Title:Curved Surface Area of Sphere
Introduction:
The concept of the Curved Surface Area of Sphere plays a vital role in various branches of mathematics and physics, as well as in real-world applications, from calculating the surface of planets and stars to designing domes and sports equipment. In this article, I will try to give a deep idea into understanding the Curved Surface Area of Sphere, its derivation, real-life applications, and how this formula is applied in various fields.
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Explanation:
To start today’s article Curved Surface Area of Sphere let, us first know,
What is
a Sphere?
Answer:
A sphere is a perfectly symmetrical three-dimensional object where every point on its surface is equidistant from a central point. This central point is called the center, and the distance from the center to any point on it’s surface is known as the radius. Spheres are common in nature and science, from microscopic particles to celestial bodies like planets and stars.
Key Features of a Sphere:
a.It is a round three-dimensional shape.
b.Every cross-section through the center is a perfect circle.
c.All the surface points are equidistant from the center.
Mathematically, the equation of a sphere in three-dimensional space can be expressed as:
(x−x₀)² + ( y - y₀) ² + (z - z₀)² = r²
where (x₀ , y₀ , z₀) is the center, and r is the radius.
Understanding Curved Surface Area of sphere:
In geometry, the surface area of an object refers to the total area that the surface of the object occupies. For two-dimensional shapes like circles and rectangles, calculating surface area is straightforward. However, for three-dimensional shapes like spheres, cones, and cylinders, surface area calculation involves more complex formulas.
The total surface area of an object consists of both flat and curved surfaces. For a sphere, since there are no flat surfaces, the surface area is entirely curved.
Curved Surface Area of Sphere:
_20240926_175151_0000.webp "Curved Surface Area of Sphere")
The Curved Surface Area of Sphere refers to the total area covered by the surface of the sphere without any breaks or interruptions. Mathematically, the curved surface area of a sphere is given by the formula:
A = 4πr²
where:
A represents the surface area,
r is the radius of the sphere,
π(pi) is a constant approximately equal to 3.14159.
This formula signifies that the surface area is proportional to the square of the radius. As the radius increases, the surface area increases quadratically.
Derivation of the Formula:
The formula for the curved surface area of a sphere can be derived using calculus, specifically through the concept of surface integrals. However, a more intuitive way to derive the formula involves the following steps:
Step 1:Concept of
slicing the sphere
Let,us imagine we are slicing the sphere into several infinitesimally small circular bands or strips parallel to its equator. These strips can be thought of as thin rings, each with a small width Δr (delta r)
Step 2:Area of a
Circular Strip
The surface area of each strip can be approximated by the formula for the circumference of a circle. The circumference of a circle at a distance h from the center of the sphere is 2πr where r = √( r² - h²) is the radius of the circular strip at height h.
Summing Over
All Strips
By integrating the areas of all the strips from the top of the sphere (h=r) to the bottom (h=−r), we can derive the total surface area of the sphere. This process involves solving the integral:
r
A = ∫ 2π√(r² -h²)dh
-r
This integral simplifies to 4πr² which is the familiar formula for the curved surface area of a sphere.
Examples and Applications:
Example 1:Calculating the Surface
Area of a Basketball
Let,us assume a basketball has a diameter of 24 cm. To calculate its surface area, we first determine the radius, which is half of the diameter:
Diameter (d) = 24 cm
Radius (r) = 24/2 = 12 cm
Using the formula for the surface area:
A = 4πr²
= 4π (12)² [putting value of r ]
= 4 x 144 x π
= 576 π
By substituting the approximate value of π we get,
A≈576×3.1416 [taking π ≈ 3.1416]
≈ 576×3.1416 ≈1809.56cm2
Thus, the surface area of the basketball is approximately 1809.56 cm².
Example 2:Surface Area
of Earth
If we approximate the radius of Earth as 6,371 km, the surface area can be calculated using the same formula:
A=4πr²
= 4π x (6371)²
=4π×40,589,641
= 4 x 3.1416 x 40589641
= 510,065,664.6624
≈510,065,664km²
Thus, the surface area of the Earth is approximately 510 million square kilometers.
Some problems on curved surface area of sphere.
Question (1):The diameter of a spherical ball 14m,
find its curved surface area.(Take π = 22/7)
Answer:
Diameter of the sphere(d) = 14m
∴ Radius of the sphere(r) =14/2 m
= 7 m
∴ Its curved surface area = 4πr²
= (4x 22/7 x 7) m²
= ( 4 x 22) m²
= 88 m²
Question (2):The radius of a sphere is 100 cm,
find its curved surface area.(Take π = 3.14)
Answer:
Radius (r) = 100 cm
∴ C.S.A of the sphere = 4πr²
= 4 x 3.14 x 100²
= 4 x 3.14 x 100 x 100
= 400 x 314.00
= 400 x 314
= 125600 cm²
Question (3):The curved surface area of
a sphere is 154 m² , find its diameter. (Take π = 22/7)
Answer:
1st Method,
C.S.A = 154 m²
Let, the radius of the sphere = r m
∴ A/Q,
4πr² = 154
=> 4 x 22/7 x r² = 154
=> r² = ( 154 x 7 ) / (4 x 22)
=> r² = ( 7 x 7) / (2 x 2)
=> r = 7 /2
=> r = 3.5
∴ Diameter of the sphere (d) = 2 x r
= 2 x 3.5
= 7 cm
2nd Method,
C.S.A = 154 m²
Let, the radius of the sphere = r m
∴ Radius of the sphere (r)= √ (C.S.A / 4π)
= √ {154 / (4 x 22/7)}
=√ (154 x 7) / (4 x 22)
=√ (7 x 7) / (2 x 2)
= 7 /2
= 3.5 m
∴ Diameter of the sphere (d) = 2 x r
= 2 x 3.5[putting value of r]
= 7 cm
Comparison with Other Geometrical Shapes:
The surface area of a sphere is unique compared to other three-dimensional shapes because it has no edges or vertices, which leads to a minimal surface area for a given volume. For example:
A Cube with the same volume as a sphere
will have a larger surface area.
A cylinder with the same radius as a sphere will also have a larger surface area if its height equals the diameter of the sphere.
This property of spheres makes them efficient in minimizing surface area, which is why many natural objects, like bubbles and water droplets, tend to form spherical shapes.
Real-life Applications:
(a) Astronomy: In astronomy, the surface area of celestial objects like planets, stars, and moons is calculated using the formula for the curved surface area of a sphere. This is essential for understanding phenomena like heat radiation, gravity, and orbital mechanics.
(b) Physics: In thermodynamics, the surface area of a sphere is used to determine the rate of heat transfer from a spherical object. For example, blackbody radiation from stars is often modeled using spherical objects.
(c) Engineering: In engineering, the design of domes, tanks, and other curved structures often involves calculating the surface area of spherical sections to determine material usage and structural strength.
(d) Biology: Spheres appear in biology in the form of cells, bubbles, and even viruses. Understanding the surface area helps in studying processes like diffusion, osmosis, and the exchange of gases across cell membranes.
(e) Sports Equipment: Many sports balls, like soccer balls and tennis balls, are designed with a spherical shape. Knowing the surface area is important for manufacturing, ensuring durability, and predicting aerodynamic behavior.
Common Problems and Solutions:
When calculating the surface area of a sphere, several common mistakes can occur. These include:
(a) We forget to square the radius in the formula.
(b) We may confuse the diameter with the radius
(we must remember, the radius is half the diameter).
(c) Incorrectly rounding the value of π(pi)
By keeping these points in mind, one can avoid errors and ensure accurate calculations.
Conclusion:
The Curved Surface Area of Sphere is a fundamental concept with applications across various fields. Whether you're calculating the surface of a planet, designing spherical objects, or solving problems in physics and engineering, understanding this formula is essential. With a clear derivation and practical examples, this article has provided an in-depth look into the mathematical beauty and real-world relevance of the curved surface area of a sphere.
FREQUENTLY ASKED QUESTIONs ON CURVED SURFACE AREA OF SPHERE:
(i).What is the TSA and CSA of sphere?
+Answer:As sphere has no flat surface.So TSA and CSA of sphere are same.
∴ TSA = CSA of sphere = 4πr²,
where r = radius of the sphere
(ii).Curved surface area of sphere formula
+Answer:CSA of sphere = 4πr²,
where r = radius of the sphere
(iii).Lateral surface area of sphere
+Answer:LSA of sphere = 4πr²,
where r = radius of the sphere
(iv). Volume area of sphere
+Answer:Volume of sphere = 4/3 πr³,
where r = radius of the sphere
(v).What is SI unit of energy?
+Answer:As, we know the ability of doing work is called energy.
So, S.I unit of work and energy are same and it is Joule.
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