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Volume of a Sphere

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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 real-world 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.

27388_ToV_-01.svg

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 – Step-by-Step 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.

27388_ToV_-02.svg

Identify the radius of the volleyball.
Use the formula $V = \frac{4}{3} \pi r^3$ to calculate the volume.
Calculate the volume, rounding to one decimal place.

Volume of a Hemisphere – Guided Practice

Hemispheres are prevalent in various designs and natural formations, such as domed buildings and half-cut 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.

Use the formula $V = \frac{2}{3} \pi r^3$ to calculate the volume.
Calculate the volume, rounding to one decimal place.

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 $\pi$.

27388_ToV_-03.svg

Identify the moon lamp's radius.
Use the formula $V = \frac{4}{3} \pi r^3$ to calculate the volume.
Calculate the volume, leaving the answer in terms of $\pi$.

Volume of a Sphere – Real-World 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 real-world problems involving spherical objects.

27388_ToV_-04.svg

A spherical water tank has a radius of 2 metres. How much water can it hold?
A decorative garden globe has a diameter of $18$ in. What is its volume?

Volume of a Sphere – Exercises

With your understanding of the volume formula for a sphere, try these exercises to enhance your skills!

27388_ToV_-05.svg

A sphere has a radius of $10$ cm. Calculate its volume, rounding to one decimal place.
Find the volume of a sphere with a radius of 6 inches, rounding the answer to one decimal place.
A sphere has a radius of $5$ m. What is its volume, rounded to one decimal place?
Calculate the volume of a sphere in terms of $\pi$ if it has a radius of $7$ cm.
Find the volume of a sphere with a radius of $3$ inches, leaving the answer in terms of $\pi$.

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 real-world 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

What is the formula for the volume of a sphere?
How is the radius of a sphere measured?
Can I calculate the volume of a sphere using its diameter?
Why is the volume formula of a sphere $\frac{4}{3} \pi r^3$?
What units should I use for the volume of a sphere?
How do I calculate the volume of a sphere in terms of pi?
Is it necessary to round the volume of a sphere?
What are some practical applications of knowing the volume of a sphere?
How do I convert the volume of a sphere from cm³ to m³?
Can I use a calculator to find the volume of a sphere?

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 three-dimensional 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 four-thirds, 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 value. We can substitute the value of right into our formula, like this! Seven to the third power is three hundred and forty-three, which can be multiplied by the four-thirds. The product is four hundred and fifty-seven point three recurring. 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 and thirty-six point eight, and don't forget to add on your units, inches cubed. Let's practise 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 centimetres! Substitute this value for the and then evaluate. What is three-to-the-third power, multiplied by four-thirds? It is thirty-six, 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 centimetres cubed. Find the volume of the basketball in cubic centimetres. 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 centimetres. Volume equals four thirds, times pi, times twelve cubed. The product of four-thirds and twelve to the third power is two thousand, three hundred and four, pi. The volume of the basketball is approximately seven thousand, two hundred and thirty-eight point two centimetres cubed. To summarise, 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 have now practised, 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 ovoid. 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 and sixty-eight billion, eighty-two million, five hundred and seventy-three thousand, one hundred and six cubic miles.

Volume of a Sphere exercise

Would you like to apply the knowledge you’ve learnt? You can review and practice it with the tasks for the video Volume of a Sphere.
  • Determine which formula would be used to find the volume of the sphere.

    Hints

    The 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.

    Solution

    The 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.

    Hints

    The 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.

    Solution

    Step 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.

    Hints

    The radius and diameter are labelled on the sphere.

    The radius of a sphere is half the distance of the diameter.

    Solution

    The 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.

    Hints

    After 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.

    Hints

    The 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...

    $V=\frac{4}{3} \pi (5^3)$

    Solution

    The 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.

    Hints

    The 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 $1,767.1\:cm^3$.

    Solution

    There are two correct answers.

    $\text{diameter}=15\:cm$

    $\text{radius}=7.5\:cm$