What Are Concentric Circles?
Concentric circles are two or more circles that lie in the same plane and share the same centre, while each has a different radius. The word breaks into con- (together) and centre, so it literally means "sharing a centre." A dartboard, a set of ripples spreading from a dropped pebble, and the grooves on a vinyl record are all everyday concentric circles.
Two rules follow straight from the definition. First, concentric circles never intersect, because at every point the outer circle is farther from the shared centre than the inner one. Second, the gap between any two of them is constant all the way around, and it equals the difference of their radii, $R - r$.
For Example: Count the rings inside a tree stump and you can read the year a forest fire struck.
Each ring is a circle, every ring shares the same centre, and no two rings ever cross. Scientists date droughts, fires, and floods by reading those nested rings, because their common centre and steadily growing radii record one year at a time. That pattern, circles with a shared centre and different sizes, is exactly what geometry calls concentric circles.
How Are Concentric Circles Different From Congruent Circles?
This is the single most-asked question about the topic, and the two words are easy to swap by accident. The difference is which quantity is shared.
Concentric circles share the same centre but have different radii. They look like a bullseye, nested inside one another.
Congruent circles share the same radius but have different centres. They look like identical twins that could be slid on top of each other and match exactly.
So concentric circles are almost never congruent (their radii differ), and congruent circles are almost never concentric (their centres differ). The only way a pair could be both is if they were the same circle, which is not usually counted as a genuine pair. If two circles have the same radius, they are congruent; if they share a centre, they are concentric.
How Do You Find the Area Between Two Concentric Circles?
The ring-shaped region trapped between two concentric circles has its own name, the annulus. To find its area, take the area of the larger circle and subtract the area of the smaller one.
Let $R$ be the radius of the outer circle and $r$ the radius of the inner circle. Then:
$$\text{Annulus area} = \pi R^2 - \pi r^2 = \pi\left(R^2 - r^2\right).$$
Variable glossary: $R$ is the outer radius, $r$ is the inner radius, and $\pi \approx 3.14159$. The width of the ring is $R - r$, but that width is not what goes inside the formula, a point worth holding onto for the mistakes section. The full ring geometry is covered under the annulus topic.
What Are the Properties of Concentric Circles?
The shared centre forces a short, reliable list of properties.
Common centre, different radii. Every circle in the family passes the same point at its middle.
They never meet. No two concentric circles intersect or are tangent.
Constant separation. The distance between two of them is $R - r$ at every point.
Same equation form, different constant. In coordinate geometry they all read $x^2 + y^2 = k$ for different positive values of $k$ (centre at the origin).
A common tangent line touches only the outer boundary at a point, and any chord of the larger circle that is tangent to the smaller one is bisected at the point of contact.
What Is the Equation of Concentric Circles?
Because concentric circles share one centre, their equations all carry the same centre coordinates and differ only in the radius. For a common centre $(h, k)$, every circle in the family reads
$$(x - h)^2 + (y - k)^2 = r^2,$$
where $r$ takes a different value for each circle. When the shared centre is the origin, $h = 0$ and $k = 0$, so the family collapses to the simpler
$$x^2 + y^2 = r^2.$$
Variable key:
Symbol | Meaning |
|---|---|
$(h, k)$ | the common centre shared by every circle |
$r$ | the radius, which changes from one circle to the next |
$x, y$ | coordinates of any point on a given circle |
The quickest way to spot concentric circles from their equations is this: the $(h, k)$ part is identical and only the right-hand constant changes. For example, $(x - 2)^2 + (y - 1)^2 = 9$ and $(x - 2)^2 + (y - 1)^2 = 25$ are concentric, both centred at $(2, 1)$ with radii 3 and 5.
Examples of Concentric Circles
The examples build from a direct area subtraction to a coordinate-geometry check.
Example 1
Two concentric circles have radii $r = 3$ cm and $R = 5$ cm. Find the area of the annulus.
$$\text{Area} = \pi(R^2 - r^2) = \pi(5^2 - 3^2) = \pi(25 - 9) = 16\pi \approx 50.27 \text{ cm}^2.$$
Final answer: $16\pi \text{ cm}^2$, about 50.27 cm².
Example 2
Two concentric circles have radii 8 cm and 5 cm. A student writes the ring area as $\pi(8 - 5)^2$. What went wrong, and what is the correct area?
Watch the wrong path first. The student squared the difference of the radii, computing $\pi(3)^2 = 9\pi$. That treats the ring as if it were a small circle of radius 3, which it is not.
The ring is the big disc minus the small disc, so each radius must be squared before subtracting.
$$\text{Area} = \pi(R^2 - r^2) = \pi(8^2 - 5^2) = \pi(64 - 25) = 39\pi \approx 122.52 \text{ cm}^2.$$
The wrong method gives $9\pi$; the right method gives $39\pi$, more than four times larger.
Final answer: $39\pi \text{ cm}^2$, about 122.52 cm².
Example 3
An annulus has area $33\pi \text{ cm}^2$ and its outer radius is 7 cm. Find the inner radius.
Start from the formula and solve for $r$.
$$\pi(R^2 - r^2) = 33\pi$$
$$49 - r^2 = 33$$
$$r^2 = 16$$
$$r = 4.$$
Final answer: the inner radius is 4 cm.
Example 4
A circular running track has an inner edge of radius 30 m and is 4 m wide. What area does the track surface cover?
The outer radius is $30 + 4 = 34$ m.
$$\text{Area} = \pi(34^2 - 30^2) = \pi(1156 - 900) = 256\pi \approx 804.25 \text{ m}^2.$$
Final answer: about 804.25 m² of track surface.
Example 5
Write the equations of two concentric circles centred at the origin with radii 3 and 5.
Circles centred at the origin follow $x^2 + y^2 = (\text{radius})^2$.
$$x^2 + y^2 = 9 \qquad \text{and} \qquad x^2 + y^2 = 25.$$
Same left side, different constant, which is the coordinate signature of concentric circles. For the general form, see the equation of a circle.
Final answer: $x^2 + y^2 = 9$ and $x^2 + y^2 = 25$.
Example 6
Are two concentric circles ever congruent? Two circles have the same centre; one has radius 6 and the other radius 6. What are they?
Concentric circles must have different radii by definition. If two circles share a centre and have the same radius 6, they are not two circles at all, they are the same circle drawn twice.
Final answer: identical radius plus identical centre means one single circle, not a concentric pair.
Where Do Concentric Circles Appear in Real Life?
Concentric circles matter because they model any process that spreads outward evenly from one source. A first-time learner often pictures them as decorative, but their real value is measurement.
Tree-ring dating reads nested annual rings to fix the year of a fire or drought, each ring a concentric circle around the pith. The method, dendrochronology, depends on the rings never crossing.
Radar and sonar display range as concentric circles around the sensor, so distance reads directly off the ring.
Archery and darts score by concentric rings, where the annulus area between rings sets how hard each band is to hit.
Ripples in water expand as concentric circles, spreading energy evenly from the point of impact.
The point is not the picture. It is that a shared centre plus a known radius lets you turn "how far out" into an exact number.
What Are the Most Common Mistakes With Concentric Circles?
Two mistakes account for most lost marks, and both come from rushing the formula.
Mistake 1: Squaring the difference of the radii
Where it slips in: annulus area problems where the ring looks thin.
Don't do this: compute $\pi(R - r)^2$. The exact first-instinct error is treating the ring width as if it were a radius.
The correct way: square each radius first, then subtract: $\pi(R^2 - r^2)$. The width $R - r$ never gets squared on its own.
Mistake 2: Confusing concentric with congruent
Where it slips in: definition questions and true-or-false items.
Don't do this: call same-size circles concentric. Same size means congruent, not concentric.
The correct way: ask which quantity is shared. Shared centre is concentric; shared radius is congruent.
Conclusion
Concentric circles share one centre and have different radii, so they never intersect.
The ring between them is an annulus, with area $\pi(R^2 - r^2)$.
Concentric means shared centre; congruent means shared radius, and the two are almost never the same.
In coordinate form they read $x^2 + y^2 = k$ for different values of $k$.
The gap between two concentric circles is the constant $R - r$.
To go deeper on circle geometry with a teacher, explore Bhanzu's geometry tutor or a middle school math tutor, or browse math classes online.
A Practical Next Step
Work through the exercises below to lock in the idea: draw three concentric circles with a compass by keeping the point fixed and changing only the radius, then compute the area of each ring between neighbours. Next, find the width of an annulus whose area is $20\pi$ and whose outer radius is 6. If the squaring step trips you, return to the annulus formula above. Want a live Bhanzu trainer to guide you through circle problems? Book a free demo class.
Read More
Circumference of a circle — measuring the distance around each ring.
Parts of a circle — centre, radius, chord, and arc defined.
Sector of a circle — the pie-slice region of a single circle.
Chord of a circle — the tangent-chord property mentioned above.
Diameter of a circle — the longest chord through the centre.
Semicircle — half a circle and its area.
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