
Height = square root of 83336.66
Height = 288.68m to nearest whole number
To calculate the area of this equilateral triangle we calculate:
Area = ½ x base x height
Area = ½ x 333.33 x 288.68m
Area = 48,113m2 (meter squared) to the nearest whole number.
Results of the Rriangles.
In the above results, I have found out that the equilateral Triangle had the largest area. I also noticed that there is a pattern. The pattern is that as the base increases and the height decreases, the area also increases.
QUADRILATERALS
I am going to investigate different quadrilaterals making sure that the perimeter is 1000 metres at all times.
The formula for the quadrilaterals:
- Perimeter = length + width + length + width.
My first shape is going to be a square:
250m
A square has 4 sides. The 4 sides of the square are all equal. Therefore that means that all the edges will have the length of 250m due to the perimeter being 1000m.
Area = length x width.
Area = 250 meters x 250 meters
Area = 62,5002 (meters squared)
The next 5 shapes are going to be different types of rectangles. They will be numbered for further investigation:
(1) 450m
50m
(2) 400m
100m
(3) 300m
200m
(4) 350m
150m
(5) 280m
220m
Results of the Quadrilaterals.
In the above results, I have found out that the Square had the largest area.
200 cm
To find out the area of a pentagon, I am going to divide the pentagon into 5 isosceles triangles.
Each length is equal to 200 cm as 1000
5
= 200
hyp hyp
h
200cm
To find out the hypotenuse, I am going to use the tangent formula
Tan = Opp
Adj
Tan 36 =100
Adj
Adj = 100
Tan (36)
=137.64(2dp)
The height = 137.64
The area of one of the five triangles that make up the pentagon is
Area = ½ x base x height
To calculate the area of this isosceles triangle we calculate:
Area = ½ x base x height
Area = ½ x 200 x 137.64m
Area = 13,764 m2 (meter squared) to the nearest whole number
Area x 5 = area of a pentagon
13,764 x 5 =68,820
Area of a regular pentagon = 68,820 m2
Now I am going to find out the area of a hexagon
I am now going to investigate circles.
CIRCLES
The formulas needed for this part of the investigation are:
Area = πr2
Perimeter = πd = 2πr
The perimeter of a circle is also known as the circumference.
I will use 3.14 as π in my calculations.
The radius is the distance from the centre point of the circle to the edge of the circle.
2r = Diameter.
To find the radius I substitute the values I have for the circumference and π into the equation for the perimeter of the circle.
Circumference = 2 x π x radius
1000 = 2 x 3.14 x radius.
1000 = D
3.14
D = 318.47 to two decimal places.
Therefore the Radius = 318.47
2
Radius =159.24 to two decimal places.
I have calculated the radius of this circle. With this I will be able to determine the area the fence will cover. I substitute the values I have for the radius and π into the equation to find the area.
Area = π x r x r
Area = 3.14 x 159.24 x 159.24
Area = 79,382.34m2
CONCLUSION
The following table consists of all the measurements to the triangles above.
Bearing the calculations in mind, I then put the Isosceles Triangles together in a table in order of size of the base. I than calculated the area for a few more Isosceles triangles with a higher Base than the Equilateral Triangle and tabulated them below in order and compared the base of the quadrilaterals with the Area.
I then put these values in a graph, shown below. I realised I had reached the peak of the area as I reached the area of the equilateral triangle.
The following table consists of all the measurements to the quadrilaterals above.
Bearing the calculations in mind, I then put the quadrilaterals in order and compared the base of the quadrilaterals with the Area. I than calculated the area for a few more quadrilaterals and tabulated them below in order and compared the base of the quadrilaterals with the Area.
I then put these values in a graph, shown below. I realised I had reached the peak of the area as I reached the area of the Square.
I put the maximum results together to compare them.
The Circle gave me a coverage area of 79,382.34 metres squared. The Square gave me a maximum coverage area of 62,5002 (metres squared). The Triangle gave me a maximum coverage area of 48,1662 (metres squared).
Comparing all the values investigated so far in comparison the Circle covers the largest farm area. Therefore I would recommend the Farmer to build her fence in a circle, as she will cover the area of 79,382.342 metres squared.