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All Circles Are
Similar

Discover why every circle in the plane is similar to every other circle, using translations and dilations to map one circle onto another.

Geometry

What Does Similar Mean in Geometry?

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 (scaling). Formally:

For polygons, similarity requires equal corresponding angles and proportional sides. But circles have no angles and no sides, so we need a different approach: we prove similarity using transformations.

What Defines a Circle?

A circle is completely determined by just two parameters:

Center \( (h, k) \)
The fixed point equidistant from every point on the circle. Determines position in the plane.
Radius \( r \)
The constant distance from the center to any point on the circle. Determines size.

Two Transformations, One Proof

To show any circle \( C_1 \) is similar to any circle \( C_2 \), we need exactly two transformations:

1
Translation: Slide \( C_1 \) so its center lands on the center of \( C_2 \). This is a rigid motion that changes position but not size or shape.
2
Dilation: Scale by the factor \( k = \dfrac{r_2}{r_1} \) centered at the now-shared center. This changes size to match \( C_2 \) exactly.

Mapping Circle 1 onto Circle 2

\(C_1\) \(r_1\) Translate \(C_1'\) \(C_2\) \(r_2\) Dilate by \(k = r_2 / r_1\)

The Theorem: All Circles Are Similar

1
Translate by vector \( \vec{v} = O_2 - O_1 \). This maps \( O_1 \rightarrow O_2 \), giving a circle of radius \( r_1 \) centered at \( O_2 \).
2
Dilate about \( O_2 \) by scale factor \( k = \dfrac{r_2}{r_1} \). Every point at distance \( r_1 \) from \( O_2 \) maps to distance \( k \cdot r_1 = r_2 \).
3
Conclude: The composition (translation then dilation) is a similarity transformation, so \( C_1 \sim C_2 \). ✓

The Scale Factor \( k \)

The dilation factor \( k \) determines how lengths, arcs, and circumferences scale between two similar circles:

\(r_1 = 3\) Small Circle \(r_2 = 9\) Large Circle \(k = \frac{9}{3} = 3\)
Radii
Scale by \( k \): \( r_2 = k \cdot r_1 \)
Circumference
Scales by \( k \): \( C_2 = k \cdot C_1 \)
Area
Scales by \( k^2 \): \( A_2 = k^2 \cdot A_1 \)

Similar vs. Congruent Circles

Similar (all circles) Congruent (special case)
Transformation Translation + Dilation Rigid motion(s) only
Scale Factor \( k = \frac{r_2}{r_1} \) (any \( k > 0 \)) \( k = 1 \) exactly
Radii Can differ Must be equal: \( r_1 = r_2 \)
Shape preserved? Yes Yes
Size preserved? No Yes

Why Proportions Stay Constant

Because all circles are similar, certain ratios are universal constants: the same for every circle regardless of size.

1
Circumference-to-diameter ratio: \( \dfrac{C}{d} = \pi \) for every circle. This is precisely why \( \pi \) exists as a constant.
2
Arc length-to-radius ratio: For a fixed central angle \( \theta \), the ratio \( \dfrac{s}{r} = \theta \) (in radians) is the same for all circles.
3
Area-to-radius-squared ratio: \( \dfrac{A}{r^2} = \pi \) for every circle. Area scales as the square of the scale factor.

How to Prove Two Circles Are Similar

1
Identify the centers \( O_1 = (h_1, k_1) \) and \( O_2 = (h_2, k_2) \), and radii \( r_1 \) and \( r_2 \).
2
Write the translation vector: \( \vec{v} = (h_2 - h_1, \; k_2 - k_1) \). This maps \( O_1 \rightarrow O_2 \).
3
Compute the scale factor: \( k = \dfrac{r_2}{r_1} \). This dilation about \( O_2 \) maps radius \( r_1 \rightarrow r_2 \).
4
State the conclusion: The composition of translation by \( \vec{v} \) and dilation by \( k \) maps \( C_1 \) onto \( C_2 \), proving \( C_1 \sim C_2 \).

Example 1: Basic Mapping

Show that the circle with center \( (0, 0) \) and radius \( 2 \) is similar to the circle with center \( (5, 3) \) and radius \( 6 \).

0 5 3 \((0,0),\; r=2\) Translate by \((5,3)\) \((5,3),\; r=6\) Dilate by \(k=3\)

Example 2: Scaling Down

Circle \( A \) has center \( (-3, 4) \) and radius \( 10 \). Circle \( B \) has center \( (1, -2) \) and radius \( 5 \). Find the similarity transformation.

1
Translation vector: \( \vec{v} = (1 - (-3), \; -2 - 4) = (4, -6) \).
2
Scale factor: \( k = \dfrac{r_B}{r_A} = \dfrac{5}{10} = \dfrac{1}{2} \). Since \( k < 1 \), this is a contraction.
3
Conclusion: Translate by \( (4, -6) \), then dilate by \( \frac{1}{2} \) about \( (1, -2) \). Circle \( A \sim \) Circle \( B \). ✓

Example 3: Arc Length Consequence

A \( 60^\circ \) arc on a circle of radius \( 4 \) has length \( s_1 \). What is the corresponding arc length on a similar circle of radius \( 12 \)?

\(r = 4\) \(60^\circ\) \(r = 12\) \(60^\circ\)

Why \( \pi \) Is a Universal Constant

The fact that all circles are similar is the fundamental reason why \( \pi \) is a single, universal number rather than changing from circle to circle.

\(\frac{C}{d} = \pi\) \(\frac{C}{d} = \pi\) \(\frac{C}{d} = \pi\) Same ratio for every circle

Common Mistakes to Avoid

✕ Confusing Similar and Congruent
All circles are similar, but only circles with equal radii are congruent. Similar allows different sizes; congruent requires the same size.
✕ Forgetting the Translation
Students often jump straight to dilation. You must first translate so the centers coincide, then dilate. The order matters.
✕ Wrong Scale Factor Direction
The scale factor is \( k = \frac{r_2}{r_1} \) (target over source), not \( \frac{r_1}{r_2} \). Reversing it maps away from the target circle.

Summary: Key Relationships

Quantity Formula How It Scales
Radius \( r \) By \( k \)
Diameter \( d = 2r \) By \( k \)
Circumference \( C = 2\pi r \) By \( k \)
Arc Length \( s = r\theta \) By \( k \)
Area \( A = \pi r^2 \) By \( k^2 \)
Sector Area \( A_s = \frac{1}{2}r^2\theta \) By \( k^2 \)

Question 1 of 5

Two circles have radii \( r_1 = 4 \) and \( r_2 = 10 \). What is the scale factor of the dilation that maps the first circle onto the second?

Question 2 of 5

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

Question 3 of 5

Circle \( P \) has area \( 9\pi \). Circle \( Q \) has area \( 81\pi \). What is the scale factor \( k \) from \( P \) to \( Q \)?

Question 4 of 5

The ratio \( \frac{C}{d} = \pi \) holds for every circle. This is a direct consequence of which fact?

Question 5 of 5

Circle \( A \) has center \( (2, -1) \) and radius \( 7 \). Circle \( B \) has center \( (-4, 5) \) and radius \( 7 \). These circles are: