141. The SSS Blueprint · Building triangles from three sides

Try to break it

Drag A, B, or C. The two arcs always stretch to meet exactly at the third corner — give the compass any three lengths and they close up into a single triangle. The arcs only miss if one side is longer than the other two combined.

How you build it

Construct a triangle from three sides.

• Mark point A — the starting vertex.
• Mark point B — the other end of the base.
• Draw the base segment AB with the given length.
• Place compass at A. Set radius to the length of AC. Draw an arc above AB.
• Place compass at B. Set radius to the length of BC (a different radius from step 4). Draw an arc. Where the two arcs cross is point C.
• Mark point C exactly where the two arcs cross.
• Draw segment AC — join vertex A to vertex C.
• Draw segment BC — join vertex B to vertex C. Triangle ABC is complete!

The proof, step by step

Prove that the constructed triangle has the three given side lengths.

• First, fix the base segment AB. This establishes the first side and its endpoints.
• Next, draw an arc from A with radius equal to AC. This marks all possible locations for C at that distance.
• Finally, draw an arc from B with radius BC. Their intersection uniquely determines point C, completing the triangle.

Worked example

When constructing a triangle with sides 5 cm, 6 cm, and 7 cm, which step comes immediately after drawing the base?

After drawing the base (one side), the next step is to draw an arc from one of its endpoints using the length of the second side. This helps locate the third vertex.

• Draw the second side directly.
• Draw an arc from one endpoint with the length of the third side.
• Draw an arc from one endpoint with the length of the second side. — correct
• Draw the third side.