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Annulus Area Calculator

Calculate the area of an annulus (ring shape) using A = PI*(R^2 - r^2), where R is the outer radius and r is the inner radius. Essential for engineering washers, pipe cross-sections, circular tracks, orbital mechanics, and ring-shaped design elements.

Reviewed by Christopher FloiedUpdated

This free online annulus area calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Radius of the outer circle

Radius of the inner circle (must be less than outer radius)

Results

Annulus Area

201.0619 sq units

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Annulus Area Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Annulus Area Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Annulus Area Calculator Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

  • •Use the Annulus Area Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

About This Calculator

The Annulus Area Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate the area of an annulus (ring shape) using A = PI*(R^2 - r^2), where R is the outer radius and r is the inner radius. Essential for engineering washers, pipe cross-sections, circular tracks, orbital mechanics, and ring-shaped design elements. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.

About Annulus Area Calculator

The Annulus Area Calculator computes the area of the ring-shaped region between two concentric circles using A = PI*(R^2 - r^2). An annulus is formed when a smaller circle is removed from the center of a larger circle. This shape appears constantly in everyday life and engineering: washers, gaskets, pipe cross-sections, circular running tracks, CDs and DVDs, tree ring cross-sections, and orbital paths of satellites. The formula subtracts the inner circle area from the outer circle area. The factored form PI*(R+r)*(R-r) is useful for computations and reveals that the annulus area depends on both the sum and difference of the radii.

The Math Behind It

The annulus area formula A = pi*(R^2 - r^2) is derived by subtracting the area of the inner circle (pi*r^2) from the area of the outer circle (pi*R^2). Factoring using the difference of squares: A = pi*(R+r)*(R-r). This factored form shows that the area depends on the average radius (R+r)/2 and the width (R-r). An interesting property: the area of an annulus can be determined by a single chord of the outer circle that is tangent to the inner circle. If this chord has length L, then the annulus area is pi*L^2/4, regardless of the individual radii. This surprising result follows from the Pythagorean theorem: L^2 = 4(R^2 - r^2), so pi*L^2/4 = pi*(R^2 - r^2). The centroid of an annulus lies at its center. The second moment of area (moment of inertia) about the center is I = pi*(R^4 - r^4)/4, which is crucial in structural engineering for analyzing hollow circular columns and shafts. Hollow shafts are more efficient than solid ones because they resist torsion with less material. In fluid mechanics, flow through an annular pipe (between an outer tube and an inner rod) follows the annular Poiseuille flow profile. The flow rate depends on R^4 - r^4 and R^2 - r^2 in specific combinations. In thermodynamics, heat conduction through a cylindrical shell involves annular cross-sections. In orbital mechanics, the area swept by a planet's orbit between two distances from the sun forms an annular sector.

Formula Reference

Annulus Area

A = pi * (R^2 - r^2)

Variables: R = outer radius, r = inner radius

Alternative (factored)

A = pi * (R + r) * (R - r)

Variables: Uses difference of squares factorization

Worked Examples

Example 1: Circular Running Track

A running track has inner radius 30 m and outer radius 35 m. Find the track area.

Step 1:R = 35 m, r = 30 m
Step 2:A = pi * (35^2 - 30^2) = pi * (1225 - 900) = pi * 325 = 1021.02

The track area is approximately 1021.02 square meters.

Example 2: Washer

A flat washer has outer radius 10 mm and inner radius 6 mm.

Step 1:R = 10 mm, r = 6 mm
Step 2:A = pi * (100 - 36) = pi * 64 = 201.06

The washer area is approximately 201.06 square mm.

Common Mistakes & Tips

  • !Subtracting radii instead of squaring them first: pi*(R-r)^2 is NOT the annulus area. You must square each radius separately and then subtract.
  • !Swapping the inner and outer radii, which would give a negative result. R must be greater than r.
  • !Using diameter instead of radius. If given diameters, divide each by 2 before applying the formula.

Related Concepts

Semicircle Area

A semicircle is half a full circle. Annulus and semicircle calculations both derive from the circle area formula.

Torus Volume

A torus is a 3D shape created by rotating an annulus's cross-section around an axis, linking annular and toroidal geometry.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Semicircle Area Calculator

Torus Surface Area Calculator

Torus Volume Calculator

Frequently Asked Questions

Can I find the annulus area from a single tangent chord?

Yes. If a chord of the outer circle is tangent to the inner circle and has length L, then the annulus area is pi*L^2/4. This works because by the Pythagorean theorem, (L/2)^2 = R^2 - r^2, so pi*(R^2 - r^2) = pi*L^2/4.

Why are hollow shafts stronger per unit weight than solid ones?

The material far from the center contributes more to torsional and bending resistance than material near the center. A hollow shaft (annular cross-section) removes the less effective central material while keeping the more effective outer material, giving a better strength-to-weight ratio.

What if the inner radius is zero?

When r = 0, the annulus becomes a full circle with area pi*R^2. The formula handles this edge case correctly.