(An archive question of the week) While gathering answers to questions about volume and surface area formulas, I ran across this question that applies to all of them: Given all the approximations and assumptions we make in the derivations we show (without calculus), how can we claim that the resulting formula is exact? Or can …
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!
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 »
We’ve been looking at measuring and drawing devices (compass, ruler, protractor); let’s move on to units of measurement. A fairly common question for students learning about measurement is, “What is the most appropriate unit for measuring ___?” The answer is not always clear, as we’ll see.
Last time we looked at how to use a ruler to measure distances. This time, we’ll consider another common question over the years: how to use a protractor to measure angles. We’ll also consider the relationship between protractors and the compass and straightedge constructions that started this series on geometry tools. And just like last …
Having just discussed why we use compass and straightedge in geometry, let’s flip that around and look at a common question at the more elementary level: How do you use a ruler to measure or draw a line of a given length? The usual issue here is working with the fractional markings on an inch …
Some time ago I looked at questions about trisecting an angle by compass and straightedge, which entailed discussing the rules for such constructions. We left open another common question: Why are such constructions important, and why do we use those particular tools? This probably isn’t explained as often as it should be. Why does it …
(A new question of the week) Today I want to look at a recent question that led into both geometrical and trigonometrical solutions, and particularly a useful perspective on the Law of Sines.
(A new question of the week) Last week we looked at a question about a triangle inscribed in a semicircle. Not long after that question, the same student, Kurisada, asked a question about triangle inscribed in a circle, which had some connections to the other. As we enjoy doing, we led the student through several …