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

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:

Applications

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 wyrmath.wordpress.com .  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.