Concentric Circle Area Calculator and Formula

A concentric circle area calculator finds the area of the ring-shaped region between two circles that share the same center. Enter the outer and inner radii in the same unit; the calculator returns the annulus area and the area of each circle.

Formula

The annulus area is the area of the outer circle minus the area of the inner circle:

A = π(R² − r²)

• A is the area of the ring-shaped region, in square units.
• R is the outer radius, in units of length.
• r is the inner radius, in the same length units as R.
• π (pi) is approximately 3.14159265.

The calculator labels the two radii “Outer radius” and “Inner radius.” Its default values are 5 and 3 units. The inner radius must not be larger than the outer radius. The area units are squared: for radii in inches, the result is square inches; for radii in centimeters, it is square centimeters.

Worked Example

Using the calculator’s default outer radius of 5 units and inner radius of 3 units:

1. Square each radius: 5² = 25 and 3² = 9.
2. Subtract the inner squared radius from the outer: 25 − 9 = 16.
3. Multiply by pi: A = π × 16 = 50.265… square units.
4. Rounded to two decimal places, the annulus area is 50.27 square units.

For context, the outer circle’s area is π × 25 = 78.54 square units, and the inner circle’s area is π × 9 = 28.27 square units. Their rounded difference is 50.27 square units.

What Concentric Circles Mean

Concentric circles have a common center but different radii. The space between their circumferences is called an annulus. A washer, a circular track around a central opening, and the cross-section of a hollow pipe are familiar examples of this geometry.

“Concentric” describes the circles’ centers, not their sizes. The inner circle can be much smaller than the outer one, and the gap between their edges can vary. If the circles do not share a center, this formula does not generally give the area of the region between them: the overlap and uncovered portions depend on the distance between their centers.

How to Use the Calculator

1. Measure or identify the outer radius, from the common center to the outer edge.
2. Enter the inner radius, from that same center to the inner edge.
3. Use the same length unit for both entries. The calculator uses that unit for the radii and its square for the areas.
4. Read the first result for the annulus area. The other results show the full outer-circle and inner-circle areas.

If your measurements are diameters rather than radii, divide each diameter by 2 before entering it. For example, an outer diameter of 10 cm and an inner diameter of 6 cm correspond to radii of 5 cm and 3 cm, giving an annulus area of 50.27 cm².

When the Result Is Useful

The annulus formula is useful whenever a flat circular region has a circular portion removed from its center. It can estimate the face area of a washer or gasket, the surface area of a circular border, or the amount of material in a flat ring, provided the measured boundaries are circular and centered.

Area is not the same as volume. For a flat sheet, the area may help determine how much surface is covered. To estimate the volume of a uniform ring-shaped solid, multiply the annulus area by its thickness or height: V = π(R² − r²)h. Here h must be a length in the same unit system, and the result is in cubic units. That extension assumes a constant cross-section and uniform thickness.

If you need the length around the ring’s boundaries rather than its surface area, calculate circumferences instead. The outer circumference is 2πR and the inner circumference is 2πr. Their sum is the total length of both circular edges; it is not the annulus area.

Common Measurement and Calculation Errors

• Mixing radii and diameters: A diameter is twice its radius. Using a diameter as if it were a radius makes the circle-area calculation four times too large.
• Using inconsistent units: Convert measurements before calculating. For instance, do not enter an outer radius in inches and an inner radius in centimeters as if they were the same unit.
• Subtracting radii instead of squared radii: The formula is π(R² − r²), not π(R − r)². A ring’s area depends on both squared radii.
• Rounding too early: Keep the measured values and intermediate calculations unrounded where possible, then round the final area to a precision appropriate for the measurements.
• Reversing the radii: The outer radius must be at least as large as the inner radius. If the input values are reversed, the result is negative and does not represent a physical ring area.

Limits and Measurement Accuracy

The formula gives an exact geometric area for two ideal circles with a shared center. Real objects may be out of round, off-center, worn, or measured imprecisely. In those cases, the calculator’s result is an estimate based on the supplied radii, not a correction for irregular edges or misalignment.

Measurement precision matters because radii are squared. A small radius error can noticeably affect the result, especially for a thin ring where the inner and outer radii are close. If high precision is needed, measure both radii from the same center and record the measurement uncertainty. Keep the units explicit when reporting the result, such as 50.27 cm² or 7.79 in², rather than simply writing “50.27.”

Frequently Asked Questions

What is the formula for the area between two concentric circles?
Subtract the inner circle’s area from the outer circle’s area: A = π(R² − r²), where R is the outer radius and r is the inner radius.

Can I use diameters instead of radii?
Yes, but convert each diameter to a radius by dividing it by 2 first. Equivalently, with outer and inner diameters D and d, the area is π(D² − d²) ÷ 4.

What if the two circles have different centers?
They are not concentric, so this annulus formula does not describe the region between their boundaries. The area of their overlap or of other regions formed by them requires the distance between the centers and a different geometric calculation.

What units does the calculator use for area?
It returns square versions of the units used for the radii. Radii in millimeters give an area in mm²; radii in feet give an area in ft². Convert both radii to one unit before calculating.

How do I calculate the area of a thick circular ring?
Use the outer radius and the radius of the opening in A = π(R² − r²). The ring’s radial thickness is R − r, but thickness alone is not enough to determine area without a radius.

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in r | Outer radius | units | 5 | 0
in a | Inner radius | units | 3 | 0
out ring | Annulus area | square units | pi * (r ^ 2 - a ^ 2) | 2
out outer | Outer circle area | square units | pi * r ^ 2 | 2
out inner | Inner circle area | square units | pi * a ^ 2 | 2