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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, then 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 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 an equilateral triangle, would fit neatly into any one of those exterior angles. Now a seven-sided shape. We haven’t really talked about this. Technically a seven-sided shape is called a regular heptagon. The reason that we haven’t talked about this is that it turns out this shape has angles that are non-integer degrees so the math involving this shape is kind of ugly and as it happens because of that the test just never asks about it, so that’s why you don’t need to worry about hexagons at all.

They never show up on the test. The regular octagon. Sum of the angle is 6 times 180, so that is 1080. And each one of the internal angles are equal, so each one must be equal to 1080 divided by 8. 1080 divided 8.

So the, cancel factor 2,that’s 540 divided 4. Cancel another factor 2.That’s 270 divided by 2. 270 divided 2 is a 135 degrees. Now that’s an interesting angle because that is the supplement of a 45 degree. 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 the regular octagon equals.

In a similar manner, we could find the individual angles in any, 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 formed, 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, so that means it must bisect the angle at A. So HAM, that little angle has to be 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 have found that angle. Now, put that aside. Now, look at quadrilateral ABCH. We note that the sum of angles in any quadrilateral’s 360.

We know that two of the angles. The angle 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 at A is 135 degrees so the angle at H is the supplement of that which is 45 degrees. So we have found this angle right here. MHA, that angle is 45 degrees. Now we’re 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 minus 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’ve found angle AMH, well the angle we’re interested in is just the vertical angle of that, so it has to be equal. So that means that x equals 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 quant section. In summary, regular polygons have all equal sides and all equal angles. We can find the sum of the angles using the n minus 2 times 180 formula and divide by n to find the measure of individual angles.

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