A sailboat leans over in the wind and then stops leaning. Two forces do it: the boat's weight pulling down through one point and the water's buoyancy pushing up through another. When the boat heels, those two points slide apart sideways, and the pair of forces twists the boat back upright. The labs below compute that twist from the shape of the hull, at every angle up to upside down.
Gravity pulls on every part of the boat, and the total acts as if it all pulled at one point, the center of gravity G. The water pushes up on the hull with a force equal to the weight of the water the hull shoves aside (Archimedes' principle). That push acts at the center of buoyancy B, the balance point of the underwater part of the hull. Upright, B sits straight below G and nothing twists. Heel the boat and the underwater shape changes: the low side digs in, the high side lifts out, and B slides toward the low side. Now weight and buoyancy push along two parallel vertical lines. The sideways gap between them is the righting arm GZ, and weight × GZ is the righting moment, the twist that rolls the boat back. Drag the mast to heel the boat, and move G with the slider.
How to read this: you are looking at a slice through the boat from behind. The shaded part is below the water, and B is its balance point. The gold bar is the sideways gap between the weight line and the buoyancy line.
Try this: pick the wide flat hull and drag the heel past 90°: B swings back past G, the arrow turns red, and the twist now rolls the boat further over.
Δ is the boat's mass in kg, g = 9.81 m/s², GZ is in meters and RM in newton-meters (N·m). x is a horizontal position measured across the water. A₀ is the underwater cross-section when upright: the boat must always displace its own weight of water, so as it heels the page solves for the waterline height that keeps that area fixed. K is the bottom of the keel, M the metacenter, I the second moment of the waterline's width and V the underwater volume. For a box of beam b and draft T, BM = b² / (12 T).
These hulls are idealized: the page treats the boat as a prism, with every slice from bow to stern the same shape as the one drawn, and the length chosen so the boat has the stated weight. Real hulls taper toward the ends, so real numbers differ, but the mechanism is the same. The heel calculation also ignores trim, the bow-down or stern-down tilt that real boats pick up as they heel.
Repeat Lab 1 at every heel angle from upright (0°) to upside down (180°) and plot GZ. Where the curve is above zero the boat twists back toward upright. Where it is below zero it twists toward upside down. The highest point is the angle of maximum stability. The angle where the curve drops through zero is the angle of vanishing stability (AVS): heel past it and the boat keeps going over. The area under the curve measures the energy a wave or gust must supply to push the boat over, and the area of the negative part measures how hard it is to flip back once upside down.
How to read this: above the zero line the boat rights itself, below it the boat rolls toward upside down. The little boat in the corner shows the heel under your cursor.
Try this: switch between the keelboat and the wide flat hull and compare the red areas: tiny for the keelboat, bigger than the green for the flat hull.
θ is the heel angle, in radians inside the integral, so the areas read in meter-radians, and multiplying by Δ·g turns them into joules. The page finds the AVS by stepping along the curve in 1° steps until GZ changes sign, then halving the bracket 40 times.
There are two ways to get a righting moment. Weight stability puts heavy ballast low in a keel, so G sits far below B and the weight line swings to the high side at every angle. Form stability comes from width: a wide, flat hull moves B a long way sideways for a small heel, so it is very stiff at first. But past the point where the deck edge goes under, B stops moving out, and the flat hull's GZ falls and turns negative. Upside down, the same width works against you: the flat deck is now the bottom, and it holds the boat inverted just as firmly as the bottom held it upright. That is why dinghies and multihulls can stay capsized, while a ballasted keelboat with a small negative area is flipped back by the next wave.