Annulus perimeter calculator

What is an annulus perimeter calculator?

An annulus perimeter calculator finds the total length of the boundary of a ring-shaped region — the shape that remains when a smaller disk is removed from a larger one that shares the same center. The boundary of this region is made up of two concentric circles, so its perimeter is simply the sum of those two circumferences.

This calculator takes the outer radius and the inner radius of the ring and returns the combined length of both circles. You can enter the radii in any common unit of length, and the result is reported in the same family of units.

Key concepts

• Outer radius (R) — the distance from the center of the annulus to its outer edge.
• Inner radius (r) — the distance from the center to the inner edge (the hole).
• Annulus — the flat region between two concentric circles. It looks like a washer or a ring.
• Perimeter (P) — the total length of the closed boundary of a shape. For an annulus the boundary has two pieces: an outer circle and an inner circle.

How does the calculator work?

The perimeter of an annulus is the sum of the lengths of its two circular boundaries. Each circle contributes a circumference equal to $2\pi$ times its radius, so the two contributions can be combined into a single linear expression in the two radii.

Formula

$$P = 2\pi R + 2\pi r = 2\pi (R + r)$$

Where $R$ is the outer radius and $r$ is the inner radius. The formula reduces to $2\pi R$ when $r = 0$ (a full disk has only the outer circle as its boundary) and to $4\pi R$ when $r = R$ (a degenerate annulus whose two circles coincide).

Worked examples

Example 1: standard ring

A washer has an outer radius of 10 cm and an inner radius of 5 cm.

$$P = 2\pi (10 + 5) = 30\pi \approx 94.248 \text{ cm}$$

Example 2: thinner ring

For an outer radius of 7 cm and an inner radius of 3 cm:

$$P = 2\pi (7 + 3) = 20\pi \approx 62.832 \text{ cm}$$

Example 3: degenerate annulus

If both radii are equal — for instance $R = r = 5$ cm — the two circles coincide but the formula still gives a finite value:

$$P = 2\pi (5 + 5) = 20\pi \approx 62.832 \text{ cm}$$

This is the limit case where the ring has zero width but the boundary is counted twice.

Example 4: full disk

When the inner radius shrinks to zero, the annulus becomes a full circle and its perimeter reduces to the circumference of the outer circle:

$$P = 2\pi (R + 0) = 2\pi R$$

Example 5: invalid geometry

If the inner radius is greater than the outer radius, the shape is not a real annulus and no perimeter is reported. For example, $R = 3$ cm and $r = 7$ cm has no solution because the inner circle cannot lie outside the outer circle.

Practical uses

• Engineering and manufacturing — estimating the cutting length needed to machine washers, gaskets, or flat ring-shaped parts.
• Construction — finding the length of edging required to trim a circular flowerbed with a path or a fountain in its center.
• Design and crafts — calculating the perimeter of frames, mirrors, or jewellery pieces shaped like a ring.
• Civil engineering — measuring the outline of circular tanks, pipes seen end-on, or annular foundations.
• Mathematics — used together with the annulus area calculator to describe ring-shaped regions completely.

Notes

• The outer radius must be greater than or equal to the inner radius. Otherwise the shape is not a valid annulus and the calculator returns no result.
• Both radii must share the same length unit; switching the unit selector reconverts the result automatically.
• Setting the inner radius to 0 collapses the annulus into a disk and the perimeter becomes simply $2\pi R$ — the circumference of the outer circle.
• The perimeter does not measure the area of the ring. For the area of the region enclosed between the two circles, use the annulus area calculator.