Main content

### Course: High school geometry > Unit 8

Lesson 1: Circle basics# Proof: all circles are similar

Watch Sal informally prove that all circles are similar by showing how we can translate then dilate any circle onto another.

## Want to join the conversation?

- I know that they are but how is this a proof might i ask?(15 votes)
- If it was me I wouldn't have put a video, I would have just had a lesson page with ALL CIRCLES ARE SIMILAR! in scary shaky letters.(16 votes)

- This proof sounds a little bit vague. I have got a question and a proposal to math community in general. Is it only me who thinks that precise definition of circle's similarity to itself should be based on its congruity? Wouldn't it be more reasonable to revise the definition of circle's similarity and be more specific on it? I propose to explore the following definition of circle's similarity:

"We may consider circles to be similar to each other if they are congruent (have equal radius); and we may consider circle to be similar to some (undiscovered / unexplored) polygon that sides are taken to its extremes."

I can't be the first man who has doubt that all kind of circles are similar to each other. There must be someone before me to appear with the same idea. I believe we may prove that NOT every circle is similar to each other if we can study the extremes of polygons. Did anyone can code Wolfram? Are there studies of extreme # of side of polygons?

Sincerely,

Sergei Tekutev

P.S. Anyone is welcomed. What i mean here is that there should be some starting point when polygon stops being a polygon and it becomes a circle with equal radius. And my guess is that initial visual appearance of a circle would be different from what we used to call a circle. If it is a legit to state that not every oval is a circle, and we say that circle is a kind of an ellipse then why not to break a circle to it is basics? Why not to try the claim that some form of polygon is a circle; if we are able to prove it than it would be evident that there are different types of circles. In this case it will be legitimate to say that circles are not similar; circles are polygons, ovals and ellipses . And circle's similarity is based on its radius.(7 votes)- The words 'circle', 'polygon', and 'similar' all have rigorous definitions already. There is no need to redefine them.

Given a point in the plane P, and a positive real number r, the circle centered at P with radius r is defined as the set of all points in the plane that are a distance r from P.

A polygon is a finite number of line segments which form a closed loop.

We also know that a line segment can intersect a circle in at most two points. Therefore, a collection of n line segments can intersect a circle at most 2n times. Because there are infinitely many points on a circle, and a circle and polygon can intersect only finitely many times, it follows that for any circle and any polygon, there will be infinitely many points where they do not intersect.

That is, a circle and a polygon will never contain the exact same set of points. So a circle is never a polygon, no matter how many sides the polygon has. That is a proven fact.

Finally, two figures are similar if there is a sequence of translations, rotations, reflections, and dilations that maps one figure onto the other. This is not just for circles, this is how similarity is defined for any set of points at all.

If we have two circles (x-a)²+(y-b)²=r² and

(x-c)²+(y-d)²=t², then we can map the first circle onto the second by applying the translation

(x, y)→(x+a-c, y+b-d)

followed by a dilation by a factor of t/r about the point (c, d). Therefore, any two circles are similar to each other. That too is a proven fact.(11 votes)

- To answer my question about equilateral shapes, yes technically all equilateral shapes would be similar to each other in the same shape category. This is because according to Wikipedia: "two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other." As you can see, not all rectangles will not have the same shape because they are free to change in two directions, but a square is based off of only one measurement that repeats itself, and same for an equilateral triangle or any other shape. So yes, all equilateral shapes will have the same shape when compared to other equilateral shapes with the same number of sides(3 votes)
- Sadly, this is not true.

Equilateral means that all the sides are the same length, but for any shape with more than three sides this does not imply that the angles are all the same. For instance not all rhombuses (or rhombi) are similar.

However, if we stipulate that a polygon is "regular" - equllateral**and**equiangular - then they will be similar. For instance all regular hexagons are similar.(14 votes)

- Sal mentions that objects that can be dilated and translated onto another object are similar by definition. Do 3D figures follow this suit? If so which ones?(5 votes)
- Well, spheres would be, cubes too. pyramids based on equilateral triangles. After that I'm not entirely sure, but regular ones might be.(8 votes)

- I think I get it. All the circles have the same curve they just look different because the radius is different?(5 votes)
- All circles are made up of 360, can be partitioned with an unlimited (infinite) amount of symmetry lines. The curve that you are talking about is always the same.... except for in terms of length and area, then they can differ depending on the central angle that is being referenced and the radius of the circle in that situation.(6 votes)

- Are all squares similar as well?(5 votes)
- Yes, sides are proportional and angles always 90. That would be true of any regular polygon such as the equilateral triangle.(5 votes)

- Is the circle the only shape in which all of that shape is similar? Thanks in advance!(4 votes)
- All regular polygons (and circles) have that property. For example, all squares are similar (or congruent), and so are all regular pentagons and all regular hexagons, etc.(4 votes)

- What is the MATHEMATICAL PROOF behind all circles are similar? Please someone help me out here.(3 votes)
- A circle is the set of all points in a plane that are at a given distance from a given point (the centre). So in fact all circles are similar by definition. You can always reposition the circle's center and adjust its radius to match any given circle.

Algebraically you can achieve this by tweaking the constant terms of a circle's equation: (x-a)^2+(y-b)^2=r^2. You can translate (reposition) the circle by replacing a and b with the required centre x and y coordinates, respectively, and the radius will equal r.(4 votes)

- Does this also work for a sphere?(3 votes)
- Yes! Because a sphere is the same shape no matter what, just scaled based on the radius! Great job!(4 votes)

- I don't get this video all circle are the same.

doesn't matter what size circles are a 1-D shape with no sides,faces,vertices,nothing!!.

I understand what he trying to say but why?(4 votes)- You are thinking of a line. (Or a point) A circle has 2 dimensions, Length and width.(1 vote)

## Video transcript

We're asked to translate and dilate the
unit circle to map it onto each circle. This is the unit circle right over here;
it's centered at (0,0), it has a radius of one. That's why we
call it a unit circle. When they say translate, they say move it
around. So that would be a translation of it.
Then dilating it means making it larger. So dilating that unit circle would be
doing something like that. So we're going to translate and dilate this
unit circle to map it onto each circle. So, for example, I can translate it so
that the center is translated to the center of that magenta circle, and
then I can dilate it so that it has been mapped on to that
larger magenta circle. I can do that for a few more. I'm not
going to do it for all of them. This is just to give you an idea of what
we're talking about. So now I'm translating the center of
my -- it's no longer a unit circle -- I'm translating the center of my circle to
the center of the purple circle and now I'm going to dilate it so it has
the same radius. And notice, I can map it. And so if you can map one shape to another
through translation and dilation, then the things are, by definition they
are going to be similar. So this is really just an exercise in
seeing that all circles are similar. If you just take any circle and you make
it have the same center as another circle then you can just scale it up or down to match the circle that you moved it
to the center of. So there you have it. Hopefully this gives you a sense that
all circles are similar.