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.

Sum it up, Angle Edition part 1

Teaching Notes




  1. If angle BAE = ______, what are all the other angles.
  2. Which angle sums equal 180o?
  3. Which angle sums equal 90o?
  4. Which angle pairs/groups are equal?
  5. Find which angle pairs are supplementary, complementary and vertical
  6. Does it matter if angle BAE is acute or obtuse?

Devoted to free calculus resources for students, free and open source materials for instructors, and active engagement for all.

Math Munch

A Weekly Digest of the Mathematical Internet

Dan Ariely

My Irrational Life

Zero-Knowledge Proofs

Random thoughts about teaching math

What is 5?

Trying to change attitudes and perceptions about math, one middle schooler at a time.

sonata mathematique

mathematics + music

Talking Math with Your Kids

Because children enjoy using their minds

Questioning My Metacognition

Trying to be a better teacher


A great site

the radical rational...

in search of innovative ideas with a well-balanced approach for the math classroom

The Math Projects Journal

Innovative math lessons you can use in your classroom today

Megan Hayes-Golding

Still learning.

Mr. Shauver - Learner Educator

Just an attempt to be better...


Ramblings of a HS Math Teacher

Continuous Everywhere but Differentiable Nowhere

I have no idea why I picked this blog name, but there's no turning back now

Who's a Math Nerd? *raising hand*

Helping all to find their inner math nerd, even if you didn't know you had one.