Ratios and Areas: An Unusual Pie Chart Problem
Here is a short problem with several levels of difficulty. The problem itself is poorly designed, as we’ll see, but still provides several useful lessons, dealing with measurement, rounding, and ratios.
Geometry
How to Find Any Part of a Segment of a Circle
Recent questions have dealt with calculations of various parts of a segment of a circle (chord, arc, sagitta, etc.). How, for example, can you find the arc length if you know the chord length and the height? The definitive explanation of these questions is found in a classic page from Ask Dr. Math, written by …
Finding Length of a Roll: Facing Reality
Last time we looked at how to find the length of material on a roll, making some necessary simplifications. Here, I want to look at some variations on that: first, about carpet in particular, and then about wire on a spool.
Finding the Length of a Rolled-up Carpet
One of the more frequent questions we had on Ask Dr. Math was about how to find the length of material (carpet, paper, wire, etc.) on a roll, knowing only the inner and outer diameters and something else: the thickness of the material, or the number of turns, or the original size of the roll. …
Finding a Locus: Algebra and Geometry
Last time we looked at the meaning of the concept of locus. This time, we’ll explore seven examples, from two students. We’ll look at both algebraic (equation) and geometric (description) perspectives.
What is the Meaning of “Locus”?
A recent question asked about an interesting locus, which led me to realize we haven’t talked about that topic in general. Here we’ll look at what a locus is, using three simple examples, and then dig into a question about the wording.
Looks Like a Frustum, But …
Last time we looked at how to find the volume of a frustum of a pyramid or cone. But sometimes what looks at first like a rectangular frustum actually isn’t. This case turns out to have a more general formula almost as nice as what we have for an actual frustum. We’ll discover that the …
Frustums: Not Frustrating but Fascinating
We’ve looked in the past at volumes and surface areas of familiar geometric shapes like spheres, pyramids, and cones; but more can be done. If we cut parallel to the base of a pyramid or cone, the result is called a frustum (no, not a frustrum!). Let’s derive some formulas, which will be remarkably simple.
Probability That a Random Triangle is Acute
Some time ago we looked into the probability that a random set of sides (from, say, a broken stick) form a triangle. A recent question asked about the probability that a random triangle is acute (all angles acute) or obtuse (at least one angle obtuse), which led to more discussion of what it means for …
Finding an Unknown Angle: Trig or Geometry
I am always interested in problems that can be solved in different ways, particularly because this can give a student a chance to be creative, as well as learning from experience that you don’t have to do it “the teacher’s way”. Here we’ll use trigonometry, and two different ways to add lines to a figure …