Concentric Circles — Definition, Formula & Examples
Concentric circles are two or more circles that share the same center point but have different radii. Think of a bullseye target or the rings of a tree trunk — each ring is a separate circle, all centered at the same spot.
Two or more circles are said to be concentric if and only if they are coplanar and share a common center. Concentric circles never intersect, and the region bounded between any two of them is called an annulus.
Key Formula
Where:
- = Radius of the larger circle
- = Radius of the smaller circle
- = Area of the ring-shaped region between the two concentric circles
How It Works
To identify concentric circles, check whether the circles have the exact same center coordinates. The radii must be different — if two circles share the same center and the same radius, they are the same circle, not concentric circles. A common task involving concentric circles is finding the area of the ring-shaped region (annulus) between them. You subtract the area of the smaller circle from the area of the larger circle. Concentric circles appear in coordinate geometry problems where circles are written in standard form and you compare their center points.
Worked Example
Problem: Two concentric circles have radii of 10 cm and 6 cm. Find the area of the region between them.
Step 1: Identify the larger and smaller radii.
Step 2: Write the annulus area formula and substitute the values.
Step 3: Compute the squares and subtract.
Step 4: Approximate the result.
Answer: The area between the two concentric circles is cm².
Another Example
Problem: Circle A has center (3, −2) and radius 5. Circle B has center (3, −2) and radius 8. Are these circles concentric?
Step 1: Compare the centers of both circles.
Step 2: The centers are identical. Check whether the radii differ.
Step 3: Since the circles share the same center but have different radii, they are concentric.
Answer: Yes, Circle A and Circle B are concentric circles.
Why It Matters
Concentric circles show up frequently in middle-school and high-school geometry when you calculate shaded-region areas on tests. In real life, engineers and designers use them to model targets, washers, pipes, and circular tracks. Understanding them also prepares you for coordinate geometry in Algebra 2, where comparing circle equations by their centers and radii is a standard skill.
Common Mistakes
Mistake: Forgetting to subtract the smaller area and instead just computing the area of the larger circle.
Correction: The region between concentric circles is a ring (annulus). You must subtract the inner circle's area from the outer circle's area: .
Mistake: Subtracting the radii first, then squaring: .
Correction: This gives the wrong answer. You need to square each radius separately, then subtract: . For example, with and , the correct calculation gives , while the incorrect one gives .