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 ↓


Surface Area and Volume of a Sphere and Hemisphere


In mathematics, the concepts of surface area and volume play an important role in understanding the geometry of various shapes. Here, we explore these ideas specifically in relation to spheres and hemispheres. This topic is important because it provides information about the properties and measurements of these shapes, which have real-world applications in various fields such as physics, engineering, and architecture.

Understanding the region

The sphere is a perfectly symmetric three-dimensional geometric object where every point on its surface is the same distance from its center. Because of its uniformity, the sphere represents an example of simplicity and symmetry in geometry.

Visual representation

Circle

Surface area of a sphere

The surface area of a sphere is the total area enclosed by the surface of the sphere. It can be calculated using the formula:

Surface Area of a Sphere = 4πr²

Here, r represents the radius of the sphere, which is the distance from the center of the sphere to any point on its surface. π (pi) is a constant that is approximately equal to 3.14159.

Example

If you have a sphere with a radius of 5 cm, the surface area of the sphere can be calculated as follows:

Surface Area = 4 × π × (5)² = 4 × π × 25 = 100π cm² ≈ 314.16 cm²

Volume of a sphere

The volume of a sphere is the amount of space it occupies, which can be determined using the following formula:

Volume of a Sphere = (4/3)πr³

Again, r is the radius, and π is the mathematical constant pi.

Example

To find the volume of a sphere of radius 5 cm, use the formula as shown below:

Volume = (4/3) × π × (5)³ = (4/3) × π × 125 = (500/3)π cm³ ≈ 523.33 cm³

Understanding hemispheres

A hemisphere is half of a sphere. You can think of it as cutting a sphere into two equal halves along its diameter.

Visual representation

Hemisphere

Surface area of a hemisphere

The surface area of a hemisphere has two parts: the curved surface area and the base area. The formula for the curved surface area of a hemisphere is half the surface area of a whole sphere:

Curved Surface Area of a Hemisphere = 2πr²

Add the area of the circular base, which is given by:

Base Area = πr²

Thus, the total surface area of a hemisphere is:

Total Surface Area of a Hemisphere = 2πr² + πr² = 3πr²

Example

Consider a hemisphere of radius 5 cm. The total surface area of the hemisphere is calculated as:

Total Surface Area = 3 × π × (5)² = 3 × π × 25 = 75π cm² ≈ 235.62 cm²

Volume of a hemisphere

The volume of a hemisphere is half the volume of a perfect sphere. Therefore, the formula for the volume of a hemisphere is:

Volume of a Hemisphere = (2/3)πr³

Example

Let's find the volume of a hemisphere with radius 5 cm:

Volume = (2/3) × π × (5)³ = (250/3)π cm³ ≈ 261.67 cm³

Real-world applications

Understanding the surface area and volume of spheres and hemispheres is important in many real-world applications. For example, these calculations are important in determining the amount of material needed to build spherical objects such as balls, spherical tanks, or domes. Architects use these calculations to design structures with curved surfaces.

Engineers often rely on knowledge of spherical volume to calculate the capacity of containers. The principles discussed here are also essential in subjects such as astronomy and geography, where the curvature of the Earth and various celestial bodies is estimated using spherical models.

Practice problems

  1. Find the surface area of a sphere of radius 10 cm.
  2. Find the volume of a sphere of diameter 24 cm.
  3. The radius of a hemisphere is 7 cm. Find its total surface area.
  4. What is the volume of a hemisphere of radius 12 cm?
  5. If the spherical tank can hold a volume of 2000 litres, find its radius.

Solution

  1. Surface Area = 4π(10)² = 400π cm² ≈ 1256.64 cm²
  2. Radius = Diameter / 2 = 24 / 2 = 12 cm
    Volume = (4/3)π(12)³ = (6912/3)π cm³ = 2304π cm³ ≈ 7238.23 cm³
  3. Total Surface Area = 3π(7)² = 147π cm² ≈ 461.81 cm²
  4. Volume = (2/3)π(12)³ = 1152π cm³ ≈ 3619.11 cm³
  5. Volume = (4/3)πr³ = 2000 litres × 1000 cm³/litre = 2000000 cm³
    r³ = (3/4) × (2000000/π) ≈ 477464.83
    r ≈ 78.02 cm

Grade 9 → 13.4


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Surface Area and Volume of a Cone
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