Circle with Hole Area Calculator & Formula

This calculator works out the area of a circular ring (a circle with a circular hole in the middle) using the outer radius and the inner radius. Enter the two radii in any consistent unit (meters, feet, inches, etc.), and you get the remaining area in square units of that same unit. The result is the area of the material or region between the outer and inner circles.

Formula

The area of a circle with a hole is the area of the outer circle minus the area of the inner circle. In symbols:

A = π × (R² − r²)

• A – area of the ring (the shaded region between the two circles), in square units.
• R – radius of the outer circle, in length units (m, cm, ft, in, etc.).
• r – radius of the inner hole, in the same length units.
• Ï€ (pi) – approximately 3.14159.

The formula is exact for any two concentric circles (circles sharing the same center). If the hole is not centered, the area is still the same, as long as the hole is fully inside the outer circle – the area depends only on the two radii, not on the hole’s position.

Worked Example

Using the default values: outer radius R = 5 m, inner radius r = 2 m.

1. Square the outer radius: R² = 5² = 25 m².
2. Square the inner radius: r² = 2² = 4 m².
3. Subtract: 25 − 4 = 21 m².
4. Multiply by π: 21 × 3.14159 ≈ 65.97 m².

The calculator shows 65.97 m² (rounded to two decimals). The outer circle area is π × 25 ≈ 78.54 m², and the inner hole area is π × 4 ≈ 12.57 m²; subtracting gives the same 65.97 m².

When to Use This Calculation

You need the area of a circular ring whenever you are dealing with a flat, round object with a round hole in the middle. Common real-world cases include:

• Washers and gaskets – the metal or rubber material between the outer edge and the bolt hole.
• Donut-shaped surfaces – the top of a donut, a round table with a hole for an umbrella, or a ring-shaped cake pan.
• Landscaping or paving – a circular flower bed around a tree, or a round walkway around a fountain.
• Material estimation – how much fabric, sheet metal, or plastic you need to cut a ring of given radii.
• Physics and engineering – the cross-sectional area of a pipe (outer circle minus inner hole) for fluid flow or stress calculations.

In every case, the answer is in square units of whatever length unit you used. If you enter radii in meters, the area is in square meters; if in inches, the area is in square inches.

How to Read the Result

The main result (ring area) is the actual surface area of the ring-shaped region. The two additional outputs – outer circle area and inner circle area – are provided so you can verify the subtraction and see the individual components. For example, a washer with outer radius 5 cm and inner radius 2 cm has a contact area of about 65.97 cm². That is the area that would touch a surface or that would be covered by paint on one side.

If you need the area in a different unit (e.g., you entered inches but want square feet), convert the radii first, then compute, or convert the result using the square of the length conversion factor (1 ft = 12 in, so 1 ft² = 144 in²). The calculator does not perform unit conversions – it assumes the two radii are in the same unit.

Common Mistakes and How to Avoid Them

• Using diameters instead of radii – the most frequent error. The formula uses radius (half the diameter). If you have the diameter, divide by 2 first. For a hole, the inner radius is half the hole’s diameter.
• Mixing units – entering outer radius in meters and inner radius in centimeters will give a wrong answer. Convert both to the same unit before entering.
• Forgetting to square before subtracting – the formula is Ï€(R² − r²), not Ï€(R − r)². The latter gives a much smaller number and is a different shape (a thin ring only if R−r is small).
• Rounding too early – if you round Ï€ to 3.14, the result will be slightly off. Use Ï€ to at least 3.14159, or the calculator’s built-in value, for accuracy.
• Negative inner radius – if the inner radius is larger than the outer radius, the ring area is negative, which is physically impossible. The calculator allows it, but the result is meaningless – always ensure r < R.

Limitations of the Formula

The formula assumes the hole is perfectly circular and the ring is flat. Real objects may have irregularities – an oval hole, a rough edge, or a hole that is not centered – but as long as the hole is circular, the area is independent of position. If the hole is not circular, you cannot use this simple formula; you would need to integrate or use a different geometric formula (e.g., for an elliptical hole).

Also, the formula gives the area of the ring in a plane. For a three-dimensional object like a pipe, this is the cross-sectional area (the area of the end face), not the surface area of the pipe’s walls. For a pipe’s internal volume, you would multiply the cross-sectional area by the length.

For very large radii, the calculator’s floating-point arithmetic may produce tiny rounding errors, but for practical purposes the result is accurate to the displayed decimals.

Frequently Asked Questions

Can I use this calculator for a pipe?

Yes, if you want the cross-sectional area of the pipe’s wall (the area of the ring you see when you look at the end of the pipe). Enter the outer radius (outside diameter divided by 2) and the inner radius (inside diameter divided by 2). The result is the wall’s cross-sectional area, which is useful for stress or flow calculations.

What if the hole is not in the center?

The area of the ring does not depend on the hole’s position. As long as the hole is a circle fully inside the outer circle, the area is still π(R² − r²). The hole’s location only affects the shape’s symmetry, not its area.

How do I find the area if I only have the diameters?

Divide each diameter by 2 to get the radius, then use the formula. For example, an outer diameter of 10 cm gives R = 5 cm; an inner diameter of 4 cm gives r = 2 cm. Then A = π(5² − 2²) ≈ 65.97 cm².

What units will the area be in?

The area is in square units of the length unit you enter. If you enter radii in meters, the area is in square meters; if in feet, square feet; if in inches, square inches. The calculator does not convert units, so keep both radii in the same unit.

Why is the area not π(R − r)²?

Because π(R − r)² is the area of a circle with radius (R − r), which is much smaller than the ring area. The ring area is the difference between two full circles, so you must subtract the squares of the radii, not the square of the difference. For R=5 and r=2, π(R−r)² = π×9 ≈ 28.27, while the correct ring area is π×21 ≈ 65.97.

Can the inner radius be zero?

Yes. If r = 0, the hole has zero radius, meaning there is no hole, and the formula gives the area of the full circle: πR². The calculator handles this correctly.

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in R | Outer radius | m | 5 | 0
in r | Inner radius | m | 2 | 0
out A | Ring area | m² | pi * (R^2 - r^2) | 2
out A_outer | Outer circle area | m² | pi * R^2 | 2
out A_inner | Inner circle area | m² | pi * r^2 | 2