How To Draw A Triangle With 3 Angles

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.

Other constructions pages on this site

List of printable constructions worksheets

Lines

Introduction to constructions
Copy a line segment
Sum of n line segments
Difference of two line segments
Perpendicular bisector of a line segment
Perpendicular from a line at a point
Perpendicular from a line through a point
Perpendicular from endpoint of a ray
Divide a segment into n equal parts
Parallel line through a point (angle copy)
Parallel line through a point (rhombus)
Parallel line through a point (translation)

Angles

Bisecting an angle
Copy an angle
Construct a 30° angle
Construct a 45° angle
Construct a 60° angle
Construct a 90° angle (right angle)
Sum of n angles
Difference of two angles
Supplementary angle
Complementary angle
Constructing 75° 105° 120° 135° 150° angles and more

Triangles

Copy a triangle
Isosceles triangle, given base and side
Isosceles triangle, given base and altitude
Isosceles triangle, given leg and apex angle
Equilateral triangle
30-60-90 triangle, given the hypotenuse
Triangle, given 3 sides (sss)
Triangle, given one side and adjacent angles (asa)
Triangle, given two angles and non-included side (aas)
Triangle, given two sides and included angle (sas)
Triangle medians
Triangle midsegment
Triangle altitude
Triangle altitude (outside case)

Right triangles

Right Triangle, given one leg and hypotenuse (HL)
Right Triangle, given both legs (LL)
Right Triangle, given hypotenuse and one angle (HA)
Right Triangle, given one leg and one angle (LA)

Triangle Centers

Triangle incenter
Triangle circumcenter
Triangle orthocenter
Triangle centroid

Circles, Arcs and Ellipses

Finding the center of a circle
Circle given 3 points
Tangent at a point on the circle
Tangents through an external point
Tangents to two circles (external)
Tangents to two circles (internal)
Incircle of a triangle
Focus points of a given ellipse
Circumcircle of a triangle

Polygons

Square given one side
Square inscribed in a circle
Hexagon given one side
Hexagon inscribed in a given circle
Pentagon inscribed in a given circle

Non-Euclidean constructions

Construct an ellipse with string and pins
Find the center of a circle with any right-angled object