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 ↓


Surface Area and Volume


In this lesson, we will explore the concepts of surface area and volume. These are two important measurements in mathematics, especially in the field of geometry. Surface area refers to the total area of an object's surface, while volume refers to the amount of space occupied by an object.

Understanding surface area

Surface area is the sum of the areas of all the surfaces of a three-dimensional object. Let us understand this with some simple examples.

Surface area of a cube

A cube is a three-dimensional shape with six equal square faces. If the length of each side of the square is a, then the area of one face of the cube is a * a or a 2.

Since a cube has six faces, the total surface area (SA) of the cube is:

SA = 6 * a 2

For example, if each side of the cube is 4 cm, then the surface area is:

SA = 6 * 4 2 = 96 cm 2

Surface area of a rectangular prism

A rectangular prism, also called a cuboid, has six rectangular faces. Let its length be l, width w and height h. The surface area (SA) of a rectangular prism can be found as follows:

SA = 2(lw + lh + wh)

Here's a visual example to help you understand it:

For example, if a rectangular prism has a length of 8 cm, a width of 5 cm, and a height of 10 cm, then the surface area is:

SA = 2(8 * 5 + 8 * 10 + 5 * 10) = 2(40 + 80 + 50) = 340 cm 2

Surface area of a cylinder

A cylinder has two circular bases and a curved surface. Let's denote the radius of the base by r and the height of the cylinder by h. The surface area (SA) of the cylinder is composed of the areas of the two bases and the curved surface area:

SA = 2πr 2 + 2πrh

Here's a simple, conceptual representation:

For example, if the radius of a cylinder is 3 cm and the height is 7 cm:

SA = 2π(3) 2 + 2π(3)(7) = 2π(9) + 2π(21) = 18π + 42π = 60π cm 2

Note: Use π ≈ 3.14 to calculate approximate numerical results.

Understanding volume

Volume is a measure of the amount of space occupied by a three-dimensional object. It is expressed in cubic units.

Volume of a cube

The volume of a cube is simply the cube of the length of its side. If the side of the cube is a, then the volume (V) is:

V = a 3

For example, if each side of a cube is 5 cm, then its volume is:

V = 5 3 = 125 cm 3

Volume of a rectangular prism

The volume of a rectangular prism (cuboid) is the product of its length, width and height. Let the length be l, width w and height h. The volume (V) is:

V = l * w * h

Using an example: If a prism has a length of 10 cm, a width of 4 cm, and a height of 6 cm, then its volume is:

V = 10 * 4 * 6 = 240 cm 3

Volume of a cylinder

The volume of a cylinder is found by multiplying the area of its base by its height. The base of the cylinder is a circle, so use the formula πr 2 for the area of a circle. Therefore, the volume (V) of the cylinder is:

V = πr 2 h

Let's take an example where the radius is 4 cm and the height is 9 cm:

V = π(4) 2 (9) = π(16)(9) = 144π cm 3

Custom question

To become proficient at calculating surface area and volume, try solving these problems:

Question 1

The length of the side of a cube is 3 cm. Find its surface area and volume.

Surface Area (SA) = 6 * 3 2 = 54 cm 2
Volume (V) = 3 3 = 27 cm 3

Question 2

Find the surface area and volume of a rectangular prism of length 5 cm, width 4 cm and height 2 cm.

Surface Area (SA) = 2(5 * 4 + 5 * 2 + 4 * 2) = 2(20 + 10 + 8) = 76 cm 2
Volume (V) = 5 * 4 * 2 = 40 cm 3

Question 3

A cylinder has a radius of 2 cm and a height of 10 cm. Find its surface area and volume (use π ≈ 3.14 for calculations).

Surface Area (SA) = 2π(2) 2 + 2π(2)(10) = 8π + 40π = 48π ≈ 150.72 cm 2
Volume (V) = π(2) 2 (10) = 40π ≈ 125.6 cm 3

Conclusion

We have discussed how to find the surface area and volume of various three-dimensional shapes such as a cube, rectangular prism, and cylinder. These calculations require an understanding of geometric properties and mathematical operations such as addition, multiplication, and exponents. Practicing with various examples can further enhance the understanding and skills in solving problems involving surface area and volume in real-world scenarios.


Grade 8 → 4.2


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