Similarity of Circles Worksheets
A circle is a set of point that are all the same distance from the center. It is a pretty cool shape that way, once you determine the distance to the center (radius) it is the same everywhere. Unlike other shapes where we are calculating all types of distances of sides and angle, circles are pretty simple. The size of circles is simply dependent on the length of the radius. So in a way all circles are similar because they only differ by the distance of that measure. Circles are a scaled version of one another. We can use this property of circles to our advantage to learn a great deal about any geometric situation. This selection of worksheets and lessons help students use the fact that all circles are similar as a tool to predict translations and scaling procedures.
Aligned Standard: High School Geometry - HSG-C.A.1
- Circle Transformation Step-by-step Lesson -Explain how the triangle can bounce around the quadrants and blow up in size.
- Guided Lesson - More on find the translation rule and the scale factor that is play here.
- Guided Lesson Explanation - I always like to explain the dilation first and then I move onto the scale factor.
- Practice Worksheet - If you can fly through these, you really have a good grasp on the topics.
- Matching Worksheet - Match the translations and scale factors to the graphs.
- Numerical Practice with Big Circles Worksheet Five Pack - It is like a huge angle scavenger hunt.
- Answer Keys - These are for all the unlocked materials above.
Homework Sheets
Follow the dilation of a circle across a coordinate system.
- Homework 1 - After translating the center of F to the center of F', dilate F about its new center to contract F' onto F.
- Homework 2 - How did this all come about?
- Homework 3 - You can use notation to state the translation and the scale factor.
Practice Worksheets
I made these really big so that you could see them closely and go over them step by step.
- Practice 1 - You can transform circle F to circle F' by translating it and then performing a dilation. Find the translation rule and the scale factor of the dilation.
- Practice 2 - See if you can forecast how these movements came about.
- Practice 3 - To find the scale factor of this dilation, calculate the radio of the radii.
Math Skill Quizzes
Each problem takes advantage of a different quadrant.
- Quiz 1 - What is the translation rule that has been followed here.
- Quiz 2 - What is the scale factor of the dilation that has taken place here?
- Quiz 3 - How does Circle F compare to that of Circle F'?
What is Similarity of Circles?
When we talk about similarity in geometry, it refers to a property where at least two shapes are relative to one another. When to shapes are said to be similar, they possess the same properties in the same proportion to one another. In the context of similarity, proportionality directs to all the values being multiples of each other within a given question.
We know that circles are defined by two main properties, the circumference and radius. The radius of a circle is the distant from any end of the circle to the center. Ultimately this measure determines the size of the circumference which is the distance of the perimeter of a circle. All circles also share the same central angle of 360 degrees.
Remember that similarity does not depend on position of center or size; hence, the similarity of circles depends mainly on proportionality of the radii of the circles. We also know that radius is a constant and unchanging number, and that every constant is proportional to another constant. So, keeping this in mind, all circles must be similar!
Remember congruence? Congruent shapes are identical, just like identical twins! They have the same size, angles, proportions. Congruence, like similarity, is also independent of position, rotation and reflection. By definition, all congruent shapes are similar, but not every similar shape is congruent.
We can use this property of circles to infer measures on other circles that we either have not measured or simply do not have the ability to measure. When treasure hunts are trying to determine what is in a shipwreck, they will often use this form of math to identify round objects underwater they can see, but do not have the ability to retrieve.