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Sphere: Definition, Formulas, Examples, Shapes, Properties

Last Updated : 04 Mar, 2024
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Sphere is a three-dimensional object that is perfectly round and symmetrical in shape. It is a set of points in 3-D space that are all equidistant from a fixed point(center). The distance from the center to any point on the surface of the sphere is the same, and this distance is called the radius. A sphere is defined in 3 axis whereas a sphere is defined only in 2 axis.

In this article, we have explained everything about the Sphere from the Definition of Sphere, Volume, and Surface Area Formula, to Real-life Examples of Spheres. Let’s get a closer look at Sphere in Detail.

Sphere: Definition, Formulas, Examples, Shapes, Properties

Sphere Definition

Sphere is defined mathematically as the set of all points in space that are equidistant from a fixed center point. Each point on the surface is unique and spaced from the center by a predetermined amount. In simple words, a three-dimensional geometric shape that is perfectly round and symmetrical is called a Sphere or we can say sphere is a three-dimensional object without any edges or corners.

Shapes of Sphere

A sphere is a perfectly symmetrical and smooth three-dimensional shape. It encompasses all points within its fixed center and radius. Its surface lacks any sharp edges or corners, offering a seamless appearance. A sphere is formed by continuously curving in all directions from a single point, giving it a perfectly symmetrical and round appearance.

Key characteristics of a sphere’s shape include:

  • Radius: Radius of the Sphere is the distance from the center of the sphere to any point on its surface is called the radius.
  • Surface: The surface of a sphere is continuous, without any flat or sharp regions.
  • Volume: The interior of the sphere contains a three-dimensional space. The volume of the Sphere is calculated by the formula 4/3πr3, where r stands for radius.
  • Symmetry: A sphere exhibits rotational symmetry, meaning it looks the same from any angle of rotation around its center.

Sphere Formulas – Volume and Surface Area

Volume and surface area formula for Sphere are as follows:

Volume of a Sphere (V) = 4/3πr3

Surface Area of a Sphere (A) = 4πr2

Where,

  • π is a Mathematical Constant
  • r is Radius of Sphere

Formulas for diameter, area, and volume are given in the following table:

Sphere Formula
Surface Area (A) 4πr2
Volume (V) (4/3)πr3
Circumference of Circle 2πr
Diameter (d) d = 2r
Radius (r) r = d/2 [Given the diameter]

People also view, Sphere Formulas

Examples of Sphere

Real-life examples of the sphere include the following:

  • Basketball
  • Soccer ball
  • Tennis ball
  • Marbles
  • Moon
  • Balloons
  • Oranges

Difference between Sphere and Circle

The following is the list of Difference between Sphere and Circle:

Property

Circle

Sphere

Dimension

Two-Dimensional shape

Three-Dimensional Shape

Geometry

Closed Curve

Infinite set of points in a plane

Equation

(x – h)2 + (y – k)2 = r2

Infinite set of points in a plane

Example

Tire, Coin, Pizza, etc.

Globe, Basketball, etc.

Sphere Formulas

Some sphere formulas are added below,

Sphere-Formulas

Surface Area of a Sphere

The total surface of a sphere, including the curved surface, is the same in three dimensions. This implies that the area of a sphere’s curved surface must be utilized as the foundation rather than the contribution of its circular base in order to calculate its surface area.

Curved Surface Area of Sphere = Total Surface Area of Sphere

Surface Area of Sphere Formula

“Surface Area” represents the total surface area of the sphere’s outer surface.

Surface Area of Sphere = 4πr² square units

How to Calculate Surface Area of a Sphere?

To calculate the surface area of a sphere, you need to follow these steps:

Step 1: Determine the radius of a sphere­, if its diametr is given divide its diameter by 2.

Step 2: Use the formula 4πr² and claculate the Surface Area

Volume of a Sphere

The volume­ of a sphere indicates the­ space it occupies. Cubic units, such as cubic mete­rs (m3), cubic centimeters (cm3), and cubic inche­s (in3), are used to measure­ this quantity. A sphere, known as a three­-dimensional sphere, has uniformly space­d points from its center. Basketballs and socce­r balls serve as example­s of commonly used spheres, e­ach possessing a unique volume.

Sphere Volume Formula

Volume of a sphere is the amount of space occupied by the sphere’s interior. The following formula is applicable to spheres of various sizes and is a fundamental concept in geometry and mathematics.

Volume of Sphere = 4/3 πr3

How to Find Volume of a Sphere?

  • Check the radius of the specified sphere. To get the radius if you only have the diameter, divide it by 2.
  • Calculate the cube of this radius, represented by the symbol ‘r3‘.
  • (4/3)π, Multiply the fraction by the result of the second step.
  • The final result shows the volume of the sphere.

Read more on How to Find Volume of Sphere?

Sphere Equation in 3D

The equation for a sphere in three-dimensional space is given by:

(x – h)2 + (y – k)2 + (z – l)2 = r2

Where,

  • (x, y, z) are Coordinates of a Point in 3D space.
  • (h, k, l) are Coordinates of Center of sphere.
  • r is Radius of Sphere

This equation describes all the points (x, y, z) that are at a distance r from the center (h, k, l) in three-dimensional space. The squared terms on the left side of the equation ensure that the distance calculation is always positive.

Hemisphere Definition

The term “hemisphere” can be broken down into “hemi,” meaning half, and “sphere,” referring to a three-dimensional shape. Consequently, a hemisphere is a 3D geometric form that represents half of a sphere, with one side being flat and the other resembling a rounded bowl. It comes into existence when a sphere is sliced exactly at its center along its diameter, resulting in two identical hemispheres. The flat side of a hemisphere is often referred to as its base or face.

Hemisphere Definition

Surface Area of Hemisphere

Surface Area of Hemisphere = 3πr2

Where,

  • π is Mathematical Constant ( π = 3.142)
  • “r” is Radius of Hemisphere

Volume of Hemisphere

Volume of Hemisphere = (2πr3)/3

Where,

  • π is Mathematical Constant ( π = 3.142)
  • “r” is Radius of Hemisphere

Difference between Hemisphere and Sphere

Spheres and hemispheres are both round shapes but there are some certain distinctions between both. Some of the common differences between Spheres and Hemispheres are listed in the following table:

Characteristic

Hemisphere

Sphere

Shape

Half of a Sphere A three-dimensional ball or Globe

Surface Area Formula

2πr² 4πr²

Volume Formula

(2/3)πr³ (4/3)πr³

Faces

Curved Surface and a Flat Base Entirely Curved Surface

Example

Dome Basketball

What is Hollow Sphere?

A hollow sphere, also known as a spherical shell or simply a shell, is a three-dimensional geometric object that is similar in shape to a regular solid sphere but has an empty or hollow interior. A hollow sphere is characterized by two radii: the outer radius (R) and the inner radius (r), where R is greater than r.

Surface Area of Hollow Sphere

The surface area of a hollow sphere includes both the outer surface area and the inner surface area.

Surface Area of Hollow Sphere = 4π(R2 + r2)

where,

  • π is Mathematical Constant ( π = 3.142)
  • R is Outer Radius of Hollow Sphere
  • r is Inner Radius of Hollow Sphere

Volume of Hollow Sphere

The volume of a hollow sphere can be calculated by subtracting the volume of the inner sphere from the volume of the outer sphere.

Volume  of Hollow Sphere = (4/3)π(R3 – r3)

where,

  • π is Mathematical Constant ( π = 3.142)
  • R is Outer Radius of Hollow Sphere
  • r is Inner Radius of Hollow Sphere

Calculation of Spheres with Diameter

Calculating spheres with diameter means using the diameter measurement to find the sphere’s properties. It starts by halving the diameter to find the radius, which is often needed for calculations. With the radius, you can find the sphere’s volume, surface area, or other characteristics as required.

Volume of Sphere using Diameter

Volume­ of a sphere can be de­termined by its radius or diamete­r. When the radius is known, the formula is V = (4/3)πr³. However, if the diame­ter is given instead, we can use the formula V = (πd³)/6 to calculate the volume.

Surface Area of a Sphere using Diameter

Surface Area of Sphere when its diameter(d) is given is calculated by the formula,

Surface Area of Sphere = π(D)2

Related Resources,

Solved Examples on Sphere

Some examples of Sphere with their solutions are,

Example 1: Find the curved surface area of a sphere with a radius of 8 cm, using π as 22/7.

Solution:

  • Radius = 8cm

Total Surface Area= 4πr2

Curved Surface Area = 4 × 22/7 × 8 × 8

Curved Surface Area = 804.57cm2

Example 2: Determine the total cost needed to paint a spherical ball with a radius of 9 cm. The cost of painting the ball is INR 7.5 per square cm, and you can use π as 22/7.

Solution:

Given,

  • Radius = 9cm

Total Surface Area= 4πr2

Curved Surface Area  = 4 × 22/7 × 9 × 9

Curved surface area = 1018.28cm2

Cost of painting the ball = 1018.28 × 7.5 = 7637.1

Cost of painting the ball is Rs. 7637.1

Example 3: What is volue of sphere if its diameter is 42 cm.

Solution:

Given,

  • Diameter = 42 cm

Radius = 21 cm

Volume of Sphere(V) = 4/3.π.(r)3

V = 4/3.22/7.(21)3 = 38792 cm3

Practice Problems on Sphere

Some problems on sphere are,

P1: A gas tank has the shape of a sphere with a 14 m diameter. What is the gas tank’s capacity in cubic meters (m³)?

P2: Determine the volume of a volleyball with a radius of 4.5 decimeters, using π as 22/7.

P3: How many times will a sphere’s volume rise if the diameter grows by 10 times?

P4: Calculate the surface area of the sphere given that its volume is 803.84 cubic meters (m3).

P5: Determine the sphere’s surface area. Its volume is 33.9 cubic centimetres (cm3).

Frequently Asked Questions on Sphere

What is Sphere in 3D Geometry?

A sphere in 3D geometry is a three-dimensional object that consists of all points equidistant from a central point. It is like a 3D version of a circle.

How Many Faces Does a Sphere Have?

A sphere has zero faces, as it is a three-dimensional object with a continuous curved surface.

What is Volume of a Hollow Sphere?

Subtract the inner from the outer sphere to determine the volume of a hollow sphere.

What Is Difference Between a Sphere and a Circle?

Unlike a circle, which exists in two dimensions, a sphere exists in three dimensions.

How to Calculate Total Surface Area of a Sphere?

The Surface Area of a Sphere can be calculated using the formula A = 4πr².

How Many Faces, Edges, and Vertices Does a Sphere Have?

A sphere has zero faces, edges, and vertices, as it is a non-polyhedral, continuous surface.

What are Some Real-Life Examples of Spheres?

Real-life examples of spheres include basketballs, soccer balls, planets like Earth, and objects such as marbles and globes.

How to Calculate Volume of a Sphere?

Calculate the volume of a sphere using the formula (4/3)πr³.

How to Measure Total or Curved Surface Area of a Sphere?

To measure the total or curved surface area (CSA) of a sphere, use the formula: CSA = 4πr2.



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