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 ↓


Surface Area of a Cube Cuboid and Cylinder


Measurement is an important part of geometry in mathematics that deals with the measurement of various geometric shapes and sizes. In this detailed explanation, we will focus on understanding the surface area of cubes, cuboids, and cylinders. These are three-dimensional shapes and knowing how to calculate their surface area is very important.

Understanding surface area

The surface area of any three-dimensional figure is the total area covered by the surface of the object. Imagine if you could peel off the outer layer of a three-dimensional object and make it flat, then the area of this flat figure would be the surface area of the object.

Cube

A cube is a special type of cuboid, with all sides the same length. A cube has 6 square faces, 12 edges, and 8 corners.

+----------+
/         /|
+----------+
|     |   |
|     +---+
|    /    |
| /       |
+----------+

For a cube with side length a, the surface area (SA) is calculated using the formula:

SA = 6a^2

Example: If the side length of a cube is 3 cm, what is its surface area?

Use of the formula:

SA = 6a^2 = 6 × 3^2 = 6 × 9 = 54 cm²

The surface area of the cube is 54 cm².

Cuboid

A cuboid is a three-dimensional shape made up of rectangular faces. It has 6 rectangular faces, 12 edges, and 8 vertices.

+---------+
/        /|
+---------+
|       | |
|       + |
|      /  |
|  /      |
+---------+

The surface area of a cuboid with length l, width w and height h is calculated as follows:

SA = 2(lw + lh + wh)

Example: Consider a cuboid with length 5 cm, width 4 cm, and height 3 cm.

Use of the formula:

SA = 2(lw + lh + wh) = 2(5×4 + 5×3 + 4×3) = 2(20 + 15 + 12) = 2×47 = 94 cm²

The surface area of the cuboid is 94 cm².

Cylinder

A cylinder has two circular bases and a curved surface. It resembles the shape of a soup can.

_______
/       /
|       |
|       |
________

For a cylinder with radius r and height h, the total surface area (TSA) is calculated as the sum of the areas of the two circular bases and the curved surface area.

The formula for TSA of a cylinder is:

TSA = 2πrh + 2πr² = 2πr(h + r)

Example: Imagine a cylinder with radius 2 cm and height 7 cm.

Use of the formula:

TSA = 2πr(h + r) = 2×π×2×(7 + 2) = 4×π×9 = 36π cm²

Using the approximation of π (3.14159),

TSA ≈ 36 × 3.14159 = 113.09724 cm²

The total surface area of the cylinder is approximately 113.10 cm².

Practical applications of surface area

Understanding the concept of surface area is important in real-world scenarios. For example, a painter needs to know the surface area of a wall to estimate the amount of paint needed. Engineers use surface area calculations to determine the amount of materials for manufacturing and construction.

More practice problems

1. Find the surface area of a cube with side 10 cm.

Solution:

SA = 6a^2 = 6×(10)^2 = 6×100 = 600 cm²

2. Find the surface area of a cuboid of dimensions 6 cm by 3 cm by 4 cm.

Solution:

SA = 2(lw + lh + wh) = 2(6×3 + 6×4 + 3×4) = 2(18 + 24 + 12) = 2×54 = 108 cm²

3. A metal pipe has a radius of 5 cm and a height of 20 cm. Find its total surface area.

Solution:

TSA = 2πr(h + r) = 2×π×5×(20 + 5) = 10×π×25 = 250π cm²

Using π ≈ 3.14159,

TSA ≈ 250 × 3.14159 = 785.3975 cm²

The total surface area of the pipe is approximately 785.40 cm².

Conclusion

Understanding how to calculate the surface area of 3D objects such as cubes, cuboids, and cylinders is fundamental in solving real-world problems associated with packaging, manufacturing, and other practical activities. Learning these basic geometric principles and how to apply them allows for efficiency and accuracy in a variety of fields of work and study.

Keep practicing with different values and dimensions to strengthen your understanding of surface areas in measurement. Each of the formulas we have discussed is a powerful tool for uncovering the skin or outer layer of three-dimensional objects and translating them into measurable forms.


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Surface Areas and Volumes
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Surface Area of a Cube Cuboid and Cylinder

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