BYTETOOLS

Annulus Calculator

Work out the area of a ring from its two radii, or solve for a missing radius from a target area, with sector area, arc lengths and the sector centroid.

50.2655
Ring area (cm²)
2
Ring width (cm)
31.4159
Outer circumference (cm)
18.8496
Inner circumference (cm)

Where each number came from

  • Ring area = π(R² − r²) = 50.2655 cm²
  • Outer disc area = πR² = 78.5398 cm²
  • Inner hole area = πr² = 28.2743 cm²
  • Outer circumference = 2πR = 31.4159 cm
  • Inner circumference = 2πr = 18.8496 cm
  • Ring width = R − r = 2 cm
  • Mean radius = (R + r) / 2 = 4 cm

Solved directly from the two radii you entered.

Annular sector at 60°

  • Sector area = (θ/2)(R² − r²) = 8.3776 cm²
  • Outer arc = Rθ = 5.236 cm
  • Inner arc = rθ = 3.1416 cm
  • Sector perimeter = both arcs + two straight edges = 12.3776 cm
  • Centroid distance from the centre = 3.8993 cm

θ is the sector angle in radians. At 360° the sector area equals the whole ring area and the centroid sits exactly at the centre; at 0° there is no sector, so the centroid is shown as a dash.

What is the Annulus Calculator?

An annulus is the flat ring left when you cut a smaller circle out of a bigger one — a washer, a pipe cross-section, a running track lane.

  • Ring area, both circumferences, width and mean radius from any two inputs
  • Solves for the missing inner or outer radius from a target ring area
  • Annular sector area, inner and outer arc lengths and sector perimeter
  • Sector centroid distance for centre-of-mass work
  • Rejects an inner radius that is not smaller than the outer one, with a clear explanation
  • Scaled drawing with the chosen sector highlighted

How to use the Annulus Calculator

  1. 1

    Choose what you know: both radii, a target area with one radius, or the width and mean radius.

  2. 2

    Pick a unit and type the values into the numbered fields.

  3. 3

    Set the sector angle in degrees — leave it at 360 for the whole ring.

  4. 4

    Read the ring area, width and both circumferences in the summary tiles.

  5. 5

    Open the sector panel for arc lengths, sector perimeter and the centroid distance, then copy the results.

About the Annulus Calculator

An annulus is the flat ring left when you cut a smaller circle out of a bigger one — a washer, a pipe cross-section, a running track lane. The ByteTools Annulus Calculator gives you the ring area π(R² − r²), both circumferences, the ring width R − r and the mean radius, and it can run backwards to find whichever radius you are missing from a target area.

It also handles the annular sector, the pie slice of a ring. Enter an angle and the tool returns the sector area (θ/2)(R² − r²), the inner and outer arc lengths, the full sector perimeter and the distance from the centre to the sector's centroid, which is the figure you need for balance and centre-of-mass work.

If you enter an inner radius that is not smaller than the outer one, the tool says so plainly rather than quietly returning a negative area. Everything is computed in your browser — no uploads, no tracking, and it works offline.

Frequently asked questions

What is the formula for the area of an annulus?

Area = π(R² − r²), where R is the outer radius and r is the inner one. For a ring with a 5 cm outer radius and a 3 cm inner radius that is π × 16 = 50.27 cm². It is simply the big disc minus the hole, which is why the two radii must be squared before subtracting, not subtracted first.

How do you find the inner radius of a ring from its area?

Rearrange the formula to r = √(R² − area ÷ π). Switch the mode dropdown to Find the inner radius from a target area, enter the outer radius and the area you want, and the tool solves it — and tells you if the ring simply will not fit inside that outer circle.

What is an annular sector?

It is a slice of the ring between two radii, like a curved piece of a washer. Its area is (θ ÷ 2)(R² − r²) with θ in radians, and its perimeter is the two arcs plus the two straight edges. The calculator gives all of these for whatever angle you set.

What is the difference between an annulus and a circle?

A circle is a solid disc; an annulus has a hole in the middle, so it is bounded by two concentric circles. That is why an annulus has two circumferences, an inner one and an outer one, and a ring width equal to the difference of the radii.

How do I work out the cross-sectional area of a pipe?

Use the outer radius of the pipe and the inner radius of the bore. The metal cross-section is the annulus area π(R² − r²), while the bore area available for flow is just πr². Enter the two radii and the tool reports both, along with the wall thickness as the ring width.

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