- Concentric circles share a single common centre but have different radii, and the ring-shaped region between any two of them is called an annulus
- For a circle x2 + y2 + 2gx + 2fy + c = 0, every concentric circle keeps the same centre and changes only the constant term c
- The area of an annulus equals pi multiplied by the difference of the squares of the outer and inner radii.
What Are Concentric Circles?
Concentric circles are two or more coplanar circles that share the same centre point but have different radii. Because the radii differ, the circles never intersect; the ring-shaped region enclosed between any two of them is called an annulus. Their defining condition is a common centre with unequal radii.
Circles that share the same centre are known as concentric circles. They fit one inside another and stay an equal distance apart all the way around. Because their radii differ, the region between two concentric circles is called the annulus. The condition for two circles to be concentric is that they share a centre but must not have the same radius.
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For a circle written as x2 + y2 + 2gx + 2fy + c = 0, the centre is (-g, -f) and the radius is the square root of g squared plus f squared minus c. Any concentric circle keeps the same g and f, changing only the constant term c.
Consider a circle with centre (-g, -f) whose general equation is x2 + y2 + 2gx + 2fy + c = 0.
A circle that is concentric with it has the same centre, so only the constant term changes: x2 + y2 + 2gx + 2fy + c1 = 0, where c is not equal to c1. This different constant is the necessary condition for the two circles to be concentric, because both equations still share the common centre (-g, -f).
Using the centre-radius form, a circle with centre (h, k) and radius r is written as (x - h)2 + (y - k)2 = r2. A circle concentric with it keeps the same centre and uses a new radius r1, giving (x - h)2 + (y - k)2 = (r1)2, where r is not equal to r1.
By assigning different radius values in these equations, we obtain a whole family of concentric circles around the same centre.
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To build a concentric circle, keep the original centre and choose a new radius. Take x2 + y2 + 4x - 8y - 6 = 0, whose centre is (-2, 4) and radius is the square root of 26. Doubling the radius gives x2 + y2 + 4x - 8y - 84 = 0, sharing the same centre.
Let us work through an example. Determine the equation of the circle that is concentric with x2 + y2 + 4x - 8y - 6 = 0, given that its radius is double that of this circle.
Comparing with the general form x2 + y2 + 2gx + 2fy + c = 0, here g = 2, f = -4 and c = -6, so the centre of the given circle is (-2, 4).
Its radius is r = square root of (g squared + f squared - c) = square root of (4 + 16 + 6) = square root of 26.
Let R be the radius of the concentric circle. Since R is double the original radius, R = 2r = 2 times the square root of 26.
The concentric circle keeps the same centre (-2, 4), so (x + 2)2 + (y - 4)2 = (2 times the square root of 26) squared = 4 times 26 = 104.
Expanding gives x2 + 4x + 4 + y2 - 8y + 16 = 104, which simplifies to x2 + y2 + 4x - 8y + 20 = 104.
Therefore the required equation of the concentric circle is x2 + y2 + 4x - 8y - 84 = 0.
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Concentric circles appear widely in everyday life. Ripples spreading outward when an object is dropped into water form expanding concentric rings around the point of impact. A coaxial cable has a central core surrounded by cylindrical layers, and target-rifle diopter sights use aligned concentric rings to centre the aim point.
- Ripples: when an object is dropped into still water, it creates an expanding system of concentric circles called ripples, centred on the point where the object landed.
- Coaxial cable: the live core is surrounded by neutral and earth cores forming cylindrical shells, a clear everyday example of concentric circles.
- Diopter sights: these target-rifle sights show concentric rings, and when they are aligned the point of impact sits exactly in the centre of the front sight circle.
What Is the Difference Between Concentric and Eccentric Circles?
The difference lies in the centre. Concentric circles share a single common centre but have different radii, so they never intersect and enclose an annulus. Eccentric circles, by contrast, have different centres; one circle sits off-centre inside or beside another. In short, concentric means same centre, eccentric means off-centre.
People often confuse these two terms, so it helps to define an eccentric circle. Eccentric circles are circles that do not share a common centre; one lies off-centre relative to another, and they may even touch or intersect. Concentric circles, by contrast, always share the same centre and never meet.