# Math 3 Unit 7 – Triangles

#### Part 1

The most important thing I learned this unit was the centers of triangles. There are 3 types that can easily be mistaken for each other; circumcenter, incenter, and centroid. The additional page that was included in our task helped visualize (I am a visual learner) just where each point was and how they were positioned in relation to the sides and vertices. The circumcenter is the point where all the perpendicular bisectors of the sides meet. The incenter is where all the angle bisectors of the vertices meet. The centroid is where all the lines connected between a vertex and the midpoint of the opposite side meet. This will be important in the future as we will need to find points that need to be equidistant from different sides or angles, and/or divide areas.

#### Part 2 – Castlerigg Stone Circle Project

I realized during this project that volume of non-standard 3D shapes is often very difficult to manage. Given a 10 oz. requirement, we first had to convert that into a cubic volume requirement, then guess and check several dimensions to finally find a size that was functional. All the heights, lengths, bases, slant lines, etc. that we had to manage all at the same time was very difficult.

One specific detail I learned was how to calculate the volume of a milk carton. The bottom part was easy; a simple rectangular prism, but the top triangle part was confusing. This led to a miscalculation in volume. At first, I calculated a volume with over 40 cubic centimeters over what we needed, which left us plenty of space. Then I realized that the triangle was not a triangular prism and could not be counted as one; after a bit of thinking, I then realized that it was actually a pyramid! the slanted faces of the carton and the slanted triangle on the sides formed a square-based pyramid with the height of the triangular prism. Even after that, I realized that the diagonal folds were reducing the height, and accounting for that I found the true volume to be exactly our initial goal, with 0.1 ounces to spare. I was very lucky that I made the design larger than I thought I needed, because in the end, it was exactly what we were required to make.

That was the greatest challenge of the project: handling all the numbers. I am prone to random mistakes, and in calculating the numerous volumes and areas I made several errors. In the end, I had to go back and fix them all last minute. Next time, I will have another team member double check all my numbers before I move forward, as to prevent another total do-over again.

But in the end, our project was a success in what it outputted. Our milk carton was very solid, functional, and good looking. Our presentation was effective in presenting all our information in a concise, non-overwhelming manner. I was able to use my design not just in a slideshow, but on a physical product.