Cover Letter

Question 1: Describe the concept of area and volume in terms of efficiency. How can we measure efficiency?

 

    Efficiency in shapes can be described as the amount of area compared to the perimeter. An efficient shape maximizes the area while minimizing the perimeter.

    A shape's efficiency can be determined by its radius. For example, a square's radius in the distance from its midpoint to one of its corners. When this radius is rotated 45 degrees, it will extend outside of the perimeter of the square. If we inscribe that square inside of a circle that shares its radius, we can see that there is a lot of wasted space.

    When we repeat this process but with, say, a pentagon, the wasted space decreases. Therefore, the most efficient shape is a circle, because there is no wasted space.

    However, the circle is not the most efficient shape when used in a tiling. When tessellated, the circle leaves concave triangular gaps. In a tessellation, a hexagon is actually the most efficient. It tiles together perfectly with no gaps or overlaps and while still minimizing wasted space.

 

Question 2: How can we prove that two triangles are similar? 

 

   Two triangles are similar when they can undergo any isometry and still hold the same values. Another way we can prove two triangles to be similar is if two of the sides have the same values. For example, take triangle A and triangle B. Triangle A has a base of 6 and a height of 3. Triangle B has a base of 18 and a height of 9. Although their side lengths are different, they still have the same values, which prove them to be similar. Finally, you can prove two triangles are similar if they have two interior angles that are the same.