polygon of forces
A polygonal figure the sides of which are vectors representing several forces acting simultaneously upon one point, so that the vector necessary to make the figure closed is the resultant of those forces.
polygon of forces: graphical method for adding vectors
A polygon of forces is a graphical construction used to find the resultant of multiple forces acting on a single point. You draw each force as a line segment, scaled to represent both its magnitude and direction, then arrange them head-to-tail in sequence. The line that closes the polygon, drawn from the starting point to the end of the final force vector, represents the resultant force in both size and direction.
The method works because force is a vector quantity. When forces act together at one point, they combine according to vector addition, not simple arithmetic. A 10 N force pointing east and a 10 N force pointing north do not combine to give 20 N in any single direction; instead they produce a resultant of approximately 14.1 N pointing northeast. The polygon of forces reveals this result visually.
In practice, you need a ruler, protractor, and drawing paper to reasonable precision. Each force is represented by a line segment whose length is proportional to the force magnitude, drawn at the correct angle from a reference direction. The sequence of forces matters for organization, but not for the final answer. A system in equilibrium closes the polygon exactly; if the polygon does not close, the gap is the magnitude and direction of the unbalanced resultant.
Variants and limitations
For two forces only, the method simplifies to the parallelogram of forces, where forces share a common starting point and form two sides of a parallelogram; the diagonal is the resultant. For three or more forces, the polygon method is more practical because vectors can be arranged sequentially without redrawing. The method is strictly geometric and independent of any coordinate system.
The accuracy depends entirely on the precision of your drawing. A small error in angle or length compounds through the construction. Today, graphical methods are often replaced by component analysis: resolving each force into horizontal and vertical components, summing each component separately, then recombining to find the resultant. However, the polygon of forces remains valuable in teaching the nature of vector addition and in field situations where calculation equipment is not available. It also provides rapid visual intuition about whether forces are nearly balanced or heavily skewed in one direction.