Sphere formula
A perfectly symmetrical 3 – Dimensional circular shaped object is a Sphere. The line that connects from the center to the boundary is called radius of the square. You will find a point equidistant from any point on the surface of a sphere. The longest straight line that passes through the center of the sphere is called the diameter of the sphere. It is twice the length of the radius of the sphere.
Formulas of a Sphere
There are four main formulas for a sphere which include sphere diameter formula, sphere surface area, and sphere volume. All these formulas are mentioned in the table given below and an example is also provided here.
| Sphere Formulas | |
|---|---|
| Diameter of a Sphere | D = 2 r |
| Surface Area of a Sphere | A = 4 π r2 |
| Volume of a Sphere | V = (4 ⁄ 3) π r3 |
Solved Examples Using Formulas of a Sphere
Question: Calculate the diameter, surface area and volume of a sphere of radius 9 cm?
Solution:
Given,
r = 9 cm
Diameter of a sphere
=2r
= 2 × 9
=18 cm
Surface area of a sphere
4πr2 = 4 × π × 92
= 4 × π × 81 = 1017.87 cm
Volume of a sphere = 4/3 πr3
= 4/3 π. 93
= 3053.628 cm3
Sphere
A sphere is a three-dimensional round-shaped object. Unlike other three-dimensional shapes, a sphere does not have any vertices or edges. All the points on its surface are equidistant from its center. In other words, the distance from the center of the sphere to any point on the surface is equal. There are many real-world objects that we see around us which are spherical in shape. Our planet Earth is not in a perfect shape of a sphere, but it is called a spheroid. The reason it is called a spheroid is that it is almost similar to that of a sphere.
What is a Sphere?
In geometry, a sphere is a three-dimensional solid figure, which is round in shape. From a mathematical perspective, it is a combination of a set of points connected with one common point at equal distances in three dimensions. Some examples of a sphere include a basketball, a soap bubble, a tennis ball, etc. The important elements of a sphere are as follows:
- Radius: The length of the line segment drawn between the center of the sphere to any point on its surface. If ‘O’ is the center of the sphere and A is any point on its surface, then the distance OA is its radius (look at the image below for your reference).
- Diameter: The length of the line segment from one point on the surface of the sphere to the other point which is exactly opposite to it, passing through the center is called the diameter of the sphere. The length of the diameter is exactly double the length of the radius.
- Circumference: The length of the great circle of the sphere is called its circumference. In the figure given below, the boundary of the dotted circle or the cross-section of the sphere containing its center is known as its circumference.
- Volume: Like any other three-dimensional object, a sphere also occupies some amount of space. This amount of space occupied by it is called its volume. It is expressed in cubic units.
- Surface Area: The area occupied by the surface of the sphere is its surface area. It is measured in square units.
Sphere Formulas
As we discussed in the previous section, a sphere has a radius, diameter, circumference, surface area, and volume. Considering a sphere to have a radius of ‘r’, the following table lists the important formulas of a sphere.
| Name | Formula |
|---|---|
| Diameter | 2 × radius of the sphere |
| Circumference | 2πr, where π is a constant, which takes the value of 22/7 or 3.14 (approx) |
| Surface Area | 4πr2 |
| Volume | (4/3)πr3 |
Sphere Surface Area
The area covered by the outer surface of the sphere is known as the surface area of a sphere. It is measured in square units. Hence, the formula to find the sphere surface area is:
Surface Area of Sphere, S = 4πr2
In terms of diameter, the surface area of a sphere is given as S = 4π(d/2)2, where d is the diameter. Check out the surface area of the sphere article for more details.
Sphere Volume
The volume of a sphere is the measure of space that can be occupied by it. It is measured in cubic units. The sphere’s volume formula is given below:
Volume of Sphere, V = (4/3)πr3
where,
- V is the volume
- r is the radius, and
- π(pi) is approx. 3.14 or 22/7.
For more details, check this article on volume of sphere.
Properties of a Sphere
A sphere is a three-dimensional object that has all the points on its outer surface to be equidistant from the center. The following properties of a sphere will help you to identify a sphere easily. They are as follows:
- It is symmetrical in all directions.
- It has only a curved surface area.
- It has no edges or vertices.
- Any point on the surface is at a constant distance from the center known as radius.
- A sphere is not a polyhedron because it does not have vertices, edges, and flat faces. A polyhedron is an object that should definitely have a flat face.
- Air bubbles take up the shape of a sphere because the sphere’s surface area is the least.
- Among all the shapes with the same surface area, the sphere would have the largest volume. Sphere’s volume formula is 4/3 × πr3 cubic units.
Circumference of a Sphere
The circumference of a sphere is defined as the length of the great circle of the sphere. It is the total boundary of the great circle. The great circle is the one that contains the center and the diameter of the sphere. It is the largest possible circle that can be drawn inside a sphere. It can also be defined as the cross-section of the sphere when it is cut along its diameter. The sphere circumference can be calculated if its radius is known by using the formula 2πr units, which is the same as the circumference of circle formula.
Difference between Circle and Sphere
A circle and a sphere are two different shapes. The important differences between a circle and a sphere are as follows:
| Circle | Sphere |
|---|---|
| A circle is a two-dimensional shape. | A sphere is a three-dimensional shape. |
| A circle extends in two directions, which are the x-axis and y-axis. | It extends in three directions, which are the x-axis, y-axis, and z-axis. |
| It does not have volume. | It has volume since it occupies some space. |
| It has one flat face. | It has one curved surface. |
| The area of a circle is πr2 square units. | The surface area of a sphere is 4πr2 square units. |
Important Notes:
- The surface area of a sphere is 4πr2.
- The volume of the sphere is 4/3πr3.
- In geometry, half of a sphere is known as a “hemisphere”.
- The total surface area and the volume of a hemisphere formula are exactly half of the sphere area and sphere volume formulas.
Sphere Examples
Example 1: Find the diameter and circumference of a sphere with a radius of 7 units.
Solution:
Given, the radius = 7 units.
Diameter = 2 × radius.
Therefore, diameter = 2 × 7
= 14 units.
Circumference of the sphere = 2πr
= 2 × (22/7) × 7
= 44 units
Therefore, the diameter and the circumference of a sphere of radius 7 units are 14 units and 44 units respectively.
Example 2: Find the volume of a sphere whose radius is 8 units.
Solution:
The formula for the sphere volume is (4/3) πr3 cubic units. Take the value of π as 22/7. Given, the radius = 8 units. Substituting the value of radius in the formula, we get,
Volume = (4/3) π × 83
= (4/3) × (22/7) × 8 × 8 × 8
= 2145.52 cubic units.
Therefore, the volume of the sphere = 2145.52 cubic units.
Example 3: Find the surface area of a sphere whose radius is 5 units. Take the value of π as 22/7.
Solution:
As we know the surface area of the sphere = 4πr2 square units. Substituting the value of radius in the formula, we get,
Surface Area = 4 × (22/7) × 52
= 4 × (22/7) × 25
= 314.28 square units
Therefore, the required surface area is 314.28 square units.
FAQs on Sphere
What is a Sphere Shape?
A sphere is a 3D shape with no vertices and edges. All the points on its surface are equidistant from its center. Some real-world examples of a sphere include a football, a basketball, the model of a globe, etc. Since a sphere is a three-dimensional object, it has a surface area and volume.
What is the Diameter of a Sphere?
The length of a line segment that connects two opposite points on the surface of the sphere passing through its center is called the diameter of a sphere. It can be calculated by multiplying the radius by 2.
How to Find the Surface Area of a Sphere?
The surface area of a sphere is the area occupied by its outer surface or boundary. In simple words, the amount of material used to cover the outer part of a sphere gives its surface area. The formula to find the sphere surface area is 4πr2 square units.
How to Find the Volume of a Sphere?
The volume of a sphere is the amount of space occupied by the sphere. For example, imagine a spherical balloon. The amount of air inside the balloon is its volume. The formula for the volume of a sphere is (4/3) πr3 cubic units.
Does a Sphere Have a Face?
A face is referred to as a flat or curved surface on a three-dimensional object. For instance, a cube has 6 faces. Thereby, a sphere has only one face which is a curved surface. It does not have any flat faces.
What is the Difference Between a Circle and Sphere?
A circle and a sphere are different objects. Since both of them are circular in shape, it creates confusion as if the two shapes are similar. The differences that outline that both are different objects are as follows:
- A circle is a two-dimensional figure whereas a sphere is a three-dimensional object.
- A circle is extended in the x-axis and the y-axis whereas a sphere is extended in three directions (x-axis, y-axis, and z-axis).
- A circle has an area only but a sphere has surface area and volume.
Is Sphere Three-Dimensional?
Yes, a sphere is a three-dimensional object that occupies three axes, which are the x-axis, y-axis, and z-axis. It has a surface area and a volume like any other three-dimensional object.
What is the Difference Between a Sphere and a Spheroid?
A sphere is a three-dimensional object that is perfectly spherical in shape. The radius of the sphere is the same at all points of the sphere from its center, whereas, a spheroid resembles a sphere but the radius is not the same at all points from the center of a spheroid. Planet Earth is considered to be a spheroid in nature.
How to Find the Circumference of a Sphere?
The circumference of a sphere is the length of the boundary of the great circle of the sphere. It is the cross-section that contains the center of the sphere. It can be calculated using the formula 2πr units.
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