Constructing Triangles

A triangle is a three-sided polygon. Learn the triangle inequality, how to construct triangles from given measurements using compass and ruler, and the main types of triangles.

What Is a Triangle?

A triangle is a polygon with three sides and three angles. The three angles of any triangle always add up to 180°.

Angle sum of a triangle

\alpha + \beta + \gamma = 180°

The three interior angles of every triangle sum to 180°.

The Triangle Inequality Rule

Not every set of three lengths can form a triangle. The rule is:

Triangle inequality

a + b > c \quad \text{for every pair of sides}

The sum of any two sides must be greater than the third side.

Can sides 3 cm, 5 cm, and 9 cm form a triangle?

Check: $3 + 5 = 8$ , which is less than $9$ .

So no — these three lengths cannot form a triangle.

Can sides 4 cm, 6 cm, and 8 cm form a triangle?

Check all pairs: $4 + 6 = 10 > 8$ ✓, $4 + 8 = 12 > 6$ ✓, $6 + 8 = 14 > 4$ ✓.

Yes — a triangle is possible.

Types of Triangles

By sides:

Type	Description
Equilateral	All three sides equal; all angles = 60°
Isosceles	Two sides equal; two base angles equal
Scalene	All three sides different; all angles different

By angles:

Type	Description
Acute	All three angles less than 90°
Right	One angle is exactly 90°
Obtuse	One angle is greater than 90°

Constructing a Triangle Given Three Sides (SSS)

1
Draw the base
Draw side $a$ (e.g. 6 cm) as a horizontal segment $BC$ using a ruler.
2
Set compass for side b
Open the compass to the length of side $b$ (e.g. 5 cm). Place the tip on $B$ and draw an arc above the base.
3
Set compass for side c
Open the compass to the length of side $c$ (e.g. 4 cm). Place the tip on $C$ and draw an arc that crosses the first arc. Mark the crossing point $A$ .
4
Complete the triangle
Draw segments $AB$ and $AC$ using a ruler. Triangle $ABC$ is complete.

Constructing a triangle with sides 6 cm, 5 cm, and 4 cm

Draw $BC = 6$ cm. Arc of radius 5 cm from $B$ , arc of radius 4 cm from $C$ . They meet at $A$ . Connect $A$ to $B$ and $A$ to $C$ . Check that all three sides measure correctly.

Constructing a Triangle Given Two Sides and the Included Angle (SAS)

1
Draw the first side
Draw segment $AB$ of the given length (e.g. 7 cm) using a ruler.
2
Draw the included angle
Use a protractor to draw the given angle (e.g. 50°) at vertex $A$ .
3
Mark the second side
Measure the second given length (e.g. 5 cm) along the angle's second arm from $A$ . Mark point $C$ .
4
Complete the triangle
Draw segment $BC$ . Triangle $ABC$ is done.

Constructing a triangle with AB = 7 cm, AC = 5 cm, angle A = 50°

Draw $AB = 7$ cm. At $A$ , use the protractor to mark 50°. Measure 5 cm along that ray to get $C$ . Connect $B$ and $C$ .

Tip Always label all vertices and write the lengths next to the sides after you finish drawing. It makes checking much easier.

Real world: roof trusses

The triangular frames inside a roof (called trusses) are constructed using precise side lengths. Engineers use the SSS method to ensure every truss is the same shape and strong enough.

Real world: bridges

Many bridges use triangular supports because a triangle is a rigid shape — it cannot be pushed out of shape by weight the way a square or rectangle can. The angles and side lengths are calculated carefully before construction.

Real world: triangular tiles

Floor tiles sometimes come in triangular shapes. Two right-angled triangles placed together form a rectangle. Tile setters use the SAS method to cut and place tiles accurately.

Common Mistakes

Forgetting to check the triangle inequality before starting — some sets of lengths are impossible
Not keeping the compass width fixed while drawing the arc
Placing the protractor centre on the wrong vertex when using SAS
Measuring sides with the ruler starting at 1 cm instead of 0 cm

Practice Problems

Can a triangle have sides of 2 cm, 3 cm, and 6 cm? Explain.

Show answer

No. 2 + 3 = 5, which is less than 6. The triangle inequality is not satisfied.

A triangle has angles of 60° and 70°. What is the third angle?

Show answer

180° − 60° − 70° = 50°. The third angle is 50°.