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Volume of a Cylinder

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Basics on the topic Volume of a Cylinder

Volume of a Cylinder

The concept of volume, particularly in cylinders, is a practical and valuable skill in many real-life scenarios. For instance, knowing how to find the volume of a cylinder can help you determine how much water a bottle can hold or how many jelly beans can fit in a jar.

This knowledge is not just limited to classroom math; it applies to everyday objects and situations, from sports equipment to household items. By learning about the volume of cylinders, you gain a useful tool that enhances your understanding of the world around you.

The volume of a cylinder is the amount of space inside the cylinder. It is calculated using the formula $V = \pi r^{2} h$, where $V$ is the volume, $r$ is the radius of the cylinder's base, and $h$ is the height of the cylinder.

Understanding Volume of a Cylinder

The volume of a cylinder can be thought of as how much liquid or material it can hold. To calculate it, you need two measurements: the radius of the circular base and the height of the cylinder.

  • The radius is the distance from the center of the circular base to its edge.
  • The height is the distance from the bottom to the top of the cylinder.

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Volume of a Cylinder – Cubic Units

In calculating the Volume of Simple 3D Shapes, the use of precise formulas is key, as is the necessity to accurately label the final solutions with their correct units.

Choosing the Correct Volume Units

Volume is expressed in cubic units because it represents three-dimensional space. Common units include cubic centimeters (cm³), cubic inches (in³), and cubic meters (m³). The unit used depends on the measurement units for radius and height.

Volume of a Cylinder – Step-by-Step

Here is the process to find the volume of a cylinder step by step.

Step Number Directions Example
1 Identify the radius and height of the cylinder. Radius $r = 4$ cm, Height $h = 10$ cm
2 Substitute the values into the formula $V = \pi r^{2} h$. $V = \pi \times 4^2 \times 10$
3 Calculate the volume, paying special attention to rounding rules. $V = \pi \times 16 \times 10 = 160\pi$ cm³ approx. 502.6 cm³
4 Write the final answer with the correct units. Volume of the cylinder is 502.7 cm³

Let’s work through an example to understand how to calculate the volume of a cylinder.

Suppose a cylinder has a radius of 3 cm and a height of 10 cm. We want to find its volume.

27210_ToV-02.svg

Identify the radius and height.
Use the formula $V = \pi r^{2} h$ to calculate the volume.
Calculate the volume using pi and round to the nearest tenth.

For other shapes like Volume of Prisms, we use different formulas to find their volume. But the way we do it is still step-by-step, just like with cylinders.

Finding the Volume ‘In Terms of $\pi$'

Leaving an answer 'in terms of $\pi$' means not using a numerical approximation for $\pi$ in the calculation. This form of answer is more precise, as it does not involve rounding off $\pi$ to a decimal. It is particularly useful in mathematical and scientific contexts where exact values are important.

Let’s look at the process of finding the volume of a cylinder ‘in terms of $\pi$'

27210_ToV-03.svg

Identify the radius and height.
Use the formula $V = \pi r^2 h$ to calculate the volume.
Calculate the volume leaving the answer in terms of $\pi$.

Volume of a Cylinder – Real-World Problems

Cylinders are a common shape in our everyday lives, found in objects like soup cans, water towers, and even in the structure of some buildings. Understanding how to calculate their volume helps us estimate the capacity of these everyday cylindrical objects.

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Let’s take a look at some problems involving cylindrical objects we may encounter in the real world.

A soup can has a diameter of $6$ cm and a height of $10$ cm. How much soup can it hold? Assume the can is a perfect cylinder. Answer 1: First, find the radius, which is half the diameter: $3$ cm. Then, calculate the volume: $V = \pi \times 3^2 \times 10 = 90\pi$ cm³. This means the can can hold approximately $282.7$ cm³ of soup.

A cylindrical water tower is $20$ meters tall and has a radius of $5$ meters. How many liters of water can it store when full? (Note: 1 m³ = 1,000 liters) Calculate the volume: $V = \pi \times 5^2 \times 20 = 500 \pi$ m³. Since 1 m³ equals $1,000$ liters, the tower can store approximately $1,570,796$ liters of water.

Volume of a Cylinder – Exercises

Using what you have learned in this text, along with the formula for the Volume of a Cylinder, practice finding the volume!

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Volume of a Cylinder – Summary

Key Points from this Text:

  • The formula for calculating the volume of a cylinder is $V = \pi r^{2} h$.
  • To find the volume, identify the radius and height of the cylinder.
  • Substitute the values into the formula and use a calculator to compute, rounding to the nearest tenth or leaving in terms of $\pi$.
  • This concept is widely used in real-world scenarios such as determining the capacity of containers.

Do you know what 3D shape has a volume that is exactly one-third of a cylinder with the same height and radius? A cone! Learn how to find the Volume of a Cone!

Volume of a Cylinder – Frequently Asked Questions

What units should I use for the volume of a cylinder?
Can I use the diameter instead of the radius in the formula?
How do I handle different units for radius and height?
Is it necessary to round off the volume?
Can this formula be used for cylinders that are not right-angled?
How do I measure the radius and height in a real-world scenario?
What if the cylinder is lying horizontally?
Can I calculate the volume of a partially filled cylinder?
What does 'leaving the answer in terms of $\pi$' mean?
Why do we square the radius in the formula?

Transcript Volume of a Cylinder

Cylinders are a three-dimensional shape we see in our everyday life. They are made up of two circular bases, connected with a curved rectangular shape. The 'volume of a cylinder' measures how much space is inside the shape. The formula used is volume equals pi multiplied by the radius squared multiplied by the height. The radius is the distance from the middle of the circular base to the outside. The height is the length of the cylinder from one circular base to the other. Let's try out our first example! Find the volume of the cylinder and round the answer to the nearest tenth of a cubic inch. First, write the formula. The radius of this cylinder is six inches, so r is equal to six. The height is eleven inches, so h is equal to eleven. The values for the radius and height can now be substituted into the formula. Volume equals pi times six squared times eleven. A calculator can quickly find the product of pi, six squared, and eleven. One thousand two hundred forty-four and seven thousand sixty-nine hundred thousandths, and so on. When we multiply a rational number by pi, the answer is always irrational, which means the number does not have an end. Since calculators can't show all those numbers, they use three dots to show infinite numbers. This number rounds to one thousand two hundred forty-four and one-tenth. The volume of the cylinder is approximately one thousand two hundred forty-four and one-tenth inches cubed. Anytime we find the volume of a 3D shape, we must include the cubic units at the end. Let's look at a different example. Read the directions, paying careful attention to the rounding rule, and identify the formula being used. Volume equals pi, times the radius squared, times the height. For this example, we know the diameter, not the radius. If a radius is half of the diameter, what is the radius of this cylinder? The radius is five and the height of this cylinder is nine centimeters. Substitute the values into the formula. Pause the video here, and use a calculator to find the volume. The product is seven hundred six followed by infinite decimal numbers. The volume can be rounded to approximately seven hundred six and nine-tenths centimeters cubed. Let's try one last example! The directions state to find the volume 'in terms of pi'. This means that we will not be using pi on our calculator, but rather leaving pi as a symbol in our final answer. Pause the video here to identify the radius and height and substitute them into the formula. We will substitute three and eight in for r and h. This time instead of calculating all of the terms, we will only calculate three squared times eight, and leave pi out. Our final answer is written as volume equals seventy-two pi inches cubed. This is the most accurate answer since we do not round. Let's summarize! Cylinders are made up of two circular bases, connected with a curved rectangular shape. We can find the volume of these shapes by using the formula pi times the radius squared times the height. And that is how you CAN master the art of calculating the volume of any cylinder!