(A new question of the week) Like many questions we get, this one can be solved in many ways. We like to guide a student to whatever solution will fit what they have learned; along the way, we may find various additional methods, and side trips into other topics of interest.
(An archive question of the week) Recently we have had a number of series digging into multiple aspects of an idea like area or percentages. But I am always running into miscellaneous questions that can’t fit into such a series because of their length or uniqueness. I want to spend the summer focusing on random …
We have been looking at some classic “false proofs” or “fallacies”, where a seemingly valid proof shows something clearly false to be true. The goal is to learn from these, how to distinguish a valid proof from an error. In a post from last year, What Role Should a Figure Play in a Proof?, I …
As a penultimate post in this series on problems that trigger a lot of doubt or opposition, we’ll look at one that is usually asked just as a “How do you do this?” question, but occasionally gets responses from people who argue for alternative interpretations. Our FAQ is at Three Houses, Three Utilities, which explains …
One of the best ways to get people eager to do something is to tell them it can’t be done. When people who are not well-versed in mathematics learn that it is impossible to trisect an angle with compass and straightedge, they sometimes seem to make it their life goal to “do the impossible”. The …
Frequently Questioned Answers: Trisecting an Angle Read More »
The last four posts dealt with formulas for finding areas using lengths of sides, starting with the triangle, where that is all you need, and then quadrilaterals, where something more must be added; and then using coordinates of vertices. Now we can use those tools to solve some of the more common real-life problems we …
We’ve been collecting techniques for finding areas of polygons, mostly using their side lengths. We started with triangles (Heron’s formula), then quadrilaterals (Bretschneider’s formula and Brahmagupta’s formula), and the fact that the largest possible area is attained when the vertices lie on a circle. We’ll look at one more way to find area, using coordinates …
Last time we looked at applying Heron’s formula to problems about the area of a triangle, where knowing the side lengths is enough to determine the area; there was a passing mention of the fact that more is needed for quadrilaterals. We’ll start here with a repeat of that idea, and then look at several …
Last time we looked at a very useful formula for finding the area of any triangle, given only the lengths of its sides. Today I want to look at several problems in which the formula has been used, some of them surprising.
There is a beautiful formula for the area of a triangle, which many students unfortunately never get to see. In this post we’ll look at that formula and three ways to prove it; next time, I’ll show some examples of how useful it can be. (By the way, you’ll also see it called “Hero’s Formula”, …