Types of Triangles

For Elementary Students

What Is a Triangle?

A triangle is a shape with 3 sides and 3 corners (angles).

Think about it like this: "Tri" means three — tricycle has 3 wheels, triangle has 3 sides!

Two Ways to Sort Triangles

We can sort triangles in two different ways:

1. By looking at the sides (how long they are)
2. By looking at the angles (how wide they are)

Sorting by Sides

Equilateral Triangle — All 3 sides are the SAME length

    /\
   /  \
  /____\

  All sides equal!

Think: "Equal" is in the name — all sides are equal!

Isosceles Triangle — 2 sides are the SAME length

    /\
   /  \
  /    \
 /______\

 Two sides equal!

Memory trick: Isosceles sounds like "I saw two sides!" (two equal sides)

Scalene Triangle — All 3 sides are DIFFERENT lengths

     /\
    /  \
   /    \
  /_______\

  All sides different!

Sorting by Angles

Right Triangle — Has one square corner (90°)

    |\
    | \
    |  \
    |___\

    Square corner!

Think: Like the corner of a piece of paper!

Acute Triangle — All angles are small and pointy (less than 90°)

    /\
   /  \
  /____\

  All angles are sharp!

Obtuse Triangle — Has one BIG, wide angle (more than 90°)

      /\
     /  \
    /____\

    One wide angle!

Memory trick: Obtuse = "O" for "Open wide!"

Every Triangle Has BOTH Types!

A triangle has:

• One side type (equilateral, isosceles, OR scalene)
• One angle type (right, acute, OR obtuse)

Example: A triangle can be "right isosceles" — it has a square corner AND two equal sides!

For Junior High Students

Classifying Triangles by Sides

Equilateral Triangle:

• All 3 sides are equal
• All 3 angles are 60°
• Symbol: Tick marks show equal sides

Properties:

• Most symmetrical triangle
• Always an acute triangle
• All sides and angles are congruent

Isosceles Triangle:

• Exactly 2 sides are equal
• The 2 angles opposite the equal sides are also equal
• Has 1 line of symmetry

Properties:

• The equal sides are called "legs"
• The third side is called the "base"
• Base angles are equal

Scalene Triangle:

• All 3 sides have different lengths
• All 3 angles have different measures
• No lines of symmetry

Properties:

• Most "irregular" triangle
• No special relationships between sides or angles

Classifying Triangles by Angles

Acute Triangle:

• All 3 angles are less than 90°
• All angles are "sharp"

Example angles: 60°, 70°, 50°

Right Triangle:

• Has exactly one 90° angle
• The side opposite the 90° angle is called the hypotenuse (longest side)
• The other two sides are called legs

Special property: Pythagorean theorem applies (a² + b² = c²)

Obtuse Triangle:

• Has exactly one angle greater than 90°
• One "wide" or "blunt" angle

Example angles: 110°, 40°, 30°

Combining Classifications

Every triangle has both a side classification and an angle classification.

Possible combinations:

Note: Equilateral triangles are always acute (all 60° angles)

Triangle Angle Sum

Important rule: The sum of all 3 angles in any triangle is 180°

Formula: ∠A + ∠B + ∠C = 180°

Example: If two angles are 50° and 70°, find the third:

• 50° + 70° + ? = 180°
• 120° + ? = 180°
• ? = 60°

Triangle Side Relationships

Triangle Inequality Theorem: The sum of any two sides must be greater than the third side.

Example: Can you make a triangle with sides 3, 4, and 10?

• Check: 3 + 4 = 7, which is NOT greater than 10
• No, these sides cannot form a triangle!

Longest side opposite largest angle: In any triangle, the longest side is always across from the largest angle.

Special Right Triangles

45-45-90 Triangle:

• Isosceles right triangle
• Two legs are equal
• Angles: 45°, 45°, 90°

30-60-90 Triangle:

• Scalene right triangle
• Specific side ratios
• Angles: 30°, 60°, 90°

Identifying Triangles

To classify a triangle:

Step 1: Look at the sides

• All equal? → Equilateral
• Two equal? → Isosceles
• All different? → Scalene

Step 2: Look at the angles

• One 90°? → Right
• All < 90°? → Acute
• One > 90°? → Obtuse

Step 3: Combine both classifications

Example: A triangle with sides 5, 5, 6 and angles 70°, 70°, 40°

• Sides: Two equal → Isosceles
• Angles: All less than 90° → Acute
• Answer: Acute Isosceles Triangle

Real-Life Triangles

Equilateral: Road signs (yield sign), trusses in bridges

Isosceles: Roof trusses, pizza slices (from center)

Right: Ramps, stairs, carpenters' squares

Scalene: Most naturally occurring triangles

Important Facts

• A triangle cannot have 2 right angles (would sum to more than 180°)
• A triangle cannot have 2 obtuse angles (same reason)
• An equilateral triangle is always acute
• The hypotenuse of a right triangle is always the longest side

Practice