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Rotational Symmetry - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

Symmetry in a figure exists if there is a reflection, rotation, or translation that can be performed and the image is identical. Rotational symmetry exists when the figure can be rotated and the image is identical to the original. Regular polygons have a degree of rotational symmetry equal to 360 divided by the number of sides.

In Geometry, we can look at a figure and say that it has symmetry if there is an isometry that will map part of the figure back onto itself. Well an isometry remember is a rigid transformation that is, a translation, a rotation or a reflection. We could be more specific however for certain objects and say that they have rotational symmetry.
An object has rotational symmetry if that figure is itself after you rotate it less than 180 degrees. If it is itself after exactly 180 degrees no more no less then that figure has point symmetry. So let's look at a couple different figures here and try and determine if it has rotational symmetry and if so what the degree is, so here we have an equilateral triangle and in my hand if I rotate this, I can definitely map it back to itself, so if I draw in some lines here that'll intersect right there I see that if all 3 of these are congruent which they are since it's an equilateral triangle, then they all must be 120 degrees so yes this has rotational symmetry and after every 120 degrees of rotation it will be itself so how many degrees of rotational symmetry does this have? it has 120 degrees.
If we look at this figure right here, we have 1, 2, 3 congruent line segments intersecting each other and it's pretty clear again that I could rotate this and have it mapped back onto itself. Well since they are 6 congruent angles we're going to have to do 360 divided by 6. Well 360 divide by 6 is 60 degrees so this figure right there has 60 degrees of rotational symmetry.
Last we look at this plus sign. If I had drawn this perfectly it's pretty clear that after, that there are 4 ways that we can rotate this so if we take 360 degrees and we rotate it 4 different ways it's pretty clear that this will have 90 degrees of rotational symmetry. So again rotational symmetry what does it mean? It means that you can rotate it less than 180 degrees and the figure will be exactly the same and if you can rotate it exactly 180 degrees so let's say we had a figure kind of like this, you see that if I rotate it exactly 180 degrees it would be itself.

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