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.

Table of Content

## 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 of the Sphere is the distance from the center of the sphere to any point on its surface is called the radius.**Radius:**

The surface of a sphere is continuous, without any flat or sharp regions.**Surface:**

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.**Volume:**

A sphere exhibits rotational symmetry, meaning it looks the same from any angle of rotation around its center.**Symmetry:**

## Sphere Formulas – Volume and Surface Area

Volume and surface area formula for Sphere are as follows:

=Volume of a Sphere (V)4/3πr^{3}

Surface Area of a Sphere (A) = 4πr^{2}Where,

is a Mathematical Constantπis Radius of Spherer

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

Sphere |
Formula |
---|---|

Surface Area (A) |
4πr^{2} |

Volume (V) |
(4/3)πr^{3} |

Circumference of Circle |
2πr |

Diameter (d) |
d = 2r |

Radius (r) |
r = d/2 [Given the diameter] |

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## 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) |
Infinite set of points in a plane |

Example |
Tire, Coin, Pizza, etc. |
Globe, Basketball, etc. |

## Sphere Formulas

Some sphere formulas are added below,

## 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:

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

Use the formula 4πr² and claculate the Surface AreaStep 2:

## Volume of a Sphere

The volume of a sphere indicates the space it occupies. Cubic units, such as cubic meters (m^{3}), cubic centimeters (cm^{3}), and cubic inches (in^{3}), 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 πr^{3}

### 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 ‘r
^{3}‘. - (4/3)π, Multiply the fraction by the result of the second step.
- The final result shows the volume of the sphere.

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## 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}= r^{2}Where,

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

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.

### Surface Area of Hemisphere

Surface Area of Hemisphere= 3πr^{2}Where,

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

### Volume of Hemisphere

Volume of Hemisphere= (2πr^{3})/3Where,

- 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(R^{2}+ r^{2})where,

is Mathematical Constant ( π = 3.142)πis Outer Radius of Hollow SphereRis Inner Radius of Hollow Spherer

### 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(R^{3}– r^{3})where,

is Mathematical Constant ( π = 3.142)πis Outer Radius of Hollow SphereRis Inner Radius of Hollow Spherer

## 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

**to calculate the volume.**

**V = (πd³)/6**### 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}

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## 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= 4πr

^{2}Curved Surface Area = 4 × 22/7 × 8 × 8

Curved Surface Area = 804.57cm

^{2}

**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= 4πr

^{2}Curved Surface Area = 4 × 22/7 × 9 × 9

Curved surface area = 1018.28cm

^{2}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 cm^{3}

**Practice Problems on Sphere**

**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 (m**^{3}**).**

**P5: Determine the sphere’s surface area. Its volume is 33.9 cubic centimetres (cm**^{3}**).**

## 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?**

**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?**

**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?**

**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?**

**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?**

**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 = 4πr

^{2}.