Finding the Area of a Circle
Students often wonder where the formula for the area of a circle comes from; and knowing something about that can help make it more memorable, as I discussed previously about other basic area formulas.
Geometry
Necessary and Sufficient Conditions: If, or Only If?
Sometimes in math, we trip over words, especially when they are used in ways that differ from everyday usage, or when the associated grammar is complicated. This set of three answers from our archive, each of which is referred to by the next one, look at relationships among the ideas of “necessary and sufficient conditions”, …
Necessary and Sufficient Conditions: If, or Only If? Read More »
What is a Diamond?
(Archive Question of the Week) Having discussed various issues involving categorizing shapes, let’s take a look at a very different shape question, which didn’t fit into the last post.
What Is a Trapezoid? More on Inclusive Definitions
A month ago, I wrote about classifying shapes, discussing inclusive and exclusive definitions, and variations in different contexts. I promised to return to the subject, moving on to the specific issue of trapezoids, and some other related topics. Now is the time.
Is Area of a Square a Circular Argument?
(New Question of the Week) I love it when students want to know why something has to be the way it is, and are not satisfied just being told to use a formula. Last month, Shunya asked this kind of question, which gave me a chance to refer to our archive and go beyond it.
Constructing a Line in a Square
(New Question of the Week) Last month we had a question from a Czech student asking about a geometry problem. The discussion illustrates language issues that can arise, and how we try to guide a student to solve a problem himself. I will fill in some gaps as we examine how to approach an interesting …
Is a Square a Rectangle? Classifying Shapes
We get many questions about classifying shapes, from both elementary and high school students (or their parents or teachers). They often have trouble with the very idea of classifying items by applying definitions, and also with the fact that definitions can vary, both between everyday and technical usages, and from one textbook to another.
What Role Should a Figure Play in a Proof?
Questions about geometric proofs have often been handicapped by the inability to show us the associated figure (until we made that easier to do on this new site). In principle, that should not be a problem, because the statement to be proved should contain all the necessary information. It should never be necessary to refer …
Unfamiliar Geometry Notation
(New Question of the Week) One of the benefits of being a Math Doctor is interacting with the math of many cultures around the world, as we attract an international following. We have observed variations in terminology and notation from country to country, as well as variations in the content of math education (some better …
How Can I Remember Area Formulas?
Students often ask about formulas for areas or volumes. Sometimes they are just overwhelmed by the number of formulas they need to know; other times they are curious about how we know they are true. The answer to both questions is, in part, the same: if you know at least something of where they come …