Types of Triangles

Learn to classify triangles by their sides and angles

Classifying Triangles by Sides

Triangles can be classified based on the lengths of their sides. There are three types: equilateral, isosceles, and scalene triangles.

1. Equilateral Triangle

An equilateral triangle has 3 equal sides and 3 equal angles (each 60°).

Equilateral triangle: All sides equal, all angles = 60°

All 3 sides are equal (shown by tick marks)
All 3 angles are equal: $60° + 60° + 60° = 180°$

2. Isosceles Triangle

An isosceles triangle has 2 equal sides and 2 equal base angles.

Isosceles triangle: Two equal sides (PQ = PR), two equal base angles

2 sides are equal (PQ = PR, shown by tick marks)
The angles opposite the equal sides are also equal
$\angle Q = \angle R = 70°$ (base angles)

3. Scalene Triangle

A scalene triangle has no equal sides and no equal angles.

Scalene triangle: All sides different, all angles different

All 3 sides have different lengths (no tick marks)
All 3 angles are different: 50°, 60°, 70°

Classifying Triangles by Angles

Triangles can also be classified based on their angles. There are three types: acute-angled, right-angled and obtuse-angled triangles.

1. Acute-angled Triangle

An acute-angled triangle has all angles less than 90°.

Acute-angled triangle: All angles < 90°

All angles (65°, 55°, 60°) are less than 90°

2. Right-angled Triangle

A right-angled triangle has one angle equal to 90°.

Right-angled triangle: One angle = 90° (shown with square marker)

The 90° angle is marked with a small square
The other two angles add up to 90°: $35° + 55° = 90°$

3. Obtuse-angled Triangle

An obtuse-angled triangle has one angle greater than 90°.

Obtuse-angled triangle: One angle > 90°

The angle at I (115°) is greater than 90°

Combined Classifications

A triangle can be described using both classifications. For example:

Example: Right-angled Isosceles Triangle

This triangle is both isosceles (2 equal sides) and right-angled (has a 90° angle).

Right-angled isosceles triangle: 90° angle with two 45° angles

Has a 90° angle at L (right-angled triangle)
Two equal sides JL = KL (isosceles)
Two equal angles of 45°: $45° + 45° + 90° = 180°$

Example: Obtuse-angled Isosceles Triangle

Obtuse-angled isosceles triangle: One obtuse angle, two equal sides

Has an obtuse angle of 110°
Two equal sides (MN = MO)
Two equal base angles of 35°

Practice Problems

Practice 1: Classify this triangle

Classify this triangle by sides and angles

ShowHide solution

By sides: Equilateral (all 3 sides equal - shown by tick marks)

By angles: Acute-angled (all angles are 60°, which is less than 90°)

Answer: Equilateral and acute-angled triangle

Practice 2: Classify this triangle

Classify this triangle by sides and angles

ShowHide solution

By sides: Scalene (the three angles are all different, so no two sides are equal)

By angles: Right-angled (has a 90° angle, marked with a square)

Answer: Scalene and right-angled triangle

Practice 3: Find the unknown angles

Triangle DEF is isosceles with DE = DF. If angle E = 55°, find angle D and angle F.

Isosceles triangle with DE = DF

ShowHide solution

Step 1: In isosceles triangle DEF, DE = DF

The base angles (E and F) are equal.

$\angle F = \angle E = 55°$

Step 2: Use angle sum of triangle = 180°

$\angle D = 180° - 55° - 55° = 70°$

Answer: $\angle D = 70°$ and $\angle F = 55°$

Key Takeaways

By Sides:

Equilateral: 3 equal sides
Isosceles: 2 equal sides
Scalene: No equal sides

By Angles:

Acute-angled: All angles < 90°
Right-angled: One angle = 90°
Obtuse-angled: One angle > 90°