1. Introduction The first installment of this post investigated the question of computing the volume of a unit ball (and by extension any ball) of dimensions. The main result we obtained was that the volume can be computed recursively as , where In this post, we will use this result to prove an explicit formula for . 2. The Formula A quick search online yields the following explicit formula for : where the function extends the…
1. Introduction This post investigates an interesting question I had been thinking about: we all know that the area of a circle of radius equals , and the volume of a sphere of radius equals . But what if we enter the territory of even higher dimensions? We should be able to say something about the “volume” of an n-dimensional sphere, where “volume” is interpreted as the equivalent measure of space in that many dimensions (e.g. area for 2D, regular…
Introduction Given any triangle , suppose we take each side and and construct an equilateral triangle on it. You may notice that given a line segment, there are two ways to construct an equilateral triangle on it (one on each side); we will eventually consider both ways, but for now suppose each triangle is constructed on the outside of , i.e. they do not overlap with . We now consider finding the centers of each of these equilateral triangles, as…
Today’s post will be fairly short and simple, but may have interesting usages in other areas. Given a quadratic equation , it is well known that solutions for are given by the quadratic formula, . There are a multitude of ways to derive this: Completing the square: can be reduced to . At this point, we know easily how to solve equations of the form , and by extension , by simply taking square roots (being wary of the two…
Introduction This post is a basic derivation of the famous Heron’s formula, which allows calculation of the area of any triangle in terms of its side lengths . We refer to the following diagram: As is convention, let sides be opposite to angles . Further, let the altitude length and . Then, by the Pythagorean Theorem applied to triangles and , we have the following two equations: Subtracting the two equations cancels the and yields ….
Given an unlimited number of 2 and 5 dollar bills, what all dollar values can be created? Clearly all multiples of 2 and 5 can be created, but they can also be combined to form many other values such as 7 (2 + 5), 13 (5 + 4 * 2) and so on. In fact, there are only a finite number of values that cannot be created in such a way by combining 2 and 5 dollar bills, and in…
Suppose we try to find the circumradius of a triangle , as shown below: How can we find it? Note that the triangle is isosceles, with the equal sides being radii of the circumcircle. To find these, we only need to know the angle and side length AC. Angle intercepts the arc , with O at the center of the circle. Angle also intercepts the same arc, but with point B on the circumference of the circle. This means (1)…
Fun Facts about Pentagram Do you know that sum of all angles of a pentagram whose all five sides are extended to meet at five points is ? For example, sum of all angles , , , and in below figure of a pentagram is equal to . Proof: It is easy to prove that: (1) (2) (3) (4) (5) Let’s say sum of all angles from to is S. Adding all five equations (1)…
Calculating value of the Golden Ratio Introduction: The Golden Ratio, represented by Greek letter “phi” () exists when a line is divided into two parts and the longer part (b) divided by the smaller part (a) is equal to the sum of (a) + (b) divided by (b), which both equal 1.618. See diagram below where (1) Proof: Suppose is the ratio such that , where b > a For convenience, let’s set a = 1, so solving for…