# 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 step-by-step instructions

The above animation is available as a printable step-by-step 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.