Intro

It is very common for most people to have taken a geometry course. However, not many people know that one of the largest concepts in calculus stems from those basic geometric principles everyone learned as prepubescents. Many great civilizations from China, to India, to Egypt, to South America have many mathematically entangling roots. Egyptians used geometric principles to build the pyramids around 3000 BC (Mathnasium, 2020). Both Pythagorus (500 BC) and Euclid (300 BC) were famous Greek mathematicians whose work in geometry is still referenced and taught to this day. Calculus was invented in the late 17th century in Europe (The Great Courses, 2016). The fascinating concept is that these two fields, while seemingly completely different aspects of mathematics, are quite closely related.

Background

According to Cuemath (2020), "Geometry...is a branch of mathematics that is primarily concerned with the shapes and sizes of the objects, their relative position, and the properties of space." When most people think of geometry, they think of finding the perimeter of polygons, determining the area of common shapes, and calculating volumes of three-dimensional objects. Most people are familiar with the area of a square being equal to its length times width, and the volume of a rectangular prism is equal to length times width times height. However, geometry is so much more. It is based on Euclid's Axioms — assumptions that he held to be self-evident (Cuemath - the Math Expert, 2020). These act as the five pillars that most of geometry is based on. In simplified terms the five axioms are as such: a straight line can be made between any two points; a straight line and be projected indefinity; a circle is made by infinitely many points all the same distance from a single point; all right angles are 90 degrees; it is possible to create a parallel line. Everything known about geometry can be traced back to these basic principles.

Calculus revolves around continuous functions, usually finding the derivatives and integrals of those functions. A derivative is the rate of change of a function at any given point. When finding tangent lines, the derivative is used to calculate the slope of the line. The opposite of the derivative is the integral. An integral is a measurement of the area bounded between the function and the x-axis. Because one is finding the measurement of an area, one can start to see how basic geometry would make a great stepping stone leading to the invention of calculus.

Riemann Sums are used to estimate the area under the continuous function. However rectangles don't accurately describe the curve as they overestimate pr underestimate the area. This is combated by increasing the number of rectangles. This will decrease the width of each rectangle. As the width gets smaller and smaller, the number of rectangles grows and as the width gets infinitely small, the number of rectangles gets infinitely large. By using this method, one can perfectly calculate that area. From this use of a calculus concept of limits (as numbers approach a certain valve) derivative formulas have been derived allowing for simple computation.