Resultant Force Calculator
Enter two forces and the angle between them to get the magnitude of their resultant — the single force that has the same effect as both combined.
Two forces, one result
Enter both force magnitudes and the angle between them and the calculator returns the resultant in newtons using the law of cosines.
Angle between them
The angle runs from 0° (forces point the same way) to 180° (forces point in opposite directions) — measure it between the two vectors.
What is a resultant force?
One force that replaces two
This resultant force calculator finds the single force that has exactly the same effect as two forces acting together at an angle. When two forces pull or push on the same object, you can replace them with one equivalent force — the resultant. Its size depends on how strong each force is and, crucially, on the angle between them: forces that line up reinforce each other, forces that oppose each other cancel out, and forces at an angle land somewhere in between. Enter the two magnitudes in newtons and the angle between them, and the calculator returns the resultant in newtons.
Enter both forces in newtons and the angle between them to get the resultant force instantly.
The magnitude of the resultant of two forces follows the law of cosines: the square root of the sum of the two squared forces plus twice their product times the cosine of the angle between them.
R = √(F1² + F2² + 2·F1·F2·cosθ)The cosine term is what couples the two forces. At 0° the cosine is 1, so the cross term is at its largest and the forces add directly. At 180° the cosine is −1, so the cross term subtracts and the forces partly or fully cancel. At 90° the cosine is 0, the cross term vanishes, and the formula collapses to plain Pythagoras.
Suppose a 30 N force and a 40 N force act on the same point with a 90° angle between them.
Square each force
30² = 900 and 40² = 1600 — the squared magnitudes.
Add the cosine term
2 × 30 × 40 × cos 90° = 2400 × 0 = 0 — the cross term vanishes at 90°.
Take the square root
√(900 + 1600 + 0) = √2500 = 50 N — the resultant force.
The angle between the forces is the lever that sets the result. When the two forces point the same way (0°) the cosine equals 1 and the resultant is simply F1 + F2 — the maximum possible value, because the forces fully reinforce each other. When they point in opposite directions (180°) the cosine equals −1 and the resultant shrinks to the absolute difference |F1 − F2| — the minimum, because the smaller force is cancelling part of the larger one. The 90° case is the familiar right-angle situation: the cross term disappears and the magnitude is the Pythagorean hypotenuse √(F1² + F2²), which is why perpendicular 30 N and 40 N forces combine to exactly 50 N. Between these extremes the resultant slides smoothly with the cosine, so a wider angle always means a smaller combined force for the same two magnitudes.
The formula is exact, but it answers a specific question, so keep its scope in mind.
Two coplanar forces and the angle between them
This calculator combines exactly two forces that lie in the same plane, and the angle you enter is the angle measured between them, not from any axis. It returns only the magnitude of the resultant, not its direction. For three or more forces, or for forces in three dimensions, resolve each into components and add the components instead.