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 ↓


Geometry


Geometry is a branch of mathematics that deals with the shape, measurement, position, and dimensions of things. It is all around us, from the patterns on a tile wall to the design of a modern skyscraper. This subject helps us not only understand the space around us, but also how to navigate through it effectively.

Basic concepts of geometry

To understand geometry, we first need to become familiar with some basic concepts and terms. These basic concepts form the foundation on which more complex ideas are built. Here are some basic concepts:

Point

A point is the most basic element in geometry. It has no shape, length, width, and height. It is usually represented by a dot and labeled with a letter, such as "A". Think of a point as a location on a map, the tip of a pencil touching paper, or a standing position on a playing field.

A

Line

A line is a straight one-dimensional figure that extends infinitely in both directions. Lines are made up of an infinite number of points extending in opposite directions. A line is usually labeled with two points, such as "Line AB" or simply written as AB with a line symbol above it.

A B

Line segment

A line segment is a part of a line that has two end points. Unlike a line, it does not extend to infinity, but rather has a finite length. A line segment is represented with these end points, for example, "AB" represents the segment from point A to point B.

A B

Ray

A ray is a part of a line that starts at one point (called an endpoint) and extends to infinity in one direction. It is represented by writing its endpoint first and then any other point on the ray, such as "ray AB."

A B

Plane

A plane is a flat, two-dimensional surface that extends infinitely in all directions. Like a line, the plane contains an infinite number of points. It is usually represented as a four-sided figure, such as a table-top or a piece of paper, but remember, it extends forever.

Angles in geometry

One of the most important concepts in geometry is the angle. An angle is formed when two rays share a common end point, called the vertex of the angle. Angles are measured in degrees (°).

Types of angles

  • Acute Angle: An angle that is less than 90°.
  • Right Angle: An angle that is exactly 90°.
  • Obtuse Angle: An angle that is more than 90° but less than 180°.
  • Straight Angle: An angle that is exactly 180°.
acute angle Right Angle obtuse angle

Triangle

Triangles are the simplest polygons and have three sides and three angles. They are classified based on their side lengths and angles. Let us learn about the different types of triangles:

Types of triangles according to sides

  • Equilateral triangle: all three sides are the same length, and as a result, all angles are equal, each being 60°.
  • Isosceles Triangle: Two sides are of equal length, and the angles opposite to these sides are also equal.
  • Scalene Triangle: All sides are different lengths, and all angles are different.

Types of triangles based on angles

  • Acute triangle: All three angles are less than 90°.
  • Right angle: It has one angle exactly 90°.
  • Obtuse-angled triangle: One of its angles is more than 90°.
Equilateral Isosceles scalene

Quadrilateral

Quadrilaterals are polygons with four sides and four angles. Different types of quadrilaterals have different properties. Let's take a look at some common types:

  • Square: Four equal sides and four right angles.
  • Rectangle: Opposite sides are equal and there are four right angles.
  • Rhombus: Four equal sides with opposite angles equal, but not necessarily right angles.
  • Parallelogram: Opposite sides are equal and parallel and opposite angles are also equal.
  • Trapezoid (or trapezoid): at least one pair of parallel sides.
Social class rectangle rhombus Quadrilateral

Circle

A circle is a special type of figure in geometry. It is the set of all points in the plane that are equidistant from a given point, called the center of the circle. The main terms related to circles are as follows:

  • Radius: The line segment connecting the center of a circle to any point on its circumference.
  • Diameter: A line segment passing through the centre of a circle and connecting two points on the circumference. It is twice the radius.
  • Chord: A line segment joining two points on the circumference of a circle.
  • Arc: A portion of the circumference of a circle.
  • Circumference: The total distance around the circle.
  • Pi (π): A special mathematical constant, approximately equal to 3.14159, that represents the ratio of a circle’s circumference to its diameter.
radius Diameter Wire Center

Perimeter and area

In geometry, two important measurements of shapes are perimeter and area. These measurements can tell us how much space a shape takes up (area) and how far its boundary extends (perimeter).

Circumference

The perimeter is the total distance around a shape. For various regular shapes, you can calculate the perimeter as follows:

  • For a triangle: Add the lengths of the three sides.
  • For a rectangle: Use the formula: Perimeter = 2 * (Length + Width)
  • For square: Use the formula: Perimeter = 4 * Side
  • For a circle: The perimeter is called the circumference, it is calculated as: Circumference = 2 * π * Radius

Example: Consider a rectangle with a length of 8 and a width of 5. The perimeter is calculated as follows:

 Perimeter = 2 * (8 + 5) = 2 * 13 = 26

Area

Area is the amount of space inside a shape. The formulas for calculating the area of various regular shapes are as follows:

  • For a triangle: Use the formula: Area = 1/2 * Base * Height
  • For a rectangle: Use the formula: Area = Length * Width
  • For square: Use the formula: Area = Side * Side
  • For a circle: Use the formula: Area = π * Radius * Radius

Example: Consider a circle with radius 7. The area is calculated as follows:

 Area = π * 7 * 7 ≈ 3.14 * 49 ≈ 153.86

Pythagorean theorem

The Pythagorean Theorem is a special rule in geometry that applies to right triangles. It deals with the lengths of the sides of a right triangle. It states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b):

 c² = a² + b²

Example: Consider a right triangle where one side is 3 units and the other side is 4 units. To find the length of the hypotenuse:

 C² = 3² + 4² = 9 + 16 = 25
 c = √25 = 5

Therefore, the length of the hypotenuse is 5 units.

Coordinate geometry

Coordinate geometry or analytic geometry is the study of geometry using a coordinate system. This branch of geometry allows us to specify points in the plane using pairs of numbers, commonly called coordinates. The most common coordinate system is the Cartesian coordinate system, which uses two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).

Plotting points

Points on the coordinate plane are plotted using ordered pairs (x, y). The x-coordinate represents the horizontal position, and the y-coordinate represents the vertical position.

Example: Plot the point (3, 4) on the Cartesian plane. This means move 3 units on the x-axis and 4 units on the y-axis.

Distance formula

The distance between two points in the coordinate plane can be calculated using the distance formula:

 Distance = √((x₂ - x₁)² + (y₂ - y₁)²)

Midpoint formula

The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is calculated using the midpoint formula:

 Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

This formula helps you find the midpoint on a straight line connecting two points.

Use of geometry in real life

Geometry has many practical applications in the real world. Engineers, architects, and artists use geometric principles to create everything from bridges to buildings and beautiful artwork. Understanding geometry can also help with everyday tasks such as moving furniture, packing, or even planning a garden.

For example, knowing the area of a room can help you buy the right amount of carpet, or knowing the volume can help you understand how much to fill a swimming pool.

Conclusion

Geometry is an integral part of mathematics that helps us understand and comprehend the world around us. Through learning about points, lines, angles, shapes, and coordinates, you begin to see the beauty and practicality of mathematics. As you delve deeper into geometry, you will discover more complex and exciting concepts that will enhance your ability to reason and think critically.

By practicing geometry, you sharpen your problem-solving skills and gain insights that can be applied across a variety of disciplines. Whether you're looking at a piece of art, designing a new gadget, or exploring the world outside, geometry plays a vital role in interpreting and interacting with your environment.


Grade 8 → 3


U
username
0%
completed in Grade 8

← Prev (2.4.2)
Word Problems on Linear Equations
Next (3.1) →
Understanding Quadrilaterals

Comments 



Geometry

https://www.buddymath.com/g8/3?bmal=en