Circles Part 2: Angle Relationships

Once similarity intuition has been built with circles, we can start getting into more specific relationships with angles and segments.  This post will look at using visual information from central angles and inscribed angles.

Pi Charts

Students sometimes lack intuition for the measure of something.  Andrew Stadel has developed this idea into a thorough curriculum on estimation. In my classes we started reasoning through similar exercises.  Once we had a decent understanding of circle parts and whole, we moved on to other types of angles.

Non-Central Angles

At this point most students have the common sense that a circle has 360 degrees, and a triangle is half that at 180 degrees.  Built with this intuition in mind, we look at a triangle created by inscribed angles.

The next day we get to see the formula that collapses 3 ideas down to 1.

Dynamic Angles in Circles

So many of us have tried to organize all the formulas that go with circles.  It’s nice when textbooks or other curriculum organizers dom something like:



This is nice, but it still implies that all these relationships are discrete.  Instead it would be better to look at them as connected.

I originally thought students would look at this and see how the measures relate.  It wasn’t obvious, so I interjected with a simple exercise to the students.

Congruent Overlap

I saw two half sheets (8.25×5.5) of paper on my desk.  I held them up and asked the class, “Are these congruent?”  The response was something like, “Uh, Duh Mr. Butler.”  So I challenged them further, “Would you bet $1,000 on it?”

Now the silent sound of a gambler’s contemplation…

A more valuable decision requires some precision and accuracy.  As they were thinking I allowed additional information, “You can’t touch them, but you can ask me to do anything you want.”  They took the bait.  “Just overlap them, Butler.”  Now they’re headed in the right direction.  Overlapping layers can reveal congruence or equal measure.  Anyone could ask a similar question for an image like these puzzle pieces:

Congruent Puzzle Pieces

To support the visual information, students need to look at the measures involved and start building the relationships (SMP7 and SMP8).  After multiple true versions of the math relationships, we can return to something like the original table.

Students start to see that the table is simply 3 snapshots of this dynamic relationship. Then it is easier to organize the information in their relative schemas.  From here we go to paper practice with simple examples and eventually into a more complex question with multiple parts.


I took a discrete snapshot from this applet and used it as a paper assessment.  Students were solving for complex relationships with multiple layers, and not all with the same process.  Common Core math like this requires utilizing big picture relationships while at the same time being able to articulate the specifics within those relationships.

Next time:

This series of dynamic relationships within a circle continues next looking at segments, first with only tangents, and then into various combinations with secants and/or tangents.


Circles Part 1: Similarity Intuition

All circles are similar, right?

Okay, maybe it’s not given.  In fact, it needs to be proven.  This proof is yet another that is so easily demonstrated with dynamic math tools (like desmos and geogebra

Circle Similarity with Desmos

So long as you can move one center onto the other (translate) and dilate one radius to equal the other, similarity is achieved.  This works for every circle.  The perfect proportional balance achieved with circles lays the foundation for most of not all relationships found in them.

Proportional Measures

Similarity gives us a simple system for comparing measures in multiple figures.


You can even explore this with repetition of congruent triangles:

Dimensional Analysis with Scale


Below are a couple of applications that push further with exploring the measures of circles, proportionality, and relationships in the measures.

Pizza Pi 

This activity was first inspired by a posting on John Stevens‘ website .  It looks at proportions with a different angle (pun intended) and asks user to compare various portions of different sized (but similarly shaped) pizza.

Rolling Tires

Another great one from Andrew Stadel.  This is a 3 act math lesson looking at relationships with circumference of a tire and cumulative rotations.

Coming up next…

So where do we go from here.  Following the basic intuition of similarity and proportions in a circle, we can build into

  1. More Angle Relationships with circles
    • Central angles and/or non-central angles
    • Circles circumscribed around a triangle start to simplify some of these relationships.
  2. Segment Relationships
    • Tangents and Incircle
    • Dynamic Relationship with Intersection of Chords, Tangent-Secant, and Secant-Secant (hint: they’re all a similar relationship)

See you in Circles Part 2: Angle Relationships.