Constructions
61. The SSS Triangle Builder · Three sides, one perfect triangle
Point C always stays exactly AC away from A and BC away from B.
SSS construction (Side-Side-Side): given three side lengths, the triangle is uniquely determined. Draw side AB, then arcs of radii len_ac and len_bc from A and B respectively — they meet at C.
What this lesson covers
Try to break it
Try pulling A and B too far apart. The arcs stop crossing — so no triangle can form. That is the Triangle Inequality Rule: a triangle is possible only when the sum of any two sides is greater than the third side. Once AB grows longer than AC + BC, the arcs can never meet.
How you build it
Construct a triangle given three sides.
- Mark point A.
- Mark point B — the first side length away from A.
- Draw the base AB.
- With centre A, open the compass to the second side and draw an arc.
- With centre B, open the compass to the third side and draw an arc crossing the first.
- Mark point C where the two arcs cross.
- Draw side AC.
- Draw side BC to complete the triangle.
The proof, step by step
Prove that the constructed triangle has the three given side lengths.
- We start by drawing the longest side AB exactly as given.
- An arc from A with radius AC ensures every point on it is exactly AC distance from A.
- An arc from B with radius BC ensures every point on it is exactly BC distance from B.
- The intersection point C satisfies both conditions simultaneously.
- Thus, joining AC and BC gives the unique triangle satisfying all three side lengths.
Worked example
A student tries to construct a triangle with sides 3 cm, 4 cm, and 8 cm. The arcs never intersect. What is the reason?
Triangle inequality theorem: the sum of any two sides must be greater than the third side. Here, 3 + 4 = 7, which is less than 8, so the triangle cannot be constructed.
- The compass is broken.
- The sum of two sides is less than the third side. — correct
- The angles are too large.
- The ruler is too short.