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Circle
Similarity

Discover why every circle in the universe is similar to every other circle, and learn how translations and dilations make this possible.

Geometry

What Does Similar Mean?

Two figures are similar if one can be obtained from the other by a sequence of rigid motions (translations, rotations, reflections) followed by a dilation. Similar figures have the same shape but not necessarily the same size.

Key Definitions

Translation
Slides every point the same distance in the same direction: no change in size or shape.
Dilation
Scales every point by a factor \( k \) from a center: changes size but preserves shape.
Scale Factor
The ratio \( k = \dfrac{r_2}{r_1} \) that maps one radius to another.

Why Circles Are Special

A circle is completely determined by just two parameters: its center and its radius. Every circle has the exact same shape: a perfectly round curve where every point is equidistant from the center. This is why we can always map one circle onto another.

\( A \) \( r_1 \) \( B \) \( r_2 \) Small Circle Large Circle

Theorem: All Circles Are Similar

The proof uses two transformations:
Step 1: Translate \( C_1 \) so its center \( O_1 \) lands on \( O_2 \).
Step 2: Dilate by scale factor \( k = \dfrac{r_2}{r_1} \) centered at \( O_2 \), mapping every point of \( C_1 \) onto \( C_2 \).

The Two-Step Proof

Original After Translation After Dilation \( O_1 \) \( O_2 \) Translate \( O \) Dilate \( O \) Mapped Perfectly ✓ \( r_1 \) \( r_2 \)

Why Dilation Works

After the translation, both circles share the same center \( O \). Consider any point \( P \) on Circle \( C_1 \). The distance from \( O \) to \( P \) is exactly \( r_1 \). Applying a dilation centered at \( O \) with scale factor \( k \):

Since every point of \( C_1 \) maps onto \( C_2 \) (and vice versa with \( k^{-1} \)), the two circles are similar by definition.

Similarity vs Congruence

Similar Congruent
Definition Same shape, possibly different size Same shape and same size
Transformations Rigid motions + dilation Rigid motions only
Scale factor \( k \) \( k > 0 \) \( k = 1 \) only
Circles All circles are similar Only circles with equal radii

How to Prove Two Circles Similar

1
Identify centers and radii. Label the two circles with their centers \( O_1, O_2 \) and radii \( r_1, r_2 \).
2
Translate. Describe the translation vector \( \vec{v} = O_2 - O_1 \) that moves \( O_1 \) to \( O_2 \).
3
Compute the scale factor. Calculate \( k = \dfrac{r_2}{r_1} \).
4
Dilate. Apply a dilation centered at \( O_2 \) with factor \( k \). Every point on the translated circle maps onto the second circle.
5
Conclude similarity. Since a translation + dilation maps one circle onto the other, they are similar. ✓

Example 1: Basic Proof

Circle \( A \) has center \( (2, 3) \) and radius \( r_1 = 4 \). Circle \( B \) has center \( (7, 1) \) and radius \( r_2 = 10 \). Show they are similar.

A(2, 3) \( r_1 = 4 \) B(7, 1) \( r_2 = 10 \)

Example 2: Concentric Circles

Two concentric circles share center \( O = (0, 0) \), with \( r_1 = 3 \) and \( r_2 = 9 \). How are they similar?

\( O \) \( r_1 = 3 \) \( r_2 = 9 \) \( k = 3 \)

Example 3: Unit Circle Mapping

Show that the unit circle centered at the origin is similar to a circle with center \( (5, -3) \) and radius \( 7 \).

1
Identify: \( O_1 = (0, 0),\; r_1 = 1 \) and \( O_2 = (5, -3),\; r_2 = 7 \).
2
Translate: Apply \( \vec{v} = (5 - 0,\; -3 - 0) = (5,\; -3) \) to move the center to \( (5, -3) \).
3
Dilate: Scale factor \( k = \dfrac{7}{1} = 7 \) centered at \( (5, -3) \).
4
Verify: Every point \( P \) at distance \( 1 \) from the origin maps to a point at distance \( 7 \) from \( (5, -3) \). ✓

Proportional Relationships

Because all circles are similar, certain ratios are invariant across all circles. The most famous constant in mathematics emerges directly from circle similarity:

\( \pi \) Is Universal
$$ \frac{C}{d} = \pi \approx 3.14159 $$ The ratio of circumference to diameter is the same for every circle.
Area Scales as \( k^2 \)
$$ \frac{A_2}{A_1} = \left(\frac{r_2}{r_1}\right)^2 = k^2 $$ Similar figures have areas in the ratio of the square of the scale factor.

Arcs, Sectors, and Similarity

Since all circles are similar, corresponding arcs and sectors with the same central angle are also similar across any two circles.

\( O \) \( \theta \) \( O' \) \( \theta \) Sector, radius \( r_1 \) Sector, radius \( r_2 \) ~

Common Mistakes

✕ Confusing Similar with Congruent
Students often say two circles with different radii are "not similar." Remember: similar means same shape, not same size. All circles are similar; only equal-radius circles are congruent.
✕ Forgetting the Translation
Dilation alone only works for concentric circles. When centers differ, you must first translate to align the centers before dilating.
✕ Inverting the Scale Factor
The scale factor to map \( C_1 \) onto \( C_2 \) is \( k = \frac{r_2}{r_1} \), not \( \frac{r_1}{r_2} \). Mixing these up reverses the mapping direction.

Summary at a Glance

Concept Details
Claim All circles are similar
Step 1 Translate to align centers
Step 2 Dilate by \( k = \frac{r_2}{r_1} \)
Why it works Every point at distance \( r_1 \) maps to distance \( r_2 \)
Congruent case \( k = 1 \) (equal radii): rigid motion only
Key invariant \( \frac{C}{d} = \pi \) for all circles
Area ratio \( \frac{A_2}{A_1} = k^2 \)

Question 1 of 5

Which sequence of transformations proves that any two circles are similar?

Question 2 of 5

Circle \( P \) has radius 5 and Circle \( Q \) has radius 15. What is the scale factor of the dilation that maps \( P \) onto \( Q \)?

Question 3 of 5

If two circles are concentric (same center), which transformation is needed to prove they are similar?

Question 4 of 5

Circle \( A \) has radius 4 and Circle \( B \) has radius 12. What is the ratio of their areas?

Question 5 of 5

Why does the ratio \( \dfrac{C}{d} = \pi \) hold for every circle?