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Resultant of Two Forces Calculator

Enter the two force magnitudes and the angle between them to find the resultant.

How it is calculated

Force F1 is laid along the x-axis and F2 at angle θ from it, in degrees. The components of the sum are Rx = F1 + F2·cos θ and Ry = F2·sin θ. The magnitude is R = √(F1² + F2² + 2·F1·F2·cos θ), equal to √(Rx² + Ry²). The direction is α = atan2(Ry, Rx), the angle of the resultant measured from F1 in degrees, so a negative angle means it lies on the other side of F1. Forces are in newtons and both must be above zero, otherwise no result is shown. The angle is the one between the two force vectors drawn from the same point: 0° means the same direction, 180° opposite directions. Results have two decimals. If two equal forces act in opposite directions, the resultant is 0.00 N and the direction is shown as a dash because it has no meaning. If F2 is larger and opposite (θ = 180°), the direction is 180°, pointing against F1.

Worked example

F1 = 30 N and F2 = 40 N with 60° between them. Step 1: cos 60° = 0.5, sin 60° = 0.8660. Step 2: Rx = 30 + 40 × 0.5 = 50.00 N and Ry = 40 × 0.8660 = 34.64 N. Step 3: R = √(50² + 34.64²) = √3,700 = 60.83 N. Step 4: direction = atan2(34.64, 50) = 34.72 ° from F1. The resultant is smaller than 30 + 40 = 70 N, which you would get only if the forces pointed the same way.

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Common mistakes

  • Adding magnitudes directly. 30 N and 40 N acting 60° apart do not give 70 N; only forces in the same direction (0°) do.
  • Entering the wrong angle. It must be the angle between the force vectors drawn from the same point. If you measured between the lines of action the other way round, use 180° minus that angle.
  • Entering radians or an angle to the horizontal. The input is in degrees and is measured between F1 and F2, while the output direction is measured from F1.
  • Entering a force as a negative number to reverse it. Both forces must be positive; set the angle to 180° to make F2 act against F1.
  • Mixing units, such as one force in kilogram-force and one in newtons. Convert both to newtons first.

About this calculator

When two forces act on the same point at an angle to each other, their combined effect — the resultant — is found geometrically with the parallelogram law: place the two force vectors as adjacent sides of a parallelogram, and the diagonal from their common origin is the resultant. Algebraically, its magnitude comes from the law of cosines applied to that triangle: R = √(F1² + F2² + 2·F1·F2·cos θ), where θ is the angle between the two forces.

The resultant’s direction relative to one of the original forces is then found with the law of sines (or an equivalent tangent formula) using the same triangle. This is fundamental to statics and engineering mechanics — anywhere multiple forces (cables, struts, applied loads, wind and gravity) act on a single point and need to be reduced to one equivalent force for equilibrium or structural analysis. With the two force magnitudes and the angle between them it finds the size of the resultant and the direction it points.

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Comments (1)

  • Editorial team

    The direction shown is measured from F1, not from the horizontal. If F1 itself points at some angle on your drawing, add that angle to the result to get the resultant's bearing in your own coordinate frame. Is your problem a pair of cables, a bracket or a ramp? Describe the setup and the angle convention you use.

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