The sphere is a round 3D object. Unlike other 3D shapes, the sphere has no vertices or edges. The distance from the center of the sphere to any point on the surface is the same. Â In geometry, a sphere is a three-dimensional figure with a round shape. From a mathematical point of view, it is a combination of a set of points connected by common points equidistant in three dimensions. Examples of spheres include basketball, soap bubbles, and tennis balls.
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Sphere formulas
There are three main formulas for a sphere, including formulas for the diameter of the sphere, the surface area of ​​the sphere, and the volume of the sphere. All of these formulas are listed in the table below.
Diameter of sphere |
D=2r |
Surface area of sphere |
A=4Ï€r2 |
volume of sphere |
(4/3)Ï€r3 |
What is the Surface Area of a Sphere?
The area covered by the outer surface of the sphere is called the surface area of the sphere. A sphere is a three-dimensional shape of a circle. The main difference between a sphere and a circle is that a circle has a two-dimensional (2D) shape whereas a sphere has a three-dimensional shape.
Derivation of Surface Area of Sphere
Because a sphere is round, we associate it with a curved shape, such as a cylinder, to find the surface area. A cylinder has a curved surface as well as a flat surface. Now if the radius of the cylinder is equal to the radius of the sphere, it implies that the sphere can fit perfectly into the cylinder. This leads us to the conclusion that the height of the cylinder is equal to the height of the sphere. So this height can be called the diameter of the sphere.
We know that if the radius of the cylinder and sphere is the same thenÂ
Surface Area of Sphere = Lateral Surface Area of Cylinder                                       (Proved by Archimedes)
Now the lateral surface area of a cylinder = 2Ï€rh
and the height of the cylinder can also be called the diameter of the sphere because we are assuming that this sphere is flawlessly fit in the cylinder.
Therefore, the height of the cylinder = diameter of the sphere = 2r.
    So,   surface area of sphere is 2πrh = 2πr(2r) = 4πr2                                               ( because h=2r)
Also, the Curved surface area of the sphere is equal to 4Ï€r2 as there is no flat surface in a sphere.
Example: If the radius of a sphere is given as 14cm, then find its surface area. (you can use π = 3.14 for your convenience).
Solution:
It is given that the Radius of the sphere is 14cm.
Now, the Surface area of sphere = 4πr2 = 4 * π *(14)2 = 24 cm2
What is the Volume of the Sphere?
The volume of a sphere is the space occupied by the interior of the sphere. Draw a semicircle on a piece of paper and rotate it 360 degrees to make a sphere. There are two types of spheres which are Solid spheres and hollow spheres. The volumes of the two types of spheres are different. In the next section, we will learn about volumes.
Derivation of Volume of Sphere
As Archimedes has already proved, if a cylinder, cone, or sphere has a radius “r” and the same cross-sectional area, then their volumes are related 1:2:3.
Therefore we can say that
 Volume of Cylinder = Volume of Cone + Volume of Sphere
The volume of Sphere = Volume of Cylinder – Volume of Cone
Now from the previous knowledge we should know that volume of cylinder = πr2h and volume of cone = (1/3)πr2h
putting the values in the above expression, we get
Volume of Sphere = Ï€r2h – (1/3)Ï€r2h = (2/3)Ï€r2h
we are assuming that height of cylinder = diameter of sphere = 2r
Hence, volume of sphere is (2/3)Ï€r2h = (2/3)Ï€r2(2r) = (4/3)Ï€r3
Also, if we have a hollow sphere thenÂ
Let R = radius of the Outer sphere, r = radius of the inner sphere then
 The volume of hollow sphere = Volume of Outer Sphere – Volume of Inner Sphere
⇒ Volume of hollow sphere  =  (4/3)Ï€R3 – (4/3)Ï€r3 = (4/3)Ï€(R3 – r3)
Example 1: Find the volume of the sphere which has a radius of 6 cm.
Solution:
It is given that the radius of the sphere is 6cm.
Now, Volume of sphere = (4/3)Ï€r3 =  ((4/3) × Ï€ × (6cm)3) = 904.779 cm3Â
Example 2: Find the volume of a sphere whose inner radius is 5cm and the outer radius is 8 cm.
Solution:
Outer radius of sphere R = 8 cm.
and Inner radius of sphere r = 5 cm.
Now,  Volume of hollow sphere  = (4/3)Ï€(R3 – r3) = (4/3)Ï€((8cm)3 – (5cm)3) = 1621.062 cm3.
Hemisphere
Hemisphere is half of the sphere. In another word, if a sphere is cut into two symmetrical pieces through the center then it is called a hemisphere. Because it is half of a sphere then the volume and surface area are half of the volume and surface area of a sphere.
Volume of hemisphere = (1/2)(4/3)Ï€r3
Surface area of Sphere = (1/2)(4Ï€r2)
Hemisphere
Properties of Sphere
The following are the properties of a sphere:
- It has no vertex or edge.
- This is not a polyhedron.
- All of the points in the sphere have the same distance to the center.
- It has a curved face, not a flat face.
- It is perfectly symmetrical.
Comparison between Circle and Sphere
Circle |
Sphere |
A circle exists in a two-dimensional shape. |
A sphere is a three-dimensional shape. |
A circle can only extend in two directions, which are the x-axis and the y-axis. |
It extends in all three directions, which are the x-axis, y-axis, and z-axis. |
It does not have any volume. |
It has volume because it occupies some space. |
The area of a circle is πr2 square units. |
The surface area of a sphere is 4Ï€r2 square units. |
Sample Questions
Question 1: A baseball is 80mm in diameter. Find the baseball’s volume. (Ï€=3.14)
Solution:
 we are given that Diameter         D = 2rÂ
                              D = 80mm.
                              r = 40mm
Now, Â Â Â Â Â Â Â Â Â Â Â Â Â Volume of sphere = (4/3)Ï€r3 = (4/3)Ï€(40mm)3 = 268082.573 mm3
Question 2: Â Hollow spheres melt into the same small hollow sphere. The inner and outer radii of the larger sphere are 5 cm and 7 cm, respectively. If the inner and outer radii of the small spheres are 3 cm and 4 cm, respectively, how many small spheres can be formed? (Ï€=3.14)
Solution:
We know that Volume of the sphere = (4/3)Ï€r3
Now, Volume of the bigger sphere = volume of the sphere with outer radius – volume of the sphere with inner radius
⇒  Volume of the bigger sphere =  (4/3) * Ï€ * (7cm)3 –  (4/3)Ï€(5cm)3
⇒  Volume of the bigger sphere =  (4/3) * π * (343-125) = (4/3) * π * (218) cm3
In the same way, Volume of smaller sphere = (4/3) * Ï€ * (4cm)3  – (4/3) * Ï€ * (3cm)3
⇒  Volume of the smaller sphere =  (4/3) * π *(64-27) = (4/3) * π * (37) cm3
hence, the number of spheres that can be formed = volume of the bigger sphere/ volume of the smaller sphere
therefore, the number of spheres that can be formed = (4/3) * π * (218) cm3/ (4/3) * π * (37) cm3
⇒ the number of spheres that can be formed = 5.92 ≈ 6 spheres.
Question 3: When you change the shape of an object from a sphere to a cylinder, then the volume of the cylinder increases, decreases, or remains unchanged. (Ï€=3.14)
Solution:
Volume is a scalar quantity that describes the volume of 3D space surrounded by nearby surfaces.
When transforming a body into another body, the amount of material remains the same, so the volume of the body does not change.
Hence, the volume remain Unchanged.
Question 4: The surface of the sphere is 500 cm2. If you change the radius to reduce the area by 50%, then find the radius. (Ï€=3.14)
Solution:
Since the area get reduced by 50%, we can say that
New surface area = 50% of the original area
⇒ 4Ï€r2 = 1/2 * 500Â
⇒ r2 = ( 1/2 * 500 ) / 4π
⇒ r2=250/12.56
⇒ r2 = 19.8945Â
⇒ r = 4.46 cm
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