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 ↓


Volume of a Cone and Sphere


Understanding volume in mensuration

The concept of volume is important in mathematics, especially in the field of measurement. Volume is the amount of space occupied by a three-dimensional object or shape. When studying shapes like cones and spheres, it is important to understand how volume is calculated. Here, we will explore how to determine the volume of a cone and sphere using simple mathematical formulas and concepts. We will also look at some examples to get a clearer understanding.

Volume of a cone

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex. A simple example of a cone is an ice cream cone.

Formula for the volume of a cone

The formula for finding the volume of a cone is:

Volume of a Cone = (1/3) * π * r² * h

Here, r is the radius of the circular base of the cone, h is the height of the cone, and π (pi) is approximately 3.14159.

Visual Example

Consider a cone with base radius 3 units and height 5 units. Below is an illustration:

Calculation Example

Using the values from our visual example, radius r = 3 units and height h = 5 units. Plug these into the formula:

Volume = (1/3) * π * (3)² * 5 ≈ (1/3) * 3.14159 * 9 * 5 ≈ 47.1239 cubic units

Thus, the volume of the cone is approximately 47.1239 cubic units.

Understanding through more text examples

Consider another example: let's say a cone has a base radius of 7 cm and a height of 10 cm. Using the formula for the volume of a cone, we have:

Volume = (1/3) * π * (7)² * 10 ≈ (1/3) * 3.14159 * 49 * 10 ≈ 1144.66 cubic centimeters

This means that the volume of this cone is approximately 1144.66 cm³.

The formula for the volume of a cone uses the area of the base circle, π * r², and multiplies it by the height h before taking one-third of that product. This accounts for the tapered shape of the cone, which occupies less space than a cylinder with the same base and height.

Volume of a sphere

A sphere is a perfectly symmetrical three-dimensional figure where all points on the surface are the same distance from the center. A common example of a sphere is a basketball.

Formula for volume of sphere

The formula for finding the volume of a sphere is:

Volume of a Sphere = (4/3) * π * r³

Here, r is the radius of the sphere, and π (pi) is approximately 3.14159.

Visual Example

Consider a sphere of radius 4 units. Below is an abstract illustration:

4

Calculation Example

Using the values from our visual example, the radius r = 4 units. Insert this into the formula:

Volume = (4/3) * π * (4)³ ≈ (4/3) * 3.14159 * 64 ≈ 268.082 cubic units

Thus, the volume of the sphere is approximately 268.082 cubic units.

Understanding through more text examples

Consider another example: let's say the radius of a sphere is 6 cm. Using the formula for the volume of a sphere, we have:

Volume = (4/3) * π * (6)³ ≈ (4/3) * 3.14159 * 216 ≈ 904.78 cubic centimeters

This means that the volume of this sphere is approximately 904.78 cm³.

The formula for the volume of a sphere takes into account a spherical shape that is spread out uniformly in all three dimensions around a center point. Thus, it uses the cube of the radius r³.

Comparative example: cone vs. sphere

Let us look at a scenario where we compare the volume of a cone and a sphere having the same radius. Suppose the radius of both the cone and the sphere is 3 units, and the height of the cone is equal to its radius, i.e. 3 units.

For a cone, using its formula, we calculate:

Volume of Cone = (1/3) * π * (3)² * 3 = (1/3) * π * 27 ≈ 28.274 cubic units

For a sphere, use its volume formula:

Volume of Sphere = (4/3) * π * (3)³ = (4/3) * π * 27 ≈ 113.097 cubic units

This comparison shows that even with the same radius, a sphere occupies much more space than a cone of the same height.

Conclusion

Understanding the volume of cones and spheres helps us in many real-world situations, whether designing objects, analyzing shapes, or solving mathematical problems. Using the simple formulas given below:

  • The volume of a cone is (1/3) * π * r² * h, where r is the radius of its base and h is its height.
  • The volume of a sphere is (4/3) * π * r³, where r is its radius.

We can easily calculate how much space these shapes occupy. By looking at various examples and solving practice problems, these concepts become intuitive and useful in various applications.


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Volume of a Cone and Sphere

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