# Sphere Basics

Most three dimensional circular shapes are spherical in structure. These include a globe, a soccer ball, and even an orb.

In the simulation given below, drag the screen and carefully observe the shape of the golden ball through multiple angles.

Isn't it interesting? Let's explore further.

In this lesson, we will explore about Sphere by learning about the definition of the sphere, shape of a sphere, surface area and volume of a sphere, properties of sphere and difference between circle and sphere with the help of interesting simulations, some solved examples and a few interactive questions for you to test your understanding.

## Shape of Sphere

### Definition

A sphere, in geometry, is a three-dimensional figure that is perfectly symmetrical and round in shape like a ball or a globe.

The word 'Sphere' is taken from the Greek word 'sphaira' which means a ball or a globe.

Any point on the surface of the sphere is equidistant from the center of the sphere. We can say that Circle, in a three-dimensional space, takes the form of a sphere.

## Surface Area of a Sphere

The surface area of a sphere is the area covered by the outer surface of the sphere. The surface area of a sphere is always measured in square units.

A sphere has just one surface only and that too curved.

Therefore, the curved surface area of a sphere is the same as the total surface area of the sphere.

Curved surface area of sphere = Total surface area of a sphere

Let us directly see the formula to determine the surface area of a sphere with radius $$r$$

 Surface Area of a Sphere = $$4\pi r^2$$

Another formula to determine the surface area of sphere using diameter $$d$$ is:

 Surface Area = $$4\pi\left( \dfrac{d}{2} \right)^2$$

Try your hands at the surface area of the sphere calculator and find the area by writing the value of radius.

## The Volume of a Sphere

The volume of a sphere is the capacity that it can hold or the space it occupies. For example, if you blow a balloon, you will see that you can fill air up to a level only and after that, your balloon will burst. So, that is the volume of your spherical balloon.

It is interesting to note that the sphere is the 3D figure that has the smallest surface area for a volume. It means that for the same volume, a cube or any other 3D object will occupy more surface area than a sphere.

The volume of a sphere is always measured in cubic units.

Now let us look at the formula to calculate the volume of a sphere:

 Volume of a Sphere = \begin{align}\frac{4}{3}\pi r^3\end{align}

In the simulation given below, write the value of radius to find the volume of the sphere.

## Properties of a Sphere

Let us look at some properties of the sphere that help us to identify spherical objects from other geometrical figures.

• A sphere has no edges, no faces, and no vertices.
• It has only one curved surface.
• It is not a polyhedron as it does not have a flat face.
• Any point on the surface of the sphere is equidistant from the center of the sphere.
• It is symmetrical in shape as we get two identical halves if we cut it through its diameter. We call it the Hemisphere.

## Difference between Circle and Sphere

If you have a spherical ball that you cut into two pieces, what would be the shape of the cross-section?

You will get a circle, right

Try your hands at the simulation given below and carefully observe the shape of the cross-section of the sphere.

If we cut a sphere through a plane, we get one flat face that is circular in shape.

Thus we can say that a sphere lies in a three-dimensional space and a circle lies on a two-dimensional plane.

The difference between a sphere and a circle is tabulated below:

 Basis of Differentiation Circle Sphere Dimension A circle is a two-dimensional figure. A sphere is a three-dimensional figure. Volume It does not have volume. It has a volume that can be calculated with the help of radius. Faces It has one flat face. It has no faces, as it is not flat. Area Formula Area of a Circle= $$\pi r^2$$ Surface Area of a Sphere= $$4\pi r^2$$

Important Notes
1. If we cut a sphere through its diameter, we get two equal halves known as Hemisphere.
2. Volume of a sphere =  \begin{align}\frac{4}{3}\pi r^3\end{align}
3. In the case of a sphere, curved surface area and total surface area are the same.
4. The surface area of a sphere = $$4\pi r^2$$
5. In nature, we observe various spherical objects like water bubbles, planets, sun, moon, other celestial bodies, etc.

## Solved Examples

 Example 1

James and his friends are playing soccer. The surface area of the ball is $$616 \text{ sq. units}$$. James wants to determine the radius of the ball, can you help him?

Use $$\pi =\dfrac{22}{7}$$

Solution

The surface area of the ball is $$616 \text{ sq. units}$$

Since a soccer ball is spherical, therefore the radius of the ball is:

\begin{align} 4\pi r^2&=616\\4\times \dfrac{22}{7}\times r^2&=616\\r^2&=616\times \dfrac{7}{22}\times \dfrac{1}{4}\\r^2&=49\\r&=7\text{ units}\end{align}

 $$\therefore$$ The radius of the soccer ball is $$7\text{units}$$.
 Example 2

The cross-section of a rubber ball has an outer diameter of $$22 \text{ inches}$$

The thickness of the rubber is $$0.5 \text{ inch}$$

What is the surface area of the inner surface of the ball to the nearest sq. inches?

Solution

Outer diameter = $$22 \text{ inches}$$

Outer radius = $$\dfrac{22}{2}\text{inches}=11\text{ inches}$$

Thickness = $$0.5 \text{ inch}$$

Inner radius $$r$$ = Outer radius - Thickness = $$11-0.5$$ = $$10.5 \text{ inches}$$

The inner surface area of the ball is:

\begin{align} 4\pi (r)^2&=4\times 3.14 \times (10.5)^2\\&=4\times 3.14\times 110.25\\&=1384.74 \text{ inches}^2\end{align}

 $$\therefore$$ The required surface area is $$1384.74 \text{ inches}^2$$ .
 Example 3

Joseph is playing with a ball.

He measures the diameter of the ball as 14 inches.

He wonders how many cubic inches of air can the ball hold.

Can you help him?

Solution

The amount of air inside the ball will occupy the entire space in the ball.

So, what we need to find here is the volume of the ball.

The radius of the ball is $$\dfrac{14}{2} \; \text{inches}=7\; \text{inches}$$

The volume of the ball is:

\begin{align}\text{Volume of the ball}&=\dfrac{4 \pi r^3}{3}\\&=\dfrac{4 \times 22\times (7)^3}{7\times3} \\&=1437.3\; \text{inches}^3\end{align}

 $$\therefore$$ The amount of air that ball contains $$1437.3\; \text{inches}^3$$
 Example 4

Cantaloupe has a diameter of about 28 inches.

What would be its surface area?

Solution

Diameter = $$28 \text{ inches}$$

Radius = $$\frac{{28}}{2} =14 \text{ inches}$$

\begin{align} \text{Surface area of sphere } &= 4\pi r^2 \0.2cm] &= 4 \times \frac{{22}}{7} \times 196 \\[0.2cm] &= 88 \times28\\[0.2cm] &= 2464 \text{ sq.inches } \\ \end{align}  $$\therefore$$ Surface area of cantaloupe = 2464 sq.inches  Example 5 A spherical ball has a volume of $$343\ inches^3$$. Find the radius of the ball. Solution \[\begin{align} \text {Volume of the sphere} &= 343 \text { inches}^3 \\ \dfrac 43 \times \pi \times r^3 &= 343\text { inches}^3\\ r^3 &=\dfrac {343\times3}{4\pi}\\ r^3 &= \dfrac {343\times3}{4\times3.14}\\ r^3 &= 81.93 \text { inches}^3\\ r &= 4.34 \text { inches} \end{align}

 $$\therefore$$ The radius of the ball is $$4.34 \text { inches.}$$

Challenging Questions
1. If we double the diameter of a sphere, what would be the change in its volume?
2. Try determining the surface area of planet Earth.
3. The surface area of a sphere is 616 square inches. If its area gets reduced by 75%, what will be its new radius?

## Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

## Let's Summarize

The mini-lesson targeted the fascinating concept of the Sphere. The math journey around Sphere starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

## 1. What is a sphere?

A sphere is a three-dimensional solid object that is symmetrical and round in shape.

## 2. What is the volume of the sphere?

The volume of the sphere is the space occupied by it. It can be calculated using the following formula: $$Volume\ of\ sphere=\frac{4}{3}\pi r^3$$

## 3. How many sides does a sphere have?

A sphere does not have any sides. It has only one curved surface.

## 4. Is a sphere a 3-dimensional shape?

Yes, the sphere is a three-dimensional shape.

## 5. How many faces does a sphere have?

A Sphere has no faces. It has one curved surface only.

Mensuration and Solids
Mensuration and Solids
grade 10 | Questions Set 1
Mensuration and Solids