Calculate paper reel weight, density, width, outside diameter or core diameter from the other four values with unit conversions.
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Paper Reel Weight Formula
The paper reel is treated as a hollow cylinder. The calculator uses the outside diameter, core diameter, reel width, and density to find the reel weight, or it rearranges the same formula to solve for one missing value.
M = rho * pi/4 * (D_o^2 - D_c^2) * B
Rearranged formulas:
rho = M / ((pi/4) * (D_o^2 - D_c^2) * B)
B = M / (rho * (pi/4) * (D_o^2 - D_c^2))
D_o = sqrt(D_c^2 + (4*M)/(pi*rho*B))
D_c = sqrt(D_o^2 - (4*M)/(pi*rho*B))
- M = paper reel weight
- rho = density of the wound paper material
- D_o = outside diameter of the reel
- D_c = core diameter
- B = reel width
- pi = 3.14159265…
Use consistent units in manual calculations. The calculator converts length inputs to meters, density to kg/m³, and weight to kilograms before calculating the result, then converts the answer back to the selected output unit.
If you leave Paper Reel Weight blank, it calculates the weight from the reel volume and density. If you leave Density blank, it divides known weight by reel volume. If you leave Width, Outside Diameter, or Core Diameter blank, it rearranges the same hollow-cylinder formula to solve for that missing dimension.
Common Paper Density and Unit Reference
Density can vary by paper grade, winding tightness, moisture, coating, and air space in the reel. Use a measured or supplier-provided density when accuracy matters.
| Material or reel type | Typical density range | Notes |
|---|---|---|
| Tissue or very light paper | 200 to 500 kg/m³ | Low bulk density, often more compressible |
| General printing or writing paper | 600 to 900 kg/m³ | Common range for many paper reels |
| Coated paper | 800 to 1,200 kg/m³ | Coatings usually increase density |
| Paperboard or dense stock | 700 to 1,100 kg/m³ | Depends strongly on caliper and grade |
| Quantity | Calculator base unit | Useful conversion |
|---|---|---|
| Length | meter, m | 1 inch = 0.0254 m |
| Density | kg/m³ | 1 g/cm³ = 1,000 kg/m³ |
| Density | kg/m³ | 1 lb/ft³ = 16.0185 kg/m³ |
| Weight | kilogram, kg | 1 lb = 0.45359237 kg |
Example Calculations
Example 1: Calculate paper reel weight
Suppose a reel has an outside diameter of 1.2 m, a core diameter of 0.15 m, a width of 1.0 m, and a density of 800 kg/m³.
M = 800 * pi/4 * (1.2^2 - 0.15^2) * 1.0
M = 890.64 kg
The reel weight is approximately 890.64 kg.
Example 2: Calculate missing width
Suppose a reel weighs 500 kg, with an outside diameter of 1.0 m, a core diameter of 0.10 m, and a density of 750 kg/m³.
B = 500 / (750 * pi/4 * (1.0^2 - 0.10^2))
B = 0.8575 m
The required reel width is approximately 0.8575 m, or 857.5 mm.
FAQ
Does the core weight count in the paper reel weight?
No. This formula subtracts the core diameter from the outside diameter, so it calculates the weight of the paper material around the core. It does not add the physical weight of the cardboard, plastic, or metal core. If you need gross shipping weight, add the core weight separately.
Why must the outside diameter be larger than the core diameter?
The reel is modeled as a hollow cylinder. If the outside diameter is equal to or smaller than the core diameter, the calculated paper volume is zero or negative, which is not physically valid.
What density should you enter?
Enter the effective density of the wound paper reel, not just the theoretical fiber density. For the best result, use a density from your paper specification, production data, or a known reel weight and dimensions. If you only have an estimate, choose a value that matches the paper grade and winding condition.
