Compute CP from volume & midship area, from CB & CM, via offsets (Simpson), or estimate a target empirically.
CP = —
CP = —
∇ (Simpson) = — m³
AM (chosen) = — m²
CP = —
Froude number: —
Suggested target: CP ≈ — (range —)
The prismatic coefficient describes how a ship's underwater volume is distributed along its length. It compares the actual displaced volume with a prism having the same selected length and the same reference transverse sectional area.
A higher CP means that relatively large sectional areas are maintained over more of the vessel's length. A lower value means that the sectional-area curve tapers more sharply towards the bow and stern.
CP is therefore different from the block coefficient. CB describes the total fullness of the underwater hull inside a length-beam-draft block, while CP concentrates on the longitudinal distribution of underwater volume.
The standard longitudinal prismatic coefficient is written as:
CP = ∇ ÷ (L × AX)
In the standard definition, AX is the maximum immersed transverse sectional area. On many conventional merchant ships, the maximum section is at or close to midship. In that common case, AX and the immersed midship area AM are effectively the same for this calculation.
This calculator uses the entered or selected midship sectional area AM:
CP = ∇ ÷ (L × AM)
| Symbol | Meaning | Usual unit |
|---|---|---|
| CP | Longitudinal prismatic coefficient | Dimensionless |
| ∇ | Displaced underwater volume | m3 |
| L | Selected longitudinal reference, such as LPP or LWL | m |
| AX | Maximum immersed transverse sectional area | m2 |
| AM | Immersed transverse sectional area at the selected midship station | m2 |
The numerator and denominator are both volumes, so CP has no unit. The length and sectional-area inputs must use compatible units. For example, metres and square metres produce a reference prism in cubic metres.
Section reference: when the maximum immersed section is not located at midship, a coefficient calculated with AM is not numerically identical to the standard maximum-section prismatic coefficient. Record the area and length references used by the calculation.
If the vessel had the same transverse sectional area at every position along the selected length, its underwater body would form a prism and CP would be 1.00. A real ship narrows towards its ends, so its displaced volume is smaller than that reference prism.
Another way to view CP is through the sectional-area curve. The displaced volume is the area under that curve, while L × AM is the area of a surrounding rectangle drawn with length L and height AM.
A high coefficient indicates a sectional-area curve that remains close to its maximum height over a substantial part of the length. A low coefficient indicates finer ends or a shorter region of large sectional area.
Consider a vessel with:
∇ = Δ ÷ ρ = 36,420.3 ÷ 1.025 = 35,532 m3
L × AM = 180 × 263.2 = 47,376 m3
CP = 35,532 ÷ 47,376 = 0.750
The vessel's displaced volume is therefore 75.0% of the reference prism formed by the selected length and midship sectional area.
The calculator provides four routes to a result. Each method uses a different set of available data.
| Method | Main inputs | Result type |
|---|---|---|
| Direct | ∇ or Δ and ρ, L, and AM | Direct geometric CP |
| From CB and CM | CB and CM, entered or derived | Coefficient-chain result |
| Offsets and Simpson integration | L and sectional areas at equally spaced stations | Integrated volume and CP |
| Empirical target | L, service speed and ship type | Preliminary suggested value and range |
Use the direct method when the displaced volume and the immersed midship or reference sectional area are known. Displaced volume may be entered directly, or the calculator can derive it from displacement mass and water density:
∇ = Δ ÷ ρ
If AM is unavailable but CM, beam and draft are known, the calculator can derive the area from:
AM = CM × B × T
The displacement, area and length must belong to the same loading condition and dimensional convention.
When the block and midship coefficients use consistent beam, draft, length and sectional references:
CB = CP × CM
CP = CB ÷ CM
The worked example can also be checked through this relationship. For a beam of 28 m and draft of 10 m:
CM = 263.2 ÷ (28 × 10) = 0.940
CB = 35,532 ÷ (180 × 28 × 10) = 0.705
CP = 0.705 ÷ 0.940 = 0.750
Consistency check: the coefficient relationship is exact only when CB, CM and CP use compatible length, beam, draft and sectional-area definitions.
The offsets method estimates displaced volume from a sequence of immersed sectional areas measured at equally spaced stations along the selected length.
For a conventional Simpson's one-third-rule arrangement with an even number of intervals:
∇ ≈ h ÷ 3 × [A0 + An + 4(A1 + A3 + ... + An−1) + 2(A2 + A4 + ... + An−2)]
Here, h is the longitudinal station spacing and A0 to An are immersed transverse sectional areas.
An odd number of stations produces an even number of intervals, so the calculator applies Simpson's one-third rule. If an even number of stations is entered, the resulting odd number of intervals does not suit the pure one-third-rule arrangement; the existing implementation then uses a trapezoidal fallback for the complete station series.
The calculator also asks for the index of the station used as AM. Select the station that corresponds to the intended midship or reference section. Choosing an unrelated station changes the denominator and therefore changes CP.
The empirical tab uses vessel length and service speed to calculate the Froude number:
Fn = V ÷ √(gL)
When SI units are used, speed must be expressed in metres per second, length in metres and gravitational acceleration in metres per second squared. The calculator performs the necessary conversion from knots.
The calculated Froude number and selected vessel type are used by the calculator's internal heuristic to produce a preliminary target value. It also displays a guidance band of ±0.04 around that value. This method is intended for early dimension studies when a complete hull model is not yet available.
Empirical result: the suggested value is a design starting point, not proof that the hull has minimum resistance. Final volume distribution should be checked against the actual sectional-area curve and an appropriate resistance assessment.
The maximum immersed transverse section and the geometrical midship section are often the same on vessels with a parallel middle body, but they are not interchangeable by definition.
On a hull where the largest immersed section lies forward or aft of midship, the standard maximum-section coefficient uses AX. A calculation using the area at midship instead should be identified as midship-referenced.
This distinction also affects the relationship between CP and CM. If CM is based on a different section from the area used in CP, the simple coefficient chain may no longer represent the same reference geometry.
A relatively low CP indicates that the underwater sectional area decreases more rapidly away from the reference section. A relatively high value indicates fuller ends or a longer region of large sectional area.
The broad ranges below are comparison values rather than design limits. Actual values depend on speed, displacement, hull proportions, parallel middle-body length and the intended distribution between the entrance and run.
| Vessel or hull type | Indicative CP | General interpretation |
|---|---|---|
| Fine displacement craft and patrol vessels | Approximately 0.55–0.65 | Finer longitudinal volume distribution |
| Container ships and Ro-Ro vessels | Approximately 0.60–0.70 | Moderate fullness influenced by service speed |
| General cargo and multipurpose vessels | Approximately 0.62–0.74 | Moderate longitudinal fullness |
| Bulk carriers and tankers | Approximately 0.70–0.82 | Fuller ends or a longer region of high sectional area |
Vessel type alone is not enough to select a final value. Two ships in the same category may require different longitudinal volume distributions because of speed, length-displacement ratio, cargo arrangement or stern and bow design.
A single whole-ship CP does not show whether volume is concentrated forward or aft of midship. Two hulls may have the same total coefficient while having different entrances, runs and stern fullness.
Naval architects may therefore consider separate forebody and afterbody prismatic coefficients. These compare the volume of each hull portion with a prism based on its own length and reference section.
Partial coefficients are useful when comparing bow and stern volume distribution, but they must always state the dividing station, length and sectional reference used.
CP belongs to a specific immersed hull condition. When draft changes, displaced volume and immersed sectional areas both change. They do not necessarily change in the same proportion.
A coefficient calculated at design draft should not automatically be applied to ballast draft or another partial-load condition. Use displacement, sectional area and length references belonging to the same condition.
Prismatic coefficient is useful for describing longitudinal volume distribution, checking preliminary hull dimensions, comparing hull forms and providing an input to several empirical resistance methods.
It does not define the exact sectional-area curve. Two hulls can have the same CP while distributing volume differently between the forebody and afterbody.
CP alone cannot determine:
With consistent dimensions and a reference section that is not smaller than the remaining immersed sections, CP should normally be greater than 0 and no greater than 1. An out-of-range result usually indicates inconsistent units, an incorrect reference area, an incorrect station selection or mismatched loading-condition data.
The terminology and formula used on this page follow standard naval architecture references:
NauticalSolver calculators are intended for preliminary engineering, study and independent checking. Use approved hydrostatic data, vessel-specific drawings and suitable resistance or hull-design software for operational, contractual or statutory work.