Find The Shaded Area Calculator

Calculate the shaded area between two concentric circles (an annulus or ring) using our simple shaded area calculator. Enter the outer and inner radii to find the area of the ring.

Calculate Shaded Area (Annulus)

What is a Shaded Area Calculator (for an Annulus)?

A shaded area calculator for an annulus (or ring) is a tool used to determine the area of the region between two concentric circles (circles that share the same center). This shaded region looks like a flat ring or a washer. You input the radius of the larger, outer circle (R) and the radius of the smaller, inner circle (r), and the calculator finds the area of the space between them.

This type of calculator is useful in various fields, including geometry, engineering, design, and manufacturing, where you might need to calculate the surface area of a ring-shaped object or the difference in area between two circles. The shaded area calculator simplifies this by applying the correct geometric formula.

Who should use it?

• Students learning about the areas of circles and composite shapes.
• Engineers and designers working with ring-shaped components.
• Anyone needing to find the area between two concentric circles quickly.

Common Misconceptions:

• It's not just the difference in radii: The area depends on the difference of the squares of the radii, multiplied by π.
• The circles must be concentric: For the simple annulus formula, the circles must share the same center point. If they are not concentric but overlap, the calculation is more complex. Our shaded area calculator assumes concentric circles.

Shaded Area (Annulus) Formula and Mathematical Explanation

The shaded area of an annulus is the difference between the area of the outer circle and the area of the inner circle.

The area of any circle is given by the formula A = πr², where r is the radius.

1. Area of the Outer Circle (A_outer) = πR² (where R is the outer radius)
2. Area of the Inner Circle (A_inner) = πr² (where r is the inner radius)
3. Shaded Area (A_annulus) = A_outer – A_inner = πR² – πr²
4. This can be simplified to: A_annulus = π(R² – r²)

This formula is what our shaded area calculator uses.

Variables Table

Variable	Meaning	Unit	Typical Range
R	Outer Radius	Length units (e.g., cm, m, inches)	R ≥ r, R > 0
r	Inner Radius	Length units (e.g., cm, m, inches)	0 ≤ r ≤ R
Aannulus	Shaded Area (Annulus)	Area units (e.g., cm², m², inches²)	≥ 0
π (Pi)	Mathematical constant	Dimensionless	~3.14159

Explore our circle area calculator for more details on basic circle calculations.

Practical Examples (Real-World Use Cases)

Example 1: Garden Pathway

Imagine a circular garden with a radius of 3 meters, and you want to build a 1-meter wide pathway around it. The garden and pathway together form a larger circle.

• Inner Radius (r) = 3 m (radius of the garden)
• Outer Radius (R) = 3 m + 1 m = 4 m (radius of garden + pathway)

Using the shaded area calculator or formula π(4² – 3²) = π(16 – 9) = 7π ≈ 21.99 square meters. The area of the pathway is about 21.99 m².

Example 2: Washer Dimensions

A metal washer has an outer diameter of 20 mm and an inner diameter (hole) of 10 mm.

• Outer Radius (R) = 20 mm / 2 = 10 mm
• Inner Radius (r) = 10 mm / 2 = 5 mm

The area of the washer's surface is π(10² – 5²) = π(100 – 25) = 75π ≈ 235.62 square millimeters. This is useful for material calculations.

How to Use This Shaded Area Calculator

1. Enter Outer Radius (R): Input the radius of the larger circle into the "Outer Radius (R)" field.
2. Enter Inner Radius (r): Input the radius of the smaller, inner circle into the "Inner Radius (r)" field. Ensure 'r' is less than or equal to 'R' and not negative.
3. Calculate: The calculator automatically updates the results as you type. If not, click the "Calculate Area" button.
4. View Results: The "Primary Result" shows the calculated shaded area (annulus). You'll also see the areas of the outer and inner circles and the difference of their radii squared as intermediate values.
5. Visualize: The diagram below the results dynamically updates to show a representation of your input radii and the resulting shaded area.
6. Reset: Click "Reset" to clear the fields and go back to default values.
7. Copy: Click "Copy Results" to copy the calculated values to your clipboard.

Understanding the annulus area formula is key to using this calculator effectively.

Key Factors That Affect Shaded Area (Annulus) Results

• Outer Radius (R): As the outer radius increases (while the inner radius stays the same), the shaded area increases significantly because the area is proportional to the square of the radius.
• Inner Radius (r): As the inner radius increases (approaching the outer radius), the shaded area decreases, becoming zero when r = R.
• Difference between R and r: While the difference (R-r) gives the width of the ring, the area depends more on (R² – r²), which is (R-r)(R+r). So, even with the same width, rings with larger average radii have larger areas.
• Units Used: Ensure both radii are in the same units. The resulting area will be in the square of those units (e.g., if radii are in cm, the area is in cm²). Our shaded area calculator assumes consistent units.
• Concentricity: The formula π(R² – r²) strictly applies to concentric circles. If the circles are not concentric, the overlapping area calculation is different and more complex.
• Measurement Accuracy: The precision of the calculated area depends directly on the accuracy of your input radii measurements. Small errors in radii can lead to larger errors in area, especially for large radii.

For areas of other shapes, check our square area calculator or rectangle area calculator.

Frequently Asked Questions (FAQ)

What is an annulus?

An annulus is the region between two concentric circles. It's shaped like a ring or a washer.

What if the inner radius is zero?

If the inner radius (r) is 0, the shaded area becomes the area of the outer circle (πR²), as there is no hole. Our shaded area calculator handles this.

What if the inner radius is equal to the outer radius?

If r = R, the shaded area is zero because the inner circle completely fills the outer circle.

Can I use diameter instead of radius?

Yes, but you must divide the diameters by 2 to get the radii before using the calculator or the formula. Radius = Diameter / 2.

What if the circles are not concentric?

If the circles are not concentric but overlap, the area of the overlapping region (or the non-overlapping parts) requires a different, more complex formula involving the distance between the centers of the circles. This shaded area calculator is for concentric circles only.

In what units is the area calculated?

The area will be in the square of the units you used for the radii. If you entered radii in centimeters (cm), the area will be in square centimeters (cm²).

How accurate is the pi (π) value used?

The calculator uses the `Math.PI` constant in JavaScript, which provides a high-precision value of π for accurate calculations.

Can I calculate the area of a sector of an annulus?

Yes, to find the area of a sector of an annulus (a portion of the ring defined by an angle), you first find the total area of the annulus and then multiply it by the ratio of the sector angle (θ, in degrees) to 360: Area_sector = π(R² – r²) * (θ / 360).

Learn more about calculating ring area with different parameters.

Related Tools and Internal Resources

• Circle Area Calculator: Calculate the area of a single circle.
• Square Area Calculator: Find the area of a square given its side.
• Rectangle Area Calculator: Calculate the area of a rectangle.
• Triangle Area Calculator: Find the area of various types of triangles.
• Cone Volume Calculator: Calculate the volume of a cone.
• Cylinder Volume Calculator: Find the volume of a cylinder.