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All Circles Are
Similar

Discover why every circle in the plane is similar to every other circle, and how transformations prove it.

Geometry

What Does Similar Mean?

Two figures are similar if one can be mapped onto the other using a sequence of rigid motions (translations, rotations, reflections) and dilations. Similar figures have the same shape but not necessarily the same size.

Key Properties of a Circle

Center
A fixed point equidistant from every point on the circle.
Radius
The constant distance \( r \) from the center to any point on the circle.
Shape
Every circle is perfectly round: the shape is determined entirely by \( r \).

Two Circles, Different Sizes

\( r=6 \) Circle A \( r=4 \) Circle B ~

Circle A has radius \( r = 6 \) and Circle B has radius \( r = 4 \). Despite different sizes, they have the exact same shape.

The Core Theorem

The proof uses two transformations:
1. Translate one circle so its center coincides with the other's center.
2. Dilate by the scale factor \( k = \frac{r_1}{r_2} \) centered at that shared center to map Circle 2 onto Circle 1.

Proof: Translate then Dilate

Step 1: Original A B Step 2: Translate B Same center Step 3: Dilate B \( k = \frac{r_A}{r_B} \) \(A \sim B'\)

The Scale Factor

If Circle 1 has radius \( r_1 \) and Circle 2 has radius \( r_2 \), the dilation that maps Circle 2 onto Circle 1 has scale factor:

\( k > 1 \)
Circle 2 is enlarged (smaller mapped to bigger).
\( k = 1 \)
Circles are congruent (same size already).
\( 0 < k < 1 \)
Circle 2 is shrunk (bigger mapped to smaller).

Similar vs. Congruent

Similar (\(\sim\)) Congruent (\( \cong \))
Transformations Rigid motions + dilation Rigid motions only
Shape Same Same
Size Can differ Must be equal
Circles All circles are similar Only if \( r_1 = r_2 \)
Scale factor Any \( k > 0 \) \( k = 1 \) only

Why Does This Matter?

1
Proportional reasoning: Ratios like \( \frac{C}{d} = \pi \) hold for all circles because they are all similar.
2
Area scales predictably: If radii have ratio \( k \), areas have ratio \( k^2 \). Similarity guarantees this.
3
\( \pi \) is universal: The constant \( \pi \) exists precisely because all circles are similar: the circumference-to-diameter ratio never changes.

How to Prove It

1
Identify centers and radii. Let Circle A have center \( O_1 \), radius \( r_1 \). Let Circle B have center \( O_2 \), radius \( r_2 \).
2
Translate Circle B by the vector \( \vec{O_2 O_1} \) so both circles share center \( O_1 \).
3
Dilate the translated Circle B from center \( O_1 \) by scale factor \( k = \frac{r_1}{r_2} \).
4
Conclude: The image of Circle B is now Circle A, so \( A \sim B \).

Example 1: Finding the Scale Factor

Circle P has center \( (1, 7) \) and radius \( 4 \). Circle Q has center \( (5, 3) \) and radius \( 10 \). Describe the similarity transformation mapping P onto Q.

\( P: r=4 \) \( Q: r=10 \) Translate \((4,-4)\)

Example 2: Ratio of Areas

Two similar circles have radii \( r_1 = 6 \) and \( r_2 = 9 \). What is the ratio of their areas?

Example 3: Circumference Ratio

Circle M has circumference \( 8\pi \) and Circle N has circumference \( 20\pi \). Find the scale factor and the ratio of their radii.

1
Find radii: \( C = 2\pi r \), so \( r_M = \frac{8\pi}{2\pi} = 4 \) and \( r_N = \frac{20\pi}{2\pi} = 10 \).
2
Scale factor: \( k = \frac{r_N}{r_M} = \frac{10}{4} = \frac{5}{2} \).
3
Circumference ratio: \( \frac{C_N}{C_M} = \frac{20\pi}{8\pi} = \frac{5}{2} = k \). Circumferences scale by the same factor as the radii.

Consequences of Similarity

\( \pi \) is constant
\( \frac{C}{d} = \pi \) for every circle, guaranteed by similarity.
Arc length scales
Equal central angles in similar circles produce arcs in ratio \( k \).
Sector area scales
Sector areas in similar circles are in ratio \( k^2 \).

Common Mistakes

✕ Confusing similar with congruent
Similar means same shape, not same size. All circles are similar, but only circles with equal radii are congruent.
✕ Forgetting the translation step
Dilation alone works only if circles are concentric. You must first translate to align centers.
✕ Squaring the scale factor for lengths
Lengths (circumference, arc) scale by \( k \). Only areas scale by \( k^2 \). Do not square when comparing linear measurements.

Summary of Key Facts

Property How It Scales Example (\( k = 3 \))
Radius \( \times\, k \) \( 2 \to 6 \)
Diameter \( \times\, k \) \( 4 \to 12 \)
Circumference \( \times\, k \) \( 4\pi \to 12\pi \)
Area \( \times\, k^2 \) \( 4\pi \to 36\pi \)
Shape Unchanged Still a circle

Question 1 of 5

Which transformation(s) are needed to prove that all circles are similar?

Question 2 of 5

Circle A has radius 5 and Circle B has radius 15. What scale factor maps Circle A onto Circle B?

Question 3 of 5

Two circles have radii in the ratio \( 2 : 5 \). What is the ratio of their areas?

Question 4 of 5

Circle P has center \( (2, -1) \) and radius 7. Circle Q has center \( (2, -1) \) and radius 3. What single transformation maps Q onto P?

Question 5 of 5

Why is \( \pi \) the same constant for every circle?