Two forces sharing a point of application combine vectorially into a resultant. Calculate its magnitude from F₁, F₂ and the included angle α with the cosine rule applied to the force triangle – or conversely one individual force or the angle from a known resultant.
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Two forces sharing a point of application combine vectorially into a resultant. Calculate its magnitude from F₁, F₂ and the included angle α with the cosine rule applied to the force triangle – or conversely one individual force or the angle from a known resultant.
Two forces F₁ = 12 kN and F₂ = 8 kN include an angle of α = 60°. The cosine rule gives R = √(12² + 8² + 2·12·8·cos 60°) = √(144 + 64 + 96) = √304 ≈ 17.44 kN. Simple addition would have given 20 kN, i.e. 15 % too much. Checking the limiting cases: at α = 0° R = 20 kN, at α = 90° R = √(144+64) = 14.42 kN, and at α = 180° R = 12 − 8 = 4 kN.
Valid for two forces of a concurrent (planar) force system, i.e. with a common point of application or lines of action meeting at one point. Only the magnitude of the resultant is computed; its direction follows from the sine rule sin β = F₁·sin α/R and is not part of the result value. Parallel forces without a common intersection, which form a couple with a moment, more than two forces (where component-wise addition is more practical), and spatial force systems are not covered. When solving for an individual force, only the physically meaningful positive root of the quadratic equation is reported.
Two forces acting at one point do not add up to the sum of their magnitudes but to a resultant that depends on the included angle. This calculator determines that magnitude with the cosine rule and can also be rearranged for one individual force or the included angle.
Forces are vectors: they have magnitude and direction. By the parallelogram law of forces, two forces with a common point of application combine into a resultant R corresponding to the diagonal of the parallelogram they span. Drawing only half of it gives the force triangle, to which the cosine rule can be applied: R = √(F₁² + F₂² + 2·F₁·F₂·cos α). The resultant replaces both forces in their effect completely – it is statically equivalent.
Two people pull a boat with two ropes. Pulling in the same direction, their forces add. Pulling at right angles, the result is noticeably less than the sum – part of the effort cancels out. And pulling directly against each other, only the difference remains. The included angle therefore decides how much of the applied effort actually arrives.
R = √(F₁² + F₂² + 2·F₁·F₂·cos α)
Magnitude: R = √(F₁² + F₂² + 2·F₁·F₂·cos α)Direction relative to F₂: sin β = F₁·sin α/RAligned (α = 0°): R = F₁ + F₂Perpendicular (α = 90°): R = √(F₁² + F₂²)Opposing (α = 180°): R = |F₁ − F₂|Validity band: |F₁ − F₂| ≤ R ≤ F₁ + F₂| Symbol / input | Meaning |
|---|---|
| Resultant R | The magnitude of the resultant force that exactly replaces both individual forces in their effect. It always lies between the difference |F₁−F₂| (opposing forces, α = 180°) and the sum F₁+F₂ (aligned forces, α = 0°). Note: the resultant is never simply the sum of the magnitudes unless the forces are exactly aligned – a very common mistake in load assumptions. |
| Force F₁ | The magnitude of the first force, always entered as positive. Its direction is fully contained in the included angle α, not in a sign. Both forces must share a point of application, or their lines of action must intersect at one point – only then do they form a concurrent force system without an additional moment. |
| Force F₂ | The magnitude of the second force, likewise entered as positive. The order of F₁ and F₂ does not matter for the magnitude of the resultant, since the formula is symmetric in both; for the direction of the resultant it does matter. |
| Included angle α | The angle between the lines of action of the two forces, measured such that α = 0° means aligned forces (their magnitudes then add) and α = 180° exactly opposing forces (they then subtract). At α = 90° the Pythagorean theorem applies. Source: the drawing, the geometry of the attachment points, or the difference of two measured direction angles. |
F₁ and F₂ are the magnitudes of the two forces, always entered as positive; direction is fully contained in the included angle, not in a sign. α is the angle between the two lines of action, measured such that α = 0° means aligned forces. Source: the drawing, the geometry of the attachment points, or the difference of two measured direction angles.
Enter both force magnitudes and determine the included angle. The angle definition matters: if α is measured from the common direction, aligned forces are α = 0°. If instead you have direction angles against a reference axis, α is their difference. As a plausibility check, verify that the result lies between |F₁−F₂| and F₁+F₂ – no resultant of two forces can fall outside that band. For the direction of the resultant, additionally evaluate the sine rule sin β = F₁·sin α/R.
Two forces F₁ = 12 kN and F₂ = 8 kN include α = 60°. The cosine rule gives R = √(12² + 8² + 2·12·8·cos 60°) = √(144 + 64 + 96) = √304 ≈ 17.44 kN. Simply adding them would give 20 kN and therefore 15 % too much – for a load assumption that means unnecessarily expensive oversizing, or in the reverse case an unsafe one. The limiting cases confirm the formula: α = 0° gives 20 kN, α = 90° gives √(144+64) = 14.42 kN (Pythagoras), α = 180° gives 12 − 8 = 4 kN.
The resultant always lies between |F₁−F₂| and F₁+F₂. Up to about α = 30° the difference from simple addition is small (under 4 %); from α = 60° it becomes significant, and at α = 120° the resultant is already smaller than the larger individual force. In practice: the more two forces diverge, the more force is applied without arriving in the resultant – the remainder loads the structure as a mutually cancelling component. This is exactly why shallow sling angles are so unfavourable in lifting equipment.
The calculation runs in newtons and radians internally. Forces may be entered in N, kN, lbf or kip, and the angle in degrees or radians. Since all three forces appear in the same equation, the result is independent of the force unit system chosen; degrees are converted to radians automatically.
Combining two rope, cylinder or spring forces at a joint; load assumptions on eyes, eye bolts and lifting points; support reactions from two inclined loads; deflection forces at pulleys and guide plates; checking graphical force-polygon constructions; back-calculating an unknown individual force from a measured total force.
Valid for two forces of a concurrent planar force system – they must share a point of application or their lines of action must meet at one point. Only the magnitude is computed; the direction of the resultant is not part of the result. Parallel forces without an intersection, which form a couple with a moment, more than two forces (where component-wise addition via Σ Fx and Σ Fy is more practical), and spatial force systems are not covered. When solving for an individual force the quadratic has two roots; only the positive, physically meaningful one is reported.
Common mistake: By far the most common error is simply adding the magnitudes – admissible only for exactly aligned forces. Second most common is the wrong angle definition: if α is taken as the angle between the forces instead of the difference of direction angles (or vice versa), the sign of the cosine term flips and the result is substantially wrong. A quick check helps: at α = 90° the Pythagorean value must come out exactly. Finally, the formula must not be applied to parallel, non-intersecting forces – there an additional moment arises that the resultant alone does not describe.
Combining two rope, cylinder or spring forces at a joint; load assumptions on eyes, eye bolts and lifting points; support reactions from two inclined loads; deflection forces at pulleys and guide plates; checking graphical force-polygon constructions; back-calculating an unknown individual force from a measured total force.
F₁ and F₂ are the magnitudes of the two forces, always entered as positive; direction is fully contained in the included angle, not in a sign. α is the angle between the two lines of action, measured such that α = 0° means aligned forces. Source: the drawing, the geometry of the attachment points, or the difference of two measured direction angles.
Valid for two forces of a concurrent planar force system – they must share a point of application or their lines of action must meet at one point. Only the magnitude is computed; the direction of the resultant is not part of the result. Parallel forces without an intersection, which form a couple with a moment, more than two forces (where component-wise addition via Σ Fx and Σ Fy is more practical), and spatial force systems are not covered. When solving for an individual force the quadratic has two roots; only the positive, physically meaningful one is reported.