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 maths; it's about relating to everyday objects, such as estimating the air in a football 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 centre. This uniformity leads to its unique formula.
Volume of a Sphere – Cubic Units
Finding the volume of 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 centimetres, then the volume will be in cubic centimetres (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 one decimal place 
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.
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 practise 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 one decimal place 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
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 centre.
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 find 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 labelled 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 centre and the radius is the distance from the centre 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 centimetres and a radius of 8 centimetres.

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 centre 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.
$\text{diameter}=15\:cm$
$\text{radius}=7.5\:cm$