Andromeda
Note

Relative Positions of Two Circles

Definition

The relative position of two circles in the same plane is determined by the distance between their centers (dd) relative to their radii (RR and rr).

  • How to read: “The d is the distance between centers; R and r are the radii.”
  • Meaning: Compare dd to sums and differences of radii to classify how two circles interact.

Why It Matters

In mechanical design, even a millimeter of error in the interaction of circular components leads to friction, wear, and system failure. It is the geometric foundation of precision in everything from gearboxes to satellite orbits.

Core Concepts

  • Line of Centers: The line joining the centers of two circles.

  • Tangent Externally:

    • The circles touch at exactly one point.
    • The distance between centers is equal to the sum of the radii: d=R+rd = R + r.

  • How to read: “The D equals R plus r.”

    • Meaning: Circles just touch from the outside; centers are one radius-length apart from each edge.

  • Tangent Internally:

    • One circle is inside the other and they touch at exactly one point.
    • The distance between centers is equal to the difference of the radii: d=R−rd = R - r.

  • How to read: “The D equals R minus r” (assuming R is greater than r).

    • Meaning: Smaller circle nestles inside larger; centers are ∣R−r∣|R - r| apart.

  • Overlapping (Intersecting):

    • The circles intersect at two points.
    • The distance between centers is less than the sum but greater than the difference of the radii: R−r<d<R+rR - r < d < R + r.

  • How to read: “The R minus r is less than d, which is less than R plus r.”

    • Meaning: Circles cross at two points—neither nested nor fully separate.

  • Outside Each Other:

    • The circles have no points in common and neither is inside the other.
    • The distance between centers is greater than the sum of the radii: d>R+rd > R + r.

  • How to read: “The D is greater than R plus r.”

    • Meaning: Complete separation with a gap between circumferences.

  • One Inside the Other (Non-Tangent):

    • One circle is inside the other with no points in common.
    • The distance between centers is less than the difference of the radii: d<R−rd < R - r.

  • How to read: “The D is less than R minus r.”

    • Meaning: One circle fully contains the other without touching.
      • Concentric Circles: A special case where d=0d = 0.

  • How to read: “The D equals zero.” - Meaning: Same center; nested circles sharing a center point.

Connected Concepts

Connected notes