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Deriving Surface Area of a Sphere Using Integration
Deriving Surface Area of a Sphere Using Integration

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Surface Area of Sphere

The surface area of a sphere is the area occupied by the curved surface of the sphere. Circular shapes take the shape of a sphere when observed as three-dimensional structures. For example, a globe or a soccer ball. Let us learn about the formula of surface area of a sphere and how to calculate the surface area of a sphere in this lesson.

 1 What is the Surface Area of Sphere? 2 Derivation of Surface Area of Sphere 3 Formula of Surface Area of Sphere 4 How to Calculate the Surface Area of Sphere? 5 FAQs on Surface Area of Sphere

## What is the Surface Area of a Sphere?

The area covered by the outer surface of the sphere is known as the surface area of a sphere. A sphere is a three-dimensional form of a circle. The difference between a sphere and a circle is that a circle is a 2-dimensional shape (2D shape), whereas a sphere is a 3-dimensional shape. The surface area of a sphere is expressed in square units. Observe the sphere given below which shows the center, the radius, and the diameter of a sphere.

### Sphere Definition

Sphere is a three-dimensional round-shaped object with no vertices or edges. The important aspects of this shape are radius, diameter, circumference, and volume.

## Derivation of Surface Area of Sphere

A sphere is round in shape, therefore to find its surface area, we relate it to a curved shape, such as the cylinder. A cylinder is a shape that has a curved surface along with flat surfaces. Now, if the radius of a cylinder is the same as the radius of a sphere, it means that the sphere can fit into the cylinder perfectly. This means that the height of the cylinder is equal to the height of the sphere. So, this height can also be called as the diameter of the sphere. Therefore, this fact was proved by a great mathematician, Archimedes, that if the radius of a cylinder and sphere is ‘r’, the surface area of a sphere is equal to the lateral surface area of the cylinder.

Hence, the relation between the surface area of a sphere and lateral surface area of a cylinder is given as:

Surface Area of Sphere = Lateral Surface Area of Cylinder

The lateral surface area of a cylinder = 2πrh, where ‘r’ is the radius and ‘h’ is the height of the cylinder. Now, the height of the cylinder can also be called the diameter of the sphere because we are assuming that this sphere is perfectly fit in the cylinder. Hence, it can be said that height of the cylinder = diameter of sphere = 2r. So, in the formula, surface area of Sphere = 2πrh; ‘h’ can be replaced by the diameter, that is, 2r. Hence, surface area of sphere is 2πrh = 2πr(2r) = 4πr2

## Formula of Surface Area of Sphere

The formula of the surface area of the sphere depends on the radius of the sphere. If the radius of the sphere is r and the surface area of the sphere is S. Then, the surface area of the sphere is expressed as:

Surface Area of Sphere = 4πr2; where ‘r’ is the radius of the sphere.

In terms of diameter, the surface area of a sphere is expressed as S = 4π(d/2)2
where d is the diameter of the sphere.

## How to Calculate the Surface Area of Sphere?

The surface area of a sphere is the space occupied by its surface. The surface area of the sphere can be calculated using the formula of the surface area of the sphere. The steps to calculate the surface area of a sphere are given below.

Let us take an example to learn how to calculate the surface area of a sphere using its formula.

Example: Find the surface area of a spherical ball that has a radius of 9 inches.

• Step 1: Note the radius of the sphere. Here, the radius of the ball is 9 inches.
• Step 2: As we know, the surface area of sphere = 4πr2, so after substituting the value of r = 9, we get, surface area of sphere = 4πr2 = 4 × 3.14 × 92 = 4 × 3.14 × 81 = 1017.36
• Step 3: Therefore, the surface area of the sphere is 1017.36 in2

## Curved Surface Area of Sphere

The curved surface area of a sphere is the total surface area of the sphere because a sphere has just one surface which is curved. Since there is no flat surface in a sphere, the curved surface area of a sphere is equal to the total surface area of the sphere. Therefore, the formula for the curved surface area of a sphere is expressed as, Curved surface area of sphere = 4πr2; where ‘r’ is the radius of the sphere.

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## Surface Area of Sphere Examples

1. Example 1: If the radius of a sphere is 20 feet, find its surface area. (Use π = 3.14).

Solution: Given, the radius ‘r’ of the sphere = 20 feet.

The surface area of the sphere = 4πr2 = 4 × π × 202 = 5024 feet2

∴ The surface area of the sphere is 5024 feet2

2. Example 2: Find the surface area of a sphere if its radius is given as 6 units.

Solution: Given, the radius ‘r’ = 6 units. So, let us substitute the value of r = 6 units

⇒ The surface area of the sphere = 4πr2 = 4 × π × 62 = 4 × 3.14 × 36 = 452.16 unit2

∴ The surface area of the sphere is 452.16 unit2

3. Example 3: State true or false.

a.) A sphere is a three-dimensional form of a circle.

b.) The curved surface area of a sphere is the total surface area of the sphere because a sphere has just one surface which is curved.

Solution:

a.) True, a sphere is a three-dimensional form of a circle.

b.) True, the curved surface area of a sphere is the total surface area of the sphere because a sphere has just one surface which is curved.

## FAQs on Surface Area of Sphere

### What is the Surface Area of Sphere in Math?

The surface area of a sphere is the total area that is covered by its outer surface. The surface area of a sphere is always expressed in square units. The formula for the surface area of a sphere depends on the radius and the diameter of the sphere. It is mathematically expressed as 4πr2; where ‘r’ is the radius of the sphere.

Why is the Surface Area of a Sphere 4 Times the Area of a Circle?

A string that completely covers the surface area of a sphere can completely cover the area surface of exactly four circles. This way you can check that the surface area of a sphere is four times the area of a circle. When we write the formula for the surface area of a sphere, we write the surface area of a sphere = 4πr2 = 4(πr2) = 4 × area of a circle.

### How Many Sides and Vertices Does a Sphere Have?

A sphere is a three-dimensional shape that is round like a circle. Hence, it has no sides, vertices, or faces.

### Does a Sphere have Infinite Faces?

No, a sphere has no face. A face is a flat surface and a sphere has no flat surface. This makes the sphere a faceless three-dimensional shape (3D shape).

What is the Curved Surface Area and Total Surface Area of a Sphere?

A sphere has just one surface and that is curved. Since there is no flat surface in a sphere, the curved surface area of a sphere is equal to its total surface area of the sphere which is 4πr2.

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What is the Surface Area of a Sphere Formula in Terms of Diameter?

The surface area of a sphere formula in terms of diameter is given as, πD2 where ‘D’ is the diameter of the sphere. It gives the relationship between the surface area of a sphere and the diameter of the sphere.

How to Calculate the Surface Area of a Sphere With the Volume?

The surface area of a sphere can be easily calculated with the help of the volume of the sphere. In this case, we should know the value of the radius of the sphere. The radius of the sphere can be calculated from the formula of the volume of the sphere, that is, Volume of a sphere = 4/3 × πr3. From this, the radius can be calculated and then its value is substituted in the formula for the surface area. We know that the surface area of the sphere = 4πr2. Another way to follow this is as follows. From the volume formula, we can derive that, r3 = 3V/4π, or r = (3V/4π)1/3. After this, we can substitute the value of r in the surface area of the sphere formula.

### What is the Surface Area of Sphere Calculator?

Surface area of sphere calculator is an online tool available for kids to ease their calculations. It is system generated tool where the surface area formula is pre-fixed all we have to do is enter the value of the given parameters, such as radius and we get the surface area of the sphere. Try now Cuemath’s surface area of a sphere calculator and get your answers in a few seconds.

How Does the Surface Area of Sphere Change When the Radius is Halved?

The surface area of the sphere gets one-fourth when the radius is halved because r becomes r/2. As, the surface area of a sphere = 4πr2, so, if we replace ‘r’ with r/2, the formula becomes 4π(r/2)2 = πr2 which is one-fourth of the surface area. Thus, the surface area of the sphere gets one-fourth as soon as its radius gets halved.

How does the Surface Area of a Sphere Change When the Radius is Tripled?

The surface area of the sphere becomes 36πr2 when the radius is tripled because ‘r’ becomes 3r’. We know that the surface area of a sphere = 4πr2, so if we replace ‘r’ with 3r, we get the formula as, surface area = 4π(3r)2 = 4π × 9r2 = 36πr2

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