Volume of a Sphere
Basics on the topic Volume of a Sphere
Volume of a Sphere – Definition
Spheres are a common sight in daily life, from basketballs to the rounded design of some buildings. Understanding the volume of a sphere is about more than just math; it's about relating to everyday objects, such as estimating the air in a soccer ball or the quantity of ice cream in a scoop. This concept is a practical tool for making sense of the world around us, blending mathematical skills with realworld applications.
Volume of a Sphere – Formula
The volume of a sphere is the total space enclosed within the sphere, and a formula is used to calculate this measurement.
$V = \frac{4}{3} \pi r^3$
 $V$ = volume
 $r$ = radius of the sphere
Understanding the Volume of a Sphere
Grasping the volume of a sphere involves recognizing that every point on the surface is an equal distance (the radius) from the center. This uniformity leads to its unique formula.
Volume of a Sphere – Cubic Units
Finding the Volume of Simple 3D Shapes requires you to use specific formulas. But it is also important to label our final solutions with the correct units.
Like all volumes, the sphere's volume is expressed in cubic units, which depend on the radius's measurement. If the radius is in centimeters, then the volume will be in cubic centimeters (cm³).
Volume of a Sphere – StepbyStep Instructions
Calculating the volume of a sphere is a straightforward process:
Step Number  Directions  Example 

1  Identify the radius of the sphere.  Radius $r = 5$ cm 
2  Substitute the value into the formula $V = \frac{4}{3} \pi r^3$.  $V = \frac{4}{3} \pi \times 5^3$ 
3  Calculate the volume, with attention to rounding rules.  $V = \frac{4}{3} \pi \times 125 = 166\frac{2}{3}\pi$ cm³ approx. 523.6 cm³ when rounded to the nearest tenth 
4  Write the final answer with the correct units.  Volume of the sphere is approximately $523.6$ cm³ 
Volume of a Sphere – Guided Practice
Let’s work through a practical example to understand how to calculate the volume of a sphere.
Consider a volleyball with a radius of $12$ cm. Find the volume of the ball.
Besides spheres, the Volume of Prisms can be determined using distinct formulas, yet the method followed is comparably similar.
Volume of a Hemisphere – Guided Practice
Hemispheres are prevalent in various designs and natural formations, such as domed buildings and halfcut fruits. Understanding the volume of a hemisphere is crucial for practical applications in these areas.
The volume $V$ of a hemisphere with radius $r$ is: $V = \frac{2}{3} \pi r^3$.
*Let’s practice with one example of finding the volume of a hemisphere shaped bowl with a radius of 10 cm.**
Calculating Volume 'In Terms of Pi'
When calculating the volume 'in terms of $\pi$', you leave $\pi$ in the equation without converting it to a decimal number. This method is often used in mathematical and scientific settings for greater precision.
Now, let's calculate the volume of this moon lamp in the shape of a sphere, and leave the answer in terms of $\pi$.
Volume of a Sphere – RealWorld Problems
Spheres are everywhere around us in sports, nature, and even in space. Knowing their volume helps us understand and measure these objects more accurately.
Let’s solve realworld problems involving spherical objects.
Volume of a Sphere – Exercises
With your understanding of the volume formula for a sphere, try these exercises to enhance your skills!
Volume of a Sphere – Summary
Key Points from this Text:
 The formula for calculating the volume of a sphere is $V = \frac{4}{3} \pi r^3$.
 To find the volume, identify the radius of the sphere.
 Substitute the radius into the formula and calculate, rounding to the nearest tenth or leaving in terms of $\pi$.
 Spheres are common in everyday life and learning their volume is applicable in numerous realworld situations.
Finding the volume of a cylinder and the volume of a cone are two other important 3D shapes to understand, and follow a similar process.
Volume of a Sphere – Frequently Asked Questions
Transcript Volume of a Sphere
Have you ever wondered how big Planet Earth really is? Our planet looks like the shape of a sphere, a common threedimensional shape. To find out how much space is inside a sphere, we can follow the steps to find the 'Volume of a Sphere'. The formula used is fourthirds, multiplied by pi, multiplied by the radius cubed. The radius of a sphere is the distance from the middle to the outside rim of the sphere. But, before we find out the volume of the earth, let's scale it down and start with something smaller; a desk top globe. We are going to find the volume of the sphere and round the solution as stated in the directions. Always start by writing down the formula you are using. Next, identify the radius of the sphere, which will be your r value. We can substitute the value of r right into our formula, like this! Seven to the third power is three hundred fortythree, which can be multiplied by the fourthirds. The product is four hundred fiftyseven and three repeating. Next, using a calculator we will find the product of this number and pi. When rounded to the nearest tenth, the volume is approximately one thousand four hundred thirtysix and eight tenths. And don't forget to add on your units, inches cubed. Let's practice another example! Let's find the volume of this orange! When the directions say in terms of pi, it means we will not be calculating the pi on the calculator, but rather leaving it as pi. Before starting, write down the formula for the volume of a sphere. What is the radius of this sphere? Three centimeters! Substitute this value for the r and then evaluate. What is threetothethird power, multiplied by fourthirds? It is thirtysix, pi, which is a precise measurement since we did not round. And like the last one, don't forget to add your units, in this case centimeters cubed. Find the volume of the basketball in cubic centimeters. Pause the video here to find the volume, and press play when you are ready to check your solution. Write down the formula first! This time, we have the diameter and not the radius. The radius is half of the diameter, so here it is twelve centimeters. Volume equals four thirds, times pi, times twelve cubed. The product of fourthirds and twelve to third power is two thousand, three hundred four, pi. The volume of the basketball is approximately seven thousand, two hundred thirtyeight and twotenths centimeters cubed. To summarize, the volume of a sphere measures the space INSIDE, and to calculate, we use the formula volume equals four thirds, times pi, times the radius cubed. Since we now have practiced, let's see if we can answer the question, what is the volume of planet earth? "Wait, this just in! The Earth may look like a sphere, but it is actually not a perfect sphere and is considered to be more of an ellipsoid." Our earth is constantly changing, therefore its size is also ever changing. But we can still estimate! The volume of the earth is approximately two hundred sixtyeight billion, eightytwo million, five hundred seventythree thousand, one hundred six cubic miles.
Volume of a Sphere exercise

Determine which formula would be used to find the volume of the sphere.
HintsThe radius of a circular object is the measurement from the center to the outer edge, while the diameter is the measurement from one side to the other, passing through the center.
To find the volume, the radius needs to be substituted into the formula for $r$. Substitution means you are replacing a variable with a known value.
SolutionThe formula used to find the volume of the volleyball would be:
$V=\frac{4}{3}\pi (4^3)$

Determine the sequence of events to solve for the volume of a sphere.
HintsThe first step when finding the volume of any shape is to identify the formula you will use.
After identifying the formula, find the radius, $r$, and substitute it into $V=\frac{4}{3} \pi r^3$.
Don't forget... in terms of pi means you leave $\pi$ as a symbol rather than calculating it.
SolutionStep 1: Identify the formula and radius.
$V = \frac{4}{3}\pi r^3$
$r=9$
Step 2: Substitute the radius in for $r$ in the formula.
$V = \frac{4}{3}\pi (9^3)$
Step 3: Evaluate the exponent.
$V = \frac{4}{3}\pi (729)$
Step 4: Multiply the known values, other than $\pi$.
$V = 972\pi $
Step 5: Leave $\pi$ as a symbol since the directions state to leave in terms of pi, and add the appropriate units.
$V = 972\pi\:cm^3 $

Show your understanding of the relationship between a radius and a diameter of a sphere.
HintsThe radius and diameter are labeled on the sphere.
The radius of a sphere is half the distance of the diameter.
SolutionThe diameter is the distance across a sphere through its center and the radius is the distance from the center to the outside. The radius is exactly half the distance of a diameter. The formula to find the volume of a sphere is $V=\frac{4}{3}\pi r^3$, which means the radius is the information needed. Given the diameter, we can find the radius, by $\bf{\dfrac{\text{diameter}}{2}}$. The sphere has a diameter of 16 inches and a radius of 8 inches.

Find the volume of a sphere.
HintsAfter you have substituted the value of the radius in the $r$, cube it, and multiply that by $\frac{4}{3}$.
This solution should not be rounded.
The product of $r^3$ and $\frac{4}{3}$ can then be multiplied by $\pi$ using a calculator.
Solution$V=\frac{4}{3}\pi (4^3)$
$V=\frac{4}{3}\pi (64)$
$\frac{4}{3}(64) = 85 \frac{1}{3}$
$V=85\frac{1}{3}\pi$
$V=268.082573...$
$V\approx268.1\:cm^3$

Identify the formula used to find the volume of a sphere.
HintsThe Volume of a Sphere refers to the space inside the sphere.
The radius of a sphere refers to the distance from the center of the sphere to any point on its surface.
It is always the same length regardless of the direction in which it is measured.
If the radius of the sphere was 5cm, we could substitute it for the $r$ in the formula for volume, like this...
$V=\frac{4}{3} \pi (5^3)$
SolutionThe volume of a sphere can be calculated using this formula.
$V=\frac{4}{3}\pi r^3$

Demonstrate your knowledge on finding the volume of a sphere.
HintsThe volume of a sphere formula is:
$V = \frac{4}{3}\pi r^3$
The radius is half of the diameter.
Be sure to check each choice to see if the diameter or radius will yield a sphere with a volume of $1,767.1\:cm^3$.
SolutionThere are two correct answers.
$\text{diameter}=15\:cm$
$\text{radius}=7.5\:cm$