Grade 6
  1. Number System  1.1. Whole Numbers  1.1.1. Place Value and Face Value  1.1.2. Comparing and Ordering Numbers  1.1.3. Rounding Numbers  1.1.4. Properties of Whole Numbers (Associative, Commutative, Distributive)  1.2. Integers  1.2.1. Introduction to Integers  1.2.2. Addition and Subtraction of Integers  1.2.3. Multiplication and Division of Integers  1.2.4. Properties of Integer Operations  1.3. Fractions  1.3.1. Types of Fractions  1.3.2. Equivalent Fractions  1.3.3. Simplification of Fractions  1.3.4. Addition and Subtraction of Fractions  1.3.5. Multiplication and Division of Fractions  1.3.6. Comparing and Ordering Fractions  1.4. Decimals  1.4.1. Understanding Decimals  1.4.2. Conversion between Fractions and Decimals  1.4.3. Addition and Subtraction of Decimals  1.4.4. Multiplication and Division of Decimals  1.4.5. Comparing and Ordering Decimals
  2. Algebra  2.1. Basics of Algebra  2.1.1. Variables and Constants  2.1.2. Expressions and Equations  2.1.3. Algebraic Terms and Coefficients  2.2. Solving Simple Equations  2.2.1. Balancing Equations  2.2.2. Solving One-Step Equations  2.2.3. Solving Two-Step Equations  2.3. Patterns and Sequences  2.3.1. Identifying Patterns  2.3.2. Arithmetic Sequences
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Equivalent Ratios  3.1.3. Simplifying Ratios  3.2. Proportion  3.2.1. Introduction to Proportion  3.2.2. Direct and Inverse Proportions  3.3. Percentages  3.3.1. Converting Ratios to Percentages  3.3.2. Percentage Increase and Decrease  3.3.3. Discount and Profit Calculations  3.3.4. Simple Interest Calculations in Percentages in Ratio and Proportion
  4. Geometry  4.1. Basic Geometric Shapes  4.1.1. Points, Lines, and Angles  4.1.2. Types of Angles  4.1.3. Intersecting and Parallel Lines  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Properties of Triangles  4.2.3. Sum of Angles in a Triangle  4.2.4. Congruence of Triangles  4.3. Quadrilaterals  4.3.1. Types of Quadrilaterals  4.3.2. Properties of Quadrilaterals  4.4. Circles  4.4.1. Radius and Diameter  4.4.2. Circumference and Area  4.4.3. Arcs and Chords  4.4.4. Sector and Segment
  5. Mensuration  5.1. Perimeter  5.1.1. Perimeter of Simple Shapes  5.2. Area  5.2.1. Area of Rectangles and Squares  5.2.2. Area of Triangles and Parallelograms  5.2.3. Area of Composite Figures  5.2.4. Area of Trapezium  5.3. Volume  5.3.1. Volume of Cubes and Cuboids  5.3.2. Volume of Cylinders  5.3.3. Volume of Cones  5.3.4. Surface Area of Simple Solids
  6. Data Handling  6.1. Introduction to Data  6.1.1. Types of Data  6.1.2. Organizing Data  6.1.3. Frequency Distribution  6.2. Graphs and Charts  6.2.1. Bar Graphs  6.2.2. Line Graphs  6.2.3. Pictographs  6.2.4. Pie Charts  6.3. Mean, Median, and Mode  6.3.1. Calculating Mean  6.3.2. Finding Median  6.3.3. Identifying Mode  6.3.4. Range of Data in Mean, Median, and Mode
  7. Probability  7.1. Basics of Probability  7.1.1. Understanding Probability  7.1.2. Probability of Simple Events  7.1.3. Experimental and Theoretical Probability
  8. Practical Geometry  8.1. Construction of Shapes  8.1.1. Constructing Triangles  8.1.2. Constructing Quadrilaterals  8.1.3. Constructing Circles and Arcs  8.2. Symmetry  8.2.1. Line of Symmetry  8.2.2. Rotational Symmetry  8.2.3. Translational Symmetry

Grade 6 → Geometry ↓


Triangles


Triangles are a fundamental part of geometry. In this guide, we will learn about triangles in detail. You will learn about their types, properties, angles, sides, and more. Our goal is to make this explanation as simple as possible, so that everyone can understand it easily.

What is a triangle?

A triangle is a polygon with three sides and three vertices. It is one of the basic shapes in geometry. The word "triangle" comes from the Latin words "tri-" meaning three, and "angulus" meaning angle. This makes sense because a triangle has three angles.

ABC

Figure 1: A simple triangle ABC.

Parts of a triangle

A triangle has three main parts: vertices, sides, and angles.

  • Vertices: The points where the sides of a triangle meet are called vertices. For example, in triangle ABC, A, B, and C are the vertices.
  • Sides: The line segments that form a triangle are called sides. In the picture above, the sides are AB, BC and CA.
  • Angle: The space between two intersecting lines is called an angle. There are three angles in a triangle. The angles in triangle ABC are ∠A, ∠B and ∠C.

Types of triangles according to sides

Triangles are classified into different types based on their sides and angles. Let us start with the classification based on sides.

Equilateral triangle

An equilateral triangle has all three sides of equal length, and consequently, all three interior angles are also equal to 60 degrees.

ABC60°60°60°

Figure 2: An equilateral triangle ABC.

Isosceles triangle

The two sides of an isosceles triangle are of equal length. Therefore, the angles opposite to those sides are also equal.

ABCx°x°

Figure 3: An isosceles triangle ABC where AB = AC.

Scalene triangle

A scalene triangle has all its sides different lengths, which means all its angles are different measures as well.

ABCx°y°z°

Figure 4: A scalene triangle ABC.

Types of triangles based on angles

Triangles can also be classified based on their interior angles:

Acute triangle

An acute-angled triangle is a triangle whose all three interior angles are less than 90 degrees.

ABCone°B°C°

Figure 5: An acute-angled triangle ABC.

Right triangle

A right triangle is a triangle that has a right angle (90 degrees).

ABC90°d°E°

Figure 6: A right-angled triangle ABC where ∠ABC = 90°.

Obtuse-angled triangle

An obtuse-angled triangle is a triangle with one angle greater than 90 degrees.

ABCf°yes°90° < h° < 180°

Figure 7: An obtuse-angled triangle ABC.

Properties of triangles

Triangles have some interesting properties that are the same regardless of the type of triangle:

Sum of interior angles

The sum of the interior angles of any triangle is always 180 degrees. This is a universal rule for all triangles.

In mathematical terms:

∠A + ∠B + ∠C = 180°

Exterior angle property

The exterior angle of a triangle is equal to the sum of the two opposite interior angles. This is often used in proofs and problem-solving.

ABC∠D∠A + ∠C

Figure 8: Exterior angle ∠D is equal to ∠A + ∠C.

Pythagorean theorem

This theorem applies specifically to right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Mathematically this can be expressed as:

a² + b² = c²

Where 'c' is the hypotenuse, and 'a' and 'b' are the other two sides.

Perimeter of a triangle

The perimeter of a triangle is the sum of the lengths of its three sides. This is a simple concept:

Perimeter = AB + BC + CA

Area of a triangle

The area of a triangle can be calculated using the following formula:

Area = (base × height) / 2

Here, the base is a side of the triangle, and the height is the perpendicular distance from the base to the opposite vertex.

Examples and exercises

Let's look at some examples to practice what we've learned:

Example 1: Classify a triangle

Given a triangle with sides of length 7 cm, 7 cm and 5 cm, classify the triangle on the basis of its sides.

Solution:

Since two sides are equal, the triangle is an isosceles triangle.

Example 2: Find the missing angle

If two angles of a triangle are 50° and 60°, then find the third angle.

Solution:

Sum of angles = 180° 50° + 60° + ∠C = 180° ∠C = 180° - (50° + 60°) ∠C = 70°

Example 3: Calculate the area

Find the area of a triangle with base 10 cm and height 8 cm.

Solution:

Area = (base × height) / 2 Area = (10 × 8) / 2 = 40 sq. cm

Example 4: Use the Pythagorean theorem

The measure of one side of a right-angled triangle is 3 cm and the measure of the other side is 4 cm. Find the length of the hypotenuse.

Solution:

a² + b² = c² 3² + 4² = c² 9 + 16 = c² 25 = C² c = √25 c = 5 cm

Conclusion

Triangles are fascinating shapes that have unique properties and a wide variety of types. Whether classified by sides or angles, each triangle provides valuable insight into the world of geometry. Understanding the basics of triangles, including their types, properties, and calculating their perimeter and area, provides a strong foundation for further mathematical education.

Keep practicing these concepts, and soon you'll find that triangles are not only simple, but also incredibly interesting. Happy learning!


Grade 6 → 4.2


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Types of Triangles

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Triangles

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