Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Mensuration → Area and Perimeter ↓


Area of Triangles


When we talk about triangles in math, especially in measurement, we are often concerned with two main properties: their area and perimeter. Here, our focus will be on understanding how to calculate the area of triangles, which is a basic aspect in grade 8 math.

What is a triangle?

A triangle is a three-sided polygon characterized by its three edges and three vertices. The most basic and essential feature of all triangles is that the sum of their interior angles is always 180 degrees. Triangles are classified into several types based on their side lengths and angles such as equilateral, isosceles, scalene, acute-angled, right-angled, and obtuse-angled triangles.

Understanding the area of a triangle

The area of a triangle is the amount of space between its three sides. Imagine cutting a triangular shape out of a piece of paper and covering a table with it; the area is the amount of surface the paper is covering.

Formula to find the area of a triangle

The area for a triangle is calculated using the following formula:

Area = 0.5 × Base × Height

In this formula:

  • The base is any one side of the triangle. In the case of right triangles, the base is often the side on which the triangle stands.
  • The altitude, also called the height, is the perpendicular distance from the base to the opposite vertex.

This formula applies to any type of triangle, provided that you choose the base and its corresponding height correctly.

Visual example

Consider a triangle with base of 6 units and height of 4 units.

Base = 6 units Height = 4 units

The area of this triangle can be found as follows:

Area = 0.5 × 6 × 4 = 12 square units

Types of triangles and associated area calculations

Equilateral triangle

An equilateral triangle has all three sides of equal length. For such triangles, a specific formula can be used to find the area:

Area = (sqrt(3) / 4) × Side²

Example: The area of an equilateral triangle with each side 4 units is:

Area = (sqrt(3) / 4) × 4² = (sqrt(3) / 4) × 16

Isosceles triangle

An isosceles triangle has two sides of equal length. To find its area, we often draw a perpendicular to the base from the vertex opposite the base, forming two right triangles.

Example: An isosceles triangle with a base of 8 units and equal sides of 5 units. Using the Pythagorean Theorem, you can first find the height and then use the area formula:

Height = sqrt(5² - (4)²) = sqrt(25 - 16) = 3
Area = 0.5 × 8 × 3 = 12 square units

Right triangle

In right-angled triangles, one angle is 90 degrees. The two sides forming this right angle are taken as the base and height. Therefore, it is simple to calculate the area using the regular formula.

Heron's formula

It is not easy to measure the height of all triangles. In such cases, especially for scalene triangles, we use Heron's formula. It requires the three sides of the triangle instead of the base and height.

The formula is:

s = (a + b + c) / 2 Area = sqrt(s × (s - a) × (s - b) × (s - c))

where a, b, and c are the lengths of the sides of the triangle, and s is the semiperimeter.

Example: Consider a triangle with sides 5 units, 6 units, and 7 units. First, calculate the semi-perimeter:

s = (5 + 6 + 7) / 2 = 9

Then apply Heron's formula:

Area = sqrt(9 × (9 - 5) × (9 - 6) × (9 - 7)) = sqrt(9 × 4 × 3 × 2) = sqrt(216) ≈ 14.7 square units

More visual examples

Example 1

Base = 8 units Height = 14 units

Given that the base is 8 units and the height is 14 units, calculate the area:

Area = 0.5 × 8 × 14 = 56 square units

Example 2

Height

The base is 100 units, and the height is the distance from the base to the top, let's say it is 70 units. Therefore, the area will be:

Area = 0.5 × 100 × 70 = 3500 square units

Closing thoughts

Understanding how to find the area of a triangle is a useful skill that extends beyond mathematics to a variety of real-life applications, such as construction, art, and engineering. Recognizing the types of triangles and their specific formulas for calculating area can simplify the process and ensure accuracy in problem-solving.

Explore different problems and keep practicing so these calculations become second nature to you. With a solid foundation in the field of triangles, you will be able to effectively tackle more complex geometric challenges in future math studies.


Grade 8 → 4.1.2


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Area of Triangles

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