Types of Triangles: Comprehensive Notes & Examples

Table of Contents

Introduction to Triangles
Classification by Sides
Classification by Angles
Special Triangles
Properties of Triangles
Methods to Solve Triangle Problems
Interactive Triangle Calculator
Test Your Knowledge: Triangle Quiz

Introduction to Triangles

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry and has several interesting properties that make it fundamental to various mathematical concepts.

Definition: A triangle is a closed figure formed by three line segments that meet at their endpoints.

Every triangle has:

3 sides
3 angles
3 vertices (corners)

The sum of all internal angles in any triangle is always 180° (or π radians).

∠A + ∠B + ∠C = 180°

Classification of Triangles by Sides

Equilateral Triangle

Definition: A triangle with all three sides equal in length.

Properties:

All sides have equal length: a = b = c
All angles are equal (60°)
It has three lines of symmetry

Area = (√3/4) × a²

Example: A triangle with sides of 5 cm, 5 cm, and 5 cm.

Isosceles Triangle

Definition: A triangle with exactly two sides equal in length.

Properties:

Two sides have equal length
The angles opposite to the equal sides are equal
It has one line of symmetry

Area = (1/2) × b × h

Where b is the base and h is the height.

Example: A triangle with sides of 7 cm, 7 cm, and 10 cm.

Scalene Triangle

Definition: A triangle with no sides equal in length.

Properties:

All sides have different lengths: a ≠ b ≠ c
All angles have different measures
No lines of symmetry

Example: A triangle with sides of 3 cm, 4 cm, and 5 cm.

Classification of Triangles by Angles

Acute Triangle

Definition: A triangle in which all three angles are acute (less than 90°).

Properties:

All angles measure less than 90°
∠A + ∠B + ∠C = 180°

Example: A triangle with angles of 60°, 50°, and 70°.

Right Triangle

Definition: A triangle that has one angle equal to 90° (a right angle).

Properties:

One angle is exactly 90°
The other two angles sum to 90°
Follows the Pythagorean theorem: a² + b² = c²

a² + b² = c²

Where c is the hypotenuse (the side opposite the right angle).

Example: A triangle with angles of 90°, 30°, and 60°.

Obtuse Triangle

Definition: A triangle with one angle greater than 90° (obtuse).

Properties:

One angle measures more than 90°
The other two angles are acute
Still satisfies ∠A + ∠B + ∠C = 180°

Example: A triangle with angles of 30°, 40°, and 110°.

Special Triangles

30°-60°-90° Triangle

Definition: A special right triangle with angles of 30°, 60°, and 90°.

Properties:

If the shortest side has length x:
- The hypotenuse has length 2x
- The remaining side has length x√3

If shortest side = x:
Hypotenuse = 2x
Middle side = x√3

45°-45°-90° Triangle (Isosceles Right Triangle)

Definition: A special right triangle with two 45° angles.

Properties:

The two legs have equal lengths
If each leg has length x, then the hypotenuse has length x√2

If legs = x:
Hypotenuse = x√2

Properties of Triangles

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

a + b > c
b + c > a
a + c > b

Example: Consider the lengths 3, 4, and 8:

3 + 4 = 7, which is less than 8
Therefore, 3, 4, and 8 cannot form a triangle

Area Formulas

There are multiple ways to calculate the area of a triangle:

Base and Height:

Area = (1/2) × base × height
Heron's Formula (when all sides are known):

Area = √(s(s-a)(s-b)(s-c))

Where s = (a + b + c)/2 (semi-perimeter)
Using Sine (when two sides and the included angle are known):

Area = (1/2) × a × b × sin(C)

Where C is the angle between sides a and b

Centroid, Orthocenter, and Circumcenter

Centroid: The point where all three medians intersect (divides each median in a 2:1 ratio)
Orthocenter: The point where all three altitudes intersect
Circumcenter: The point equidistant from all three vertices (center of the circumscribed circle)
Incenter: The point where all three angle bisectors meet (center of the inscribed circle)

Methods to Solve Triangle Problems

Pythagorean Theorem
Trigonometry
Congruence & Similarity
Area Methods

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

c² = a² + b²

Where c is the hypotenuse (the side opposite the right angle) and a and b are the other two sides.

Example Problem:

Find the hypotenuse of a right triangle with legs measuring 6 cm and 8 cm.

Solution:

Using the Pythagorean theorem: c² = a² + b²

c² = 6² + 8²

c² = 36 + 64

c² = 100

c = 10 cm

Trigonometric Methods

Trigonometric ratios and the Law of Sines and Law of Cosines are powerful tools for solving triangles.

Basic Trigonometric Ratios (for Right Triangles)

sin(θ) = opposite/hypotenuse
cos(θ) = adjacent/hypotenuse
tan(θ) = opposite/adjacent

Law of Sines

Used when we know two angles and one side, or two sides and the angle opposite one of them.

a/sin(A) = b/sin(B) = c/sin(C)

Law of Cosines

Used when we know two sides and the included angle (SAS), or three sides (SSS).

c² = a² + b² - 2ab·cos(C)
b² = a² + c² - 2ac·cos(B)
a² = b² + c² - 2bc·cos(A)

Example Problem:

In triangle ABC, if a = 8, b = 10, and C = 60°, find the third side c.

Solution:

Using the Law of Cosines: c² = a² + b² - 2ab·cos(C)

c² = 8² + 10² - 2(8)(10)·cos(60°)

c² = 64 + 100 - 160(0.5)

c² = 164 - 80

c² = 84

c ≈ 9.17

Congruence and Similarity

Congruent Triangles

Triangles are congruent if they have exactly the same shape and size.

Criteria for congruence:

SSS (Side-Side-Side): All three pairs of corresponding sides are equal
SAS (Side-Angle-Side): Two sides and the included angle are equal
ASA (Angle-Side-Angle): Two angles and the included side are equal
AAS (Angle-Angle-Side): Two angles and a non-included side are equal
HL (Hypotenuse-Leg): For right triangles, the hypotenuse and one leg are equal

Similar Triangles

Triangles are similar if they have the same shape but possibly different sizes.

Criteria for similarity:

AAA or AA (Angle-Angle-Angle or Angle-Angle): All corresponding angles are equal (two angles are sufficient since the third is determined)
SSS (Side-Side-Side): All corresponding sides are in proportion
SAS (Side-Angle-Side): Two sides are proportional and the included angles are equal

For similar triangles, if the scale factor is k:
Ratio of sides = k
Ratio of perimeters = k
Ratio of areas = k²