Force polygon

Draw every force on a joint tip to tail. If the joint is balanced, you end up where you started

Last updated 2026-09-25

Draw every force on a joint tip to tail. If the joint is in equilibrium, you end up exactly where you started. That closed shape is the force polygon, and it lets you check a joint, or even solve it, with a ruler instead of equations.

Adding forces tip to tail

To add two forces by drawing, start the second arrow where the first one ends. The arrow from the very start to the very end is their sum (also called the resultant).

It's the same addition as in force components, just drawn instead of calculated. There, you add the sideways parts together and the up-down parts together. Here, each arrow does both at once: walking 3 kN to the right and then 4 kN up is the same as adding x with x and y with y.

It works for any number of forces. Keep going tip to tail, and the sum is still the arrow from the start to the end.

Why the polygon closes

A joint in a truss doesn't move, so all the forces on it add up to nothing:

ΣFx = 0 and ΣFy = 0

Nothing sideways, nothing up or down: the last arrow ends exactly where the first one started. The distance between the start and the end is the sum of the forces, and for a balanced joint that's zero. A closed polygon is the same thing as ΣFx = 0 and ΣFy = 0, just drawn.

If the polygon doesn't close, the joint isn't balanced. The gap shows exactly what's missing: the force that would close it has the gap's length and direction.

The force triangle

With three forces, the polygon is a triangle. The top joint of the 3-4-5 truss from Support reactions has exactly three: the 8 kN load, and the two sloped members, 5 kN each.

Try it in the demo: pick the top node, and drag it around. The triangle changes shape with the truss, but it always closes. Pick the supports too: every joint has its own closed triangle.

A bigger truss

The rest of this article uses a slightly bigger truss: three triangles side by side, called a Warren truss, with 8 kN hanging from its middle joint C. Its supports each take 4 kN.

The order doesn't matter

You can draw the forces in any order. A different order gives a differently shaped polygon, but it always closes, because the sum doesn't depend on the order you add things in.

These are the five forces on joint C: the load, and the four members meeting there.

Solving a joint by drawing

The polygon can do more than check a joint. It can find two unknown member forces.

You don't know how big a member's force is, but you do know its direction: it always runs along the member. So:

1. Draw the forces you know, tip to tail.
2. From the end of the last one, draw a long guide line in the direction of one unknown member.
3. From the very start, draw a guide line in the direction of the other one.
4. Where the two lines cross, the polygon closes. The two new sides are the two member forces: measure them.

That's joint A of the Warren truss, solved without a single equation: AB comes out 5 kN and AC 3 kN. Go joint by joint like this and you can solve the whole truss. Method of joints does exactly the same with equations, and gets the same numbers. Draw to scale (say 1 cm for every 1 kN), and the answers are as good as your ruler.

Tension or compression?

The polygon also tells you which way each member works. Read each member's arrow in the polygon, then move it back onto the joint, along its member:

• Pointing toward the joint: the member pushes on the joint, so it's in compression.
• Pointing away from the joint: the member pulls on the joint, so it's in tension.

This is the same picture as the joints cut out on their own in Method of joints: a member in tension pulls on the joints at both of its ends, one in compression pushes on both. More on what the joint feels from each member is in Tension and compression.

Common mistakes

• Starting the next arrow in the wrong place. Each arrow starts where the previous one ends, not at the joint.
• Drawing a member force the wrong way. Draw it the way it acts on the joint you're looking at, not on the member.
• Not drawing to scale. A polygon only closes on paper if every arrow has the right length. Pick a scale and stick to it.
• Reading tension and compression from the member instead of the polygon. It's the arrow's direction in the closed polygon that tells you, once you move it back onto the joint.

Check it in Truzme

Open the Warren truss and compare: every member force on the canvas matches the polygons above.