Sphere: Definition, Formulas, Examples, Shapes, Properties

A sphere is a three-dimensional object that is perfectly round and symmetrical. It is defined as the set of all points in 3D space that are equidistant from a fixed point (the center). The distance from the center to any point on the surface is called the radius. A sphere is a 3D shape with no edges or corners.

Key characteristics of a Sphere

• Radius: The 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:

People also view: Sphere Formulas

Examples of Sphere
Real-life Applications 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:

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 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

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

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

Volume of Sphere = 4/3 πr³

Read more on How to Find the 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.

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

Read in Detail:
Surface Area of Hemisphere
Volume 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:

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

The volume of the areaHollow 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

The surface area of the Sphere, when its diameter(d) is given, is calculated by the formula,

Surface Area of Sphere = π(D)²

Related Resources,

• Cube
• Cylinder
• Frustum of Cone

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:

Given,
Radius = 8cm
Total Surface Area= 4πr²

Curved Surface Area = 4 × 22/7 × 8 × 8
CSA = 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 the value of a 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