YourHomeschool YourHomeschool Curriculum By state Workbooks Open the Tutor
Curriculum › Grade 10 › Mathematics

All Circles Are Similar

Aligned to G-C.A.1 — Common Core State Standards for Mathematics.

What this lesson teaches

Any two circles are similar because a dilation can scale one exactly onto the other. Take the two centers; translate so they coincide, then dilate by the ratio of the radii. Since a circle is determined entirely by its center and radius, matching them makes the circles coincide.

Worked example

Prove circle with radius 3 is similar to circle with radius 7. First translate the small circle so its center sits on the big circle's center. Then dilate about that shared center with scale factor 7/3. Every point 3 units from the center moves to 7 units from the center, so the small circle maps exactly onto the big one. A rigid motion plus a dilation exists, so the circles are similar.

Practice questions

  1. What scale factor maps a circle of radius 4 onto a circle of radius 10?
  2. Explain why you never need to check angles to prove two circles are similar.
  3. Two circles have radii 5 and 5 but different centers. What single transformation maps one onto the other?

Watch the lesson

Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.

▶ Watch this lesson

En español

Todos los círculos son semejantes

Dos círculos cualesquiera son semejantes porque una dilatación puede escalar uno exactamente sobre el otro. Toma los dos centros; traslada para que coincidan, luego dilata por la razón de los radios. Como un círculo queda determinado por su centro y su radio, hacerlos coincidir hace que los círculos coincidan.

Ejemplo: Demuestra que el círculo de radio 3 es semejante al de radio 7. Primero traslada el círculo pequeño para que su centro quede sobre el centro del grande. Luego dilata sobre ese centro compartido con factor de escala 7/3. Cada punto a 3 unidades del centro se mueve a 7 unidades del centro, así que el círculo pequeño se mapea exactamente sobre el grande. Existe un movimiento rígido más una dilatación, así que los círculos son semejantes.

More Circles lessons

Angles, Chords, Radii, and TangentsG-C.A.2Inscribed and Circumscribed CirclesG-C.A.3Constructing a Tangent from an External PointG-C.A.4Arc Length, Radians, and Sector AreaG-C.B.5

Teach this lesson today

An AI tutor that guides with questions instead of just giving answers — all K–12 core subjects, every child in the family.

Start free — no card required