Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Mensuration ↓


Surface Areas and Volumes


In mathematics, measurement refers to the study of geometric shapes, their dimensions, and related parameters such as area, perimeter, surface area, and volume. In this lesson, we will explore two main concepts: surface area and volume. We will explain these concepts with simple language and practical examples.

Understanding surface areas

The surface area of a three-dimensional object is the total area covered by the surfaces of that object. Imagine you are wrapping a gift box with paper; the total paper used represents the surface area of the box.

Surface area of a cube

A cube has six equal square faces. To find the surface area, we find the area of one face and then multiply it by six (because it has six faces).

Formula:

Surface Area of Cube = 6 * side 2

Example: If each side of a cube is 4 cm, find its surface area.

Surface Area = 6 * (4 cm) 2 = 6 * 16 cm 2 = 96 cm 2
Side

Surface area of a rectangular prism (cuboid)

A rectangular prism or cuboid has six rectangular faces. The surface area is calculated by finding the area of all the rectangles and summing them up.

Formula:

Surface Area = 2(lw + lh + wh)

where l is the length, w is the width, and h is the height.

Example: If the length of a box = 5 cm, width = 3 cm, and height = 4 cm, then find its surface area.

Surface Area = 2(5 cm * 3 cm + 5 cm * 4 cm + 3 cm * 4 cm) = 2(15 cm 2 + 20 cm 2 + 12 cm 2) = 2 * 47 cm 2 = 94 cm 2
l And H

Surface area of a sphere

A sphere is perfectly round. Finding its surface area is very easy.

Formula:

Surface Area = 4πr 2

where r is the radius of the sphere.

Example: If the radius of a sphere is 7 cm, find its surface area.

Surface Area = 4π * (7 cm) 2 = 4π * 49 cm 2 = 196π cm 2
R

Surface area of a cylinder

A cylinder has two circular bases and one curved surface. To find its surface area, sum the areas of the two circular bases and the curved surface.

Formula:

Surface Area = 2πr(h + r)

where r is the radius, and h is the height.

Example: Find the surface area of a cylinder with height 10 cm and radius 3 cm.

Surface Area = 2π * 3 cm * (10 cm + 3 cm) = 2π * 3 cm * 13 cm = 78π cm 2
R H

Understanding volume

Volume refers to the amount of space occupied by a three-dimensional object. Think of filling a swimming pool with water; the volume is the amount of water the pool can hold.

Volume of a cube

To find the volume of a cube, multiply the length of its side by twice its side. This tells us how much space is inside the cube.

Formula:

Volume of Cube = side 3

Example: If each side of a cube is 3 cm, find its volume.

Volume = (3 cm) 3 = 27 cm 3

Volume of a rectangular prism (cuboid)

The volume of a rectangular prism can be calculated by multiplying its length, width, and height.

Formula:

Volume = lwh

Example: Find the volume of a rectangular box whose length = 8 cm, width = 5 cm, and height = 2 cm.

Volume = 8 cm * 5 cm * 2 cm = 80 cm 3

Volume of a sphere

The volume of a sphere can be calculated using its radius, which is provided by the following formula.

Formula:

Volume = (4/3)πr 3

Example: If the radius of a sphere is 6 cm, find its volume.

Volume = (4/3)π * (6 cm) 3 = (4/3)π * 216 cm 3 = 288π cm 3

Volume of a cylinder

A cylinder is made up of two circular bases and a height. Its volume can be found using its radius and height.

Formula:

Volume = πr 2 h

Example: What is the volume of a cylinder with radius 3 cm and height 7 cm?

Volume = π * (3 cm) 2 * 7 cm = π * 9 cm² * 7 cm = 63π cm 3

Conclusion

Understanding surface areas and volumes is important in geometry and real-world applications. From wrapping gifts to filling pools, these calculations help us solve everyday problems. This guide will help you understand these concepts easily through the explanations and examples provided. Practice with additional problems to reinforce these ideas!


Grade 10 → 6.2


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Surface Area of a Cube Cuboid and Cylinder

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Surface Areas and Volumes

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