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Concentric

Concentric

Similar geometric figures that share a common center.

 

 

Three concentric circles sharing a common center point, each ring progressively larger than the one inside it.

Worked Example

Problem: Two concentric circles are centered at the origin. The inner circle has a radius of 3 and the outer circle has a radius of 5. Find the area of the region between them (the annulus).
Step 1: Find the area of the outer circle.
Aouter=π(5)2=25πA_{\text{outer}} = \pi(5)^2 = 25\pi
Step 2: Find the area of the inner circle.
Ainner=π(3)2=9πA_{\text{inner}} = \pi(3)^2 = 9\pi
Step 3: Subtract to find the area of the region between the two concentric circles.
Aannulus=25π−9π=16πA_{\text{annulus}} = 25\pi - 9\pi = 16\pi
Answer: The area between the two concentric circles is 16π≈50.2716\pi \approx 50.27 square units.

Why It Matters

Concentric shapes appear frequently in real life and in geometry problems. Dartboards, tree rings, and circular targets all consist of concentric circles. In coordinate geometry, concentric circles centered at the origin differ only in their radius, which simplifies equations and area calculations.

Common Mistakes

Mistake: Thinking concentric figures must be the same size.
Correction: Concentric figures share the same center but typically have different sizes. If they were the same size and shape, they would simply be identical, not just concentric.

Related Terms

  • Similar — Concentric figures are often similar shapes
  • Geometric Figure — The shapes that can be concentric
  • Annulus — Region between two concentric circles
  • Circle — Most common concentric figure
  • Radius — Distinguishes concentric circles from each other