# Formulas

## Distances on Earth 1: The Cosine Formula

Many students study trigonometry, but few get to spherical trigonometry, the study of angles and distances on a sphere. This is particularly useful in dealing with measurements on the earth (though it is not a perfect sphere). In this series, we will derive and use three different formulas for the distance between points identified by …

## Cutting Up Space Using n Planes

As the capstone of this series on counting, lets look at something that’s a little harder to count by drawing: What is the maximum number of regions into which all of 3-dimensional space can be divided by n planes? We’ll look at two significantly different perspectives. Working up through the dimensions The first question is …

## Cutting Up a Circle II: Using n Points

Last week we looked at counting the maximum number of pieces into which a circle can be cut by n chords (straight lines). Here we will look at a similar-sounding problem where we use all the chords formed by n points on the circle. We’ll also see an important example of why we shouldn’t jump …

## Cutting Up a Circle I: Using n Chords

We’ve looked at how to count diagonals in a polygon; this week and next, I want to consider two different problems (though they look similar at first) dealing with chords of a circle (which are practically the same thing as diagonals of a polygon). In each, what we count will be the regions into which …

## How Much Does the Earth Curve?

We’ve looked at how to find the circumference of the earth, and how far we can see over the horizon. Another kind of question we’ve had about the curvature of the earth is, how much does it curve over a given distance? That has been asked in several different ways, which lead to some intriguing …

## Logarithmic Graphing

We’ve been looking at the math underlying some of the graphs associated with the COVID-19 pandemic, starting with exponential growth, and then logistic growth. I want to look in more detail at a feature I mentioned in the first post, viewing a graph logarithmically. This is a powerful technique that goes far beyond a button …

## Logistic Growth

Last time we looked at the meaning of exponential growth, a term commonly used in describing the initial spread of a virus such as the current SARS2 (which causes COVID-19). But exponential growth can’t continue forever, as it would soon exceed the total population. A slightly more complicated model for growth that takes into account …

## Area of Pyramids and Cones

We’ve looked at the volume of a pyramid, the formula for which can be found geometrically by a couple very different methods. Cones can be handled the same way, so we can skim over them. Let’s finish up by considering the surface areas of both cones and pyramids. Volumes and areas I’ll start with a …

## Volume of a Pyramid – Without Calculus

Last week we looked at ways to derive the formulas for volume and surface area of a sphere, without using calculus. Let’s do the same this time for a pyramid. We’ll be seeing one method that comes very close to calculus (slicing and infinite series), and another that is fully geometrical (dissection, which we’ll do …

## Making a Sphere from Flat Material

(A new question of the week) A recent question asking how to make a sphere out of flat material called for a look at an old question on the same topic, and some new ideas, including thoughts about approximation. And we actually get to see the physical result of our assistance, which is rare! Designing …