All circles are similar
All Circles Are Similar
All Circles Are Similar is the opening topic of Circles in the Common Core State Standards. It is aligned to Standard G-C.1, which requires students to prove that all circles are similar.
All circles are similar — any circle can be mapped onto any other by a translation and a dilation.
Every question with a fully worked solution.
- Proof: all circles are similar | Mathematics II | High School Math | Khan Academy Watch
Theory
A consequence is that the ratio of circumference to diameter is the same constant, \(\pi\), for every circle.
The constant and the scale factor:
How to compare two circles
- Find the scale factor as the ratio of their radii.
- Lengths (radius, circumference) scale by \(k\).
- Areas scale by \(k^2\).
The scale factor is the ratio of radii.
| \(k\) | \(=\) | \(\dfrac{12}{3}=4\) |
A dilation centered appropriately maps one circle exactly onto the other, and dilations produce similar figures.
The ratio of circumference to diameter is always \(\pi\).
| \(\dfrac{C}{d}\) | \(=\) | \(\pi\) |
Circumference scales with the radius (linear).
| \(\dfrac{C_2}{C_1}\) | \(=\) | \(\dfrac{15}{5}=3\) |
Common pitfalls
Frequently asked questions
Why are all circles similar?
Because a dilation can map any circle onto any other; they differ only in size.
What is the scale factor between two circles?
The ratio of their radii (or diameters).
What ratio is the same for all circles?
Circumference divided by diameter, which always equals \(\pi\).
Are all circles congruent?
Only if they have the same radius. In general they are similar but not congruent.