# Simpson's rules (ship stability)

Simpson's rules are a set of rules used in ship stability and naval architecture, to calculate the areas and volumes of irregular figures.

## Simpson's 1st rule

Also known as the 1–4–1 rule (after the multipliers used ).

${\text{Area}}={\frac {h}{3}}(a+4b+c).$

## Simpson's 2nd rule

Also known as the 1–3–3–1 rule, Simpson's second rule is a simplified version of Simpson's 3/8 rule.

${\text{Area}}={\frac {3h}{8}}(a+3b+3c+d).$

## Simpson's 3rd rule

Also known as the 5–8–1 rule, SImpson's third rule is used to find the area between two consecutive ordinates when three consecutive ordinates are known.

${\text{Area}}={\frac {h}{12}}(5a+8b-c).$

## Use of Simpsons rules

Simpson's rules are used to calculate the volume of lifeboats, and by surveyors to calculate the volume of sludge in a ship's oil tanks. For instance, in the latter, Simpson's 3rd rule is used to find the volume between two co-ordinates. To calculate the entire area / volume, Simpson's first rule is used.

Simpson's rules are used by a ship's officers to check that the area under the ship's GZ curve complies with IMO stability criteria.