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 …
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 …
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 …
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 …
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 …
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 …
(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 …
We often get questions about deriving formulas for area and volume; usually when the question is about a sphere, the context is calculus, so we talk about integration, the usual modern method. But for students who only know geometry, “wait until you learn calculus” can be unsatisfying. Fortunately, there are a couple ways to do …
Volume and Surface Area of a Sphere – Without Calculus Read More »
In searching for answers about counting divisors over the last couple weeks, I found a few that are about the similar question of finding the sum of a number’s divisors. In fact, a couple questions and answers confuse the two problems. Let’s finish off the topic by looking at these. (Keep in mind that “divisor” …
(A new question of the week) Sometimes a topic you think you know looks entirely different in an unfamiliar context. Here, a returning student faces a quadratic equation where x and y are reversed, making the quadratic formula seem foreign! But this is when real learning happens, when you are forced to think through the …