
Triangle given three sides (SSS)
This page shows how to construct a triangle given the length of all three sides, with compass and straightedge or ruler. It works by first
copying one of the line segments to form one side of the triangle. Then it finds the third
vertex
from where two
arcs
intersect
at the given distance from each end of it.
Multiple triangles possible
It is possible to draw more than one triangle that has three sides with the given lengths.
For example in the figure below, given the base AB, you can draw four triangles that meet the requirements.
All four are correct in that they satisfy the requirements, and are
congruent to each other.
Note: This construction is not always possible
See figure on the right.
If two sides add to less than the third, no triangle is possible.
Printable stepbystep instructions
The above animation is available as a
printable stepbystep instruction sheet, which can be used for making handouts
or when a computer is not available.
Proof
The image below is the final drawing above with the red items added.

Argument 
Reason 
1 
Line segment LM is congruent to AB. 
Drawn with the same compass width. See Copying a line segment 
2 
The third vertex N of the triangle must lie somewhere on arc P. 
All points on arc P are distance AC from L since the arc was drawn with the compass width set to AC. 
3 
The third vertex N of the triangle must lie somewhere on arc Q. 
All points on arc Q are distance BC from M since the arc was drawn with the compass width set to BC. 
4 
The third vertex N must lie where the two arcs intersect 
Only point that satisfies 2 and 3. 
5 
Triangle LMN satisfies the three side lengths given.
LM is congruent to AB,
LN is congruent to AC,
MN is congruent to BC,
 
 Q.E.D
Try it yourself
Click here for a printable worksheet containing two triangle construction problems where you are given the three side lengths.
When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.
Other constructions pages on this site
Lines
Angles
Triangles
Right triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
NonEuclidean constructions
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