Enter the magnitude of one force and the angles opposite the other two forces (the angles in the force triangle, which sum to 180°) into the calculator to determine the magnitudes of the other two forces using Lami’s Theorem. This calculator assumes that the forces are in equilibrium.

Lami’s Theorem Calculator

Enter Angle B and Angle C (opposite Forces B and C). Then enter any one force (A, B, or C) and leave the other force fields blank to calculate them.

Lami’s Theorem Formula

Lami’s theorem is a compact statics relationship for a body in equilibrium under exactly three forces. When the three forces are coplanar, concurrent, and non-collinear, each force is proportional to the sine of the angle opposite that force. This makes the theorem useful for solving cable tensions, ring-and-hook force systems, pin joints, suspended loads, and other three-force equilibrium problems.

\frac{A}{\sin(\alpha)}=\frac{B}{\sin(\beta)}=\frac{C}{\sin(\gamma)}

On this calculator, the angles entered are the angles opposite the corresponding forces. If Force A is known and the opposite angles for Forces B and C are entered, the remaining opposite angle for Force A is found first.

\alpha = 180^\circ - \beta - \gamma

Then the two unknown forces are solved directly from Lami’s theorem:

B = A \cdot \frac{\sin(\beta)}{\sin(\alpha)}
C = A \cdot \frac{\sin(\gamma)}{\sin(\alpha)}

Because the relationship is symmetric, any one known force can be used to solve the other two as long as the opposite angles are known and the system is truly in equilibrium.

Variable Definitions

Term Meaning
Force A, Force B, Force C Magnitudes of the three forces acting on the body
α, β, γ Angles opposite Forces A, B, and C
Force unit Any consistent unit such as N, kN, MN, or lbf
Angle unit Degrees

When Lami’s Theorem Applies

  • Exactly three forces: the theorem is only for three-force systems.
  • Coplanar forces: all forces must lie in the same plane.
  • Concurrent forces: the lines of action must meet at a single point.
  • Non-collinear forces: the forces cannot all lie on one straight line.
  • Static equilibrium: the resultant force must be zero.

How to Use the Calculator

  1. Enter one known force magnitude.
  2. Enter the angle opposite Force B and the angle opposite Force C.
  3. Make sure the two entered angles form a valid three-force geometry so the remaining angle is positive.
  4. Choose the desired force unit. The calculated forces will remain in that same unit basis.
  5. Click calculate to solve the unknown force values.

If your known force is measured in Newtons, the outputs are in Newtons. If your known force is entered in kilonewtons or pound-force, the outputs stay in that selected unit system.

Example

Suppose Force A is 50 N, the angle opposite Force B is 60°, and the angle opposite Force C is 45°. First calculate the remaining opposite angle:

\alpha = 180^\circ - 60^\circ - 45^\circ = 75^\circ

Now solve for Force B and Force C:

B = 50 \cdot \frac{\sin(60^\circ)}{\sin(75^\circ)} \approx 44.83 \text{ N}
C = 50 \cdot \frac{\sin(45^\circ)}{\sin(75^\circ)} \approx 36.60 \text{ N}

So the three equilibrium forces are approximately 50 N, 44.83 N, and 36.60 N, each opposite its corresponding angle.

Common Mistakes

  • Using the wrong angle definition: Lami’s theorem uses the angle opposite each force, not just any measured angle in the sketch.
  • Entering an impossible angle pair: if the entered angles are too large, the remaining angle becomes zero or negative and the setup is not valid.
  • Applying the theorem to more than three forces: larger systems usually need force resolution into components before using equilibrium equations.
  • Ignoring equilibrium: if the body is accelerating, Lami’s theorem does not apply.
  • Mixing units: use one consistent force unit throughout the calculation.

Practical Applications

  • Tension in two cables supporting a single load
  • Force analysis at rings, shackles, and lifting hooks
  • Pin-jointed statics problems in mechanics courses
  • Brackets and support members with three-force interaction
  • Guy-wire and anchored support arrangements in 2D

Frequently Asked Questions

Can Lami’s theorem be used in three dimensions?
No. It is a planar theorem for coplanar forces. A true 3D force system requires vector methods.
Do the forces have to be tensions?
No. The forces can be tensions, reactions, pushes, pulls, or any other forces, as long as the three-force equilibrium conditions are satisfied.
Why does a very small angle sometimes create a very large force?
Because the sine of a very small angle is also small, which increases the force ratio and makes the result sensitive to angle measurement error.
What happens if the two entered angles add up to 180° or more?
The third opposite angle is no longer physically valid for a three-force equilibrium triangle, so the result should not be used.