(h x b) ÷ 2
(223.607 x 400) ÷ 2
89,442.7191 ÷ 2
area = 44721.360m²
Therefore the area of the isosceles above (not drawn to scale) has an area of 44721.360m². Now I must do this for lots of triangles so that I can eventually find the triangle with the largest area. I will start off with the base of the triangle increasing by 50m each time. Then I will zoom in until I find the right triangle. My table with the results are on the next page.
Below is a table showing the isosceles triangles’ base, side, height and area in metres and metres².
Looking at these results, it seems like as the base increases, the area also increases. However as the base is increasing, the height is decreasing. This makes the area decrease back again. The area is largest somewhere around the 300m-400m so I’m going to zoom in around that point and do exactly the same as I did in the table above except this time I am going to go up by 10m.
The area is largest around the 330m-340m point so I shall zoom in again.
The area has gone back down somewhere in between the 333m-334m so we know that the area for the largest isosceles triangle has a base between 333m and 334m. I shall zoom in one last time to find out the exact triangle with the largest area. This time I am going to go up by 0.1 of a metre each time.
In conclusion to isosceles triangles, my investigation shows that the largest isosceles triangle has a base of 333⅓m. Coincidentally, this triangle happens to be an equilateral triangle. This shows that an equilateral triangle has the largest area. However, there is also another type of triangle; scalene. Now I must investigate which triangle has the larger surface area; an equilateral triangle or a scalene triangle. A scalene is a type of triangle in which all the side lengths are different. The diagram below proves that the equilateral triangle has the larger surface area.
Triangles: Scalene
Looking at this diagram, there is no need to draw out tables to find out whether or not a scalene triangle is bigger than an equilateral in terms of area. Logically, we know that no matter how high, or how far the scalene triangles go, they will never have the same area as an equilateral (provided that the perimeters for all of the triangles add up to 1000m) and the diagram above proves it all. In conclusion, my investigation has shown that out of all the three types of triangle, equilateral has the largest surface area.
