Key takeaways
- Damaged quantity = 4T(D − T) × W ÷ (D² − C²), where T is radial depth of damage, D reel diameter, C core diameter and W reel weight.
- The formula is simply the damaged outer ring's area divided by the whole paper annulus. It needs no density figure, only geometry and weight.
- It assumes the damage is a uniform ring around the circumference. Edge (axial) damage or a single dent needs a different measurement.
Why reels need a formula
Newsprint and paper reels travel in large numbers and carry a high value per tonne. Water, crushing and handling hooks damage the outside of the reel first. The inside is usually sound. A surveyor who writes ‘reel damaged, total loss’ overstates the claim, and one who writes ‘surface damage only’ understates it. A geometric method gives both sides a number they can check.
Cargo damage in this trade is not only the value of the paper. It includes replacement, line stoppages at the press and customer complaints. A transparent quantity is the starting point for every one of those conversations.
The formula
Quantity damaged = { 4T × (D − T) × W } ÷ ( D² − C² )
- T: radial depth of damage, measured from the outer surface towards the core, in inches.
- D: overall diameter of the reel.
- C: diameter of the core on which the paper is wound.
- W: weight of the reel. The result is in the same unit as W.
Where it comes from
A reel is a cylinder of paper wound on a core, so the paper occupies a ring (an annulus) with area proportional to D² − C². If the outer layers are ruined to a depth T, the ruined paper is a thinner ring between diameters D and D − 2T. Its area is proportional to D² − (D − 2T)², which simplifies to 4T(D − T).
Paper density and width cancel out because both rings share them. The damaged share of the reel is therefore 4T(D − T) ÷ (D² − C²), and multiplying by the reel weight gives the damaged weight.
A worked example
| Input | Value |
|---|---|
| Reel diameter D | 40 in |
| Core diameter C | 4 in |
| Radial depth of damage T | 2 in |
| Reel weight W | 1,000 kg |
Numerator: 4 × 2 × (40 − 2) = 304. Denominator: 40² − 4² = 1,584. Damaged share = 304 ÷ 1,584 = 19.2%. Damaged quantity = 192 kg of the 1,000 kg reel.
Illustrative numbers for the method only. They are not drawn from any real claim.
Limits of the method
- Uniform ring: it holds when damage runs around the reel. A local dent, a torn edge or water wicking in from one end is not a ring.
- Homogeneous paper: it assumes the same density through the reel. Wet paper that has gained weight should be weighed against the dispatch weight first.
- One reel at a time: sum the per-reel results, rather than applying one T to the whole lot.
- Record T with a repeatable method (gauge points, photographs with a scale) so the figure survives a joint survey.
Used with care, the formula supports compensation talks, insurance claims and preventive packaging decisions with a number both sides can recompute.
Questions answered
- What is the formula for damaged paper reel quantity?
- Quantity damaged = 4T(D − T) × W ÷ (D² − C²), where T is the radial depth of damage, D the reel diameter, C the core diameter and W the reel weight.
- Does the formula need paper density?
- No. Density and width are common to the damaged ring and the whole reel, so they cancel. Only geometry and total weight are needed.
Cite this note
iCargoFinsure Research Desk. “How to Quantify Damage in Paper and Newsprint Reels.” Manthan, iCargoFinsure, 20 Oct 2026. https://www.icargofinsure.com/manthanblog/paper-reel-damage-formula.html