Kiss those Math Headaches GOODBYE!

Posts tagged ‘Diagonals’

ANSWER TO FRIDAY’S PROBLEM:


ANSWER TO FRIDAY’S PROBLEM:

The problem, once again:

For any polygon, a “diagonal” is defined as a line segment that runs from one vertex  to another, running  through the polygon’s interior. Find a formula that determines the number of diagonals in any convex polygon with n sides. Once you have the formula, use it to figure out the number of diagonals in a convex polygon with 1,000 sides (don’t try this by hand! — that’s why algebra was invented).

The winning answer was provided by Chris Mark. The formula, for a convex polygon with n sides, is this:  Number of Diagonals =  [n(n– 3)]/2. For n = 1000, the number of diagonals = 498,500.

The reasoning behind the formula. A polygon has as many vertices as sides. So a polygon with n sides also has n vertices. Now, consider any vertex of the n-gon. From that vertex the number of diagonals that can be drawn is (n – 3). That is because we cannot draw a diagonal to 3 vertices:  the vertex chosen, and the two adjacent vertices. So the expression (n – 3) = the number of diagonals that can be drawn from any vertex. We multiply (n – 3) by n to obtain the total number of diagonals that can be drawn from all n vertices. But if we simply multiply n by (n – 3), we’d be counting each diagonal twice. To eliminate that problem, we divide the product, [ n(n – 3)], by 2, and that provides the correct formula:

Diagonals  =  [n(n – 3)]/2

Applying this formula to a convex polygon with 1,000 sides, we see that the number of diagonals =  498,500.

In addition to providing the answer, Chris pointed out that the problem need not be restricted to regular polygons, as it was when posted. This formula works for all convex polygons, regular or not.

Thanks, Chris. And thanks to everyone who submitted answers.

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Challenge Problem – Polygon Diagonal Formula


Here’s a challenge problem for anyone who’d like to try it.

On Monday I will post the answer and the names of the first five people who got this right. So good luck, everyone.

A regular polygon is a polygon all of whose sides are congruent and all of whose  angles are congruent. For any polygon, a “diagonal” is defined as a line segment that runs from one vertex of the polygon to another, and which runs through the interior of the polygon.

Find a formula that tells how to determine the number of diagonals there are in any regular convex polygon with n sides.

Once you have the formula, use it to figure out the number of diagonals in a regular convex polygon with 1,000 sides (don’t try this by hand! — that’s why algebra was invented).

Good luck!

Tutor Tales #1: Color in Geometry


Hi,

This if the first in what I hope will be a long series of brief blogs called Tutor Tales blogs.

The idea is that while I’m tutoring I get ideas or insights on how to help students, and then I write up a short blog entry on that experience, preferably on the day that the event occurred.

I hope that these Tutor Tales will give you examples of approaches to math that help students (or that do not help, depending on what I did), and that they give you a chance to reflect on your own teaching.

For the first Tutor Tale entry, I just noticed how useful it can be to use color in geometry.

The girl I was tutoring had a problem:  Find out how many diagonals can be drawn inside a regular, convex nine-sided polygon.

I’ve already noticed that this girl likes color, and she is 17 years old. So I had a hunch that she would be open to trying a color-approach.

We created the non-agon by first drawing a circle, and then marking off nine points on the circle. Then we connected the points sequentially.

To find out how many diagonals we could draw for such a figure, we chose one color for the top point, green, and drew all of the diagonals we could for that point, in green. It turned out that there were 6 diagonals, so we put a big 6 in green at this vertex. Then we tried the next vertex, which we colored pink. We found that we could create 6 additional diagonals from this vertex, and we colored these pink. So we put a big pink 6 by this vertex. We went around the circle in a clockwise way, using a different color for each vertex. All in all we found that the pattern of diagonals was:  6, 6, 5, 4, 3, 2, 1, 0, 0, for a grand total of 27 diagonals.

Here’s the image of the figure we worked on.

non-agon-problem

One thing to consider, especially if you teach geometry, is how many opportunities there are in geometry to use color to separate different concepts and to relate similar concepts. Check it out and see what you discover.

Happy Teaching!

—  Josh