Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 ↓


Surface Areas and Volumes


Understanding the concepts of surface areas and volumes is very important in mathematics, especially when dealing with three-dimensional shapes. These concepts help us measure the extent of surfaces and the space occupied by solids. Let's dive into the fascinating world of surface areas and volumes with simple definitions, formulas, and examples that will make learning both fun and effortless.

Basic definitions

Before we explore the world of three-dimensional shapes, let's understand some basic definitions:

Surface area

Surface area is the total area that the surface of an object occupies. It is measured in square units (e.g., square centimeters, square meters). Imagine you are wrapping a gift. The amount of paper you need to cover the entire surface of the box is its surface area.

Volume

Volume is the space an object occupies. It is measured in cubic units (e.g., cubic centimeters, cubic meters). Think of filling a swimming pool with water. The total amount of water needed to fill the pool is its volume.

Cuboid

A cuboid is a three-dimensional figure with six rectangular faces. It looks like a box or a brick. Let's understand how to find its surface area and volume.

Surface area of cuboid

The surface area of a cuboid can be found by finding the areas of its six rectangular faces and adding them together.

The formula for the surface area of a cuboid is:

Surface Area = 2 * (length * width + width * height + height * length)

For example, consider a cuboid with length = 3 cm, width = 2 cm, and height = 5 cm:

Surface Area = 2 * (3 * 2 + 2 * 5 + 5 * 3) = 2 * (6 + 10 + 15) = 2 * 31 = 62 cm²

Volume of a cuboid

The volume of a cuboid is found by multiplying its length, breadth and height.

The formula for the volume of a cuboid is:

Volume = length * width * height

Use the same dimensions as before (length = 3 cm, width = 2 cm, height = 5 cm):

Volume = 3 * 2 * 5 = 30 cm³

Cylinder

A cylinder is a three-dimensional shape with two parallel circular bases and a curved surface connecting them, such as a soup can or drum. Here's how we can find its surface area and volume.

Surface area of a cylinder

The surface area of a cylinder is the sum of the areas of its two circular bases and the curved surface.

The formula for the surface area of a cylinder is:

Surface Area = 2 * π * r * (r + h)

where r is the radius of the base and h is the height of the cylinder.

For example, consider a cylinder with radius = 4 cm and height = 10 cm:

Surface Area = 2 * π * 4 * (4 + 10) = 2 * π * 4 * 14 = 112π cm²

Volume of a cylinder

The volume of a cylinder is the product of the area of its base and its height.

The formula for the volume of a cylinder is:

Volume = π * r² * h

Use the same dimensions as before (radius = 4 cm, height = 10 cm):

Volume = π * 4² * 10 = 160π cm³

Circle

A sphere is a perfectly round geometric object in three-dimensional space, like a ball. Let's see how to find its surface area and volume.

Surface area of a sphere

The surface area of a sphere is calculated using the following formula:

Surface Area = 4 * π * r²

For a sphere with radius r = 3 cm:

Surface Area = 4 * π * 3² = 36π cm²

Volume of a sphere

The formula for the volume of a sphere is:

Volume = (4/3) * π * r³

Using the radius r = 3 cm:

Volume = (4/3) * π * 3³ = 36π cm³

Examples and problems

Example 1: Surface area of cuboid

Find the surface area of a cuboid with length = 8 cm, width = 5 cm, and height = 3 cm.

Surface Area = 2 * (8 * 5 + 5 * 3 + 3 * 8) = 2 * (40 + 15 + 24) = 158 cm²

Example 2: Volume of a cylinder

Find the volume of a cylinder with a diameter of 8 cm and a height of 15 cm. Note: the diameter is twice the radius.

First, determine the radius: radius = diameter / 2 = 8 / 2 = 4 cm.

Volume = π * 4² * 15 = 240π cm³

Practice problems

  1. Surface area of a sphere: Find the surface area of a sphere of radius 7 cm.
  2. Volume of a cuboid: A cuboid has length 10 cm, width 4 cm and height 6 cm. Find its volume.
  3. Volume of a cylinder: Determine the volume of a cylinder of radius 5 cm and height 20 cm.

Concluding remarks

Surface area and volume are important in a variety of fields, from architecture to manufacturing. By mastering these basics, you'll be well-equipped to tackle more complex problems and real-world applications. Take your time to practice with different shapes and formulas, and you'll enhance not only your math skills but also your spatial reasoning abilities. Keep exploring and have fun with geometry!


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Surface Areas and Volumes

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