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Regular Polygons


Now we can start talking about the most special and elite of all the shapes in geometry, the regular polygons. To begin, we need to discuss this very funny word, regular. In everyday life, regular means ordinary, common, not exceptional in any way. In geometry the word regular means exactly the opposite of this. And this is what's so confusing, this is one word where the use in mathematics, the use in geometry is the absolute opposite of its colloquial everyday use.

In geometry the word regular means special and elite. In particular when the word regular describes a polygon, it means that the polygon must have two properties. It must be equilateral, having all equal sides, and also equiangular, having all equal angles. In any category of polygons, the regular polygon of that category is the most symmetrical and well-balanced example of that category.

It's the most elite member of the category. The regular triangle is an equilateral triangle. The regular quadrilateral is the square, equal sides, equal angles. This is the regular pentagon. Notice that the sum of the angles in any pentagon is 540 degrees, so we could find the measure of the individual angles.

Each individual angle we would just divide 540 by 5, and that's 108. So 108 degrees is the angle in any regular pentagon. The regular hexagon. The sum of the angles here is 720. Well, to find the individual angles we'll divide 720 by 6, we get 120. Now that's an interesting number because 120 is the supplement of a 60-degree angle.

So that's good to know that a 60-degree angle, say a little equilateral triangle would fit neatly into any one of those exterior angles. Now a seven-sided shape we haven't really talked about these, so technically a seven-sided shape is called a regular heptagon. The reason that we haven't talked about this, is that turns out this shape has angles that are non-integer degrees.

So the math involved in this shape is kind of ugly and as it happens because of that the test never asks about it. So that is why you don't need to worry about heptagons at all, they never show up on the test. The regular octagon. The sum of the angles is 6 times 180, so that is 1080.

And each one of the internal angles are equal, so each one must equal 1080 divided by 80. 1080 divided by eight. So cancel a factor of two, that's 540 divided by four. Cancel another factor of two, that's 270 divided by two. 270 divided by two is 135 degrees.

Now that's an interesting angle because that is the supplement of a 45-degree angle. And so that's a fact that can be very useful. That the supplement of a 45-degree angle is exactly what every angle in a regular octagon equals. In similar manner we could find the individual angles in any regular higher polygon.

Because we know all the angles we could also find angles formed by the diagonals. So this is a relatively hard problem. Pause the video and think about this a little bit and then we'll talk about this. Okay. So what we have is a regular octagon with two diagonals drawn and we want to find the angle for it, the angle x, the angle of two of those diagonals.

Well, we have to think about this step by step. First of all, because it's a regular octagon, we know every interior angle is 135 degrees. Clearly, AE, diagonal AE, divides the octagon symmetrically in half. It's a mirror line for the whole octagon. Cuz that means it must bisect the angle at A.

So HAM, that little angle, has to be the bisected 135 degrees. Well 135 degrees divided by 2 is 67.5 degrees. So that is the angle HAM, this angle right here. So we've found that angle. Now, put that aside. Now, look at quadrilateral ABCH.

We know that the sum of the angles in any quadrilateral is 360. We know that two of the angles, the angle at A and the angle at B are 135 degrees. Well, if you look at that shape we realize that is a trapezoid, and because Equals BC it's a symmetrical trapezoid. And in fact because it's a trapezoid we know that the angle at H is just gonna be the supplement of the angle at A.

Well, the angle A is a 135 degrees, so the angle H is the supplement of F which is 45 degrees. And so we have found this angle right here, MHA, that angle's 45 degrees. Now, we're now in good shape. Because look at that triangle HAM.

We know two of the angles in that. We have this angle, and we have this angle. So we know that the sum of the three angles in that triangle is 180 degrees. We add things up. We subtract, 180- 112.5, and we get 67.5.

So turns out that in fact, that is a little isosceles triangle, believe it or not. Because two of the angles are equal. But now that we found angle AMH, well, the angle we're interested in is the vertical angle that so has to be equal. So that means that x = 67.5 degrees.

So that is an example of a very, very hard problem. A problem like that could appear among the hardest problems on the quad section. In summary, regular polygons have all equal sides and all equal angles. We can find the sum of the angles, usually using the N minus two times 180 formula, and divide by N to find the measure of the individual angles.

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