# Formula for Volume of a Rectangular Prism

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Formula for Volume of a Rectangular Prism
CCSS.MATH.CONTENT. 5.MD.5b

## Understanding the Volume of a Rectangular Prism

In our daily lives, we often come across objects shaped like rectangular prisms - think of boxes, bricks, or even parts of buildings. A rectangular prism is a 3D shape with six faces, each one a rectangle. When we talk about the volume of a rectangular prism using cubic units, we find out how much space it takes up. It's like figuring out how much water can fill a fish tank or how many books you can pack into a box. This volume helps us understand the capacity of these objects in cubic units, which measure space in three dimensions.

Volume Formula: The volume of a rectangular prism is calculated by multiplying its length ($l$), width ($w$), and height ($h$). The formula is $V = l \times w \times h$.

## Formula for Volume of a Rectangular Prism

In math, we often have more than one way to solve a problem, and that's true for finding the volume of a rectangular prism too.

The formula $V = l \times w \times h$ directly multiplies the prism's dimensions. This method is straightforward and effective for calculating volume.

Let’s learn how to find the volume of a rectangular prism with this formula.

A rectangular prism has a length of $4$ cm, width of $3$ cm, and height of $6$ cm. Calculate its volume.

• Length ($l$) = $4$ cm
• Width ($w$) = $3$ cm
• Height ($h$) = $6$ cm
• Volume = $l \times w \times h = 4 \times 3 \times 6 = 72$ cm$^3$

The volume of this prism is $72$ cubic centimeters.

We use cubic units for units of volume because volume measures three-dimensional space. Think of stacking little blocks inside a box - you're filling it lengthwise, widthwise, and heightwise. So, we multiply these three dimensions, and the result is in cubic units, like filling a box with tiny cubes.

Find the volume of a prism with a length of 5 m, a width of 2 m, and a height of 3 m.

• Length ($l$) = $5$ m
• Width ($w$) = $2$ m
• Height ($h$) = $3$ m
• Volume = $l \times w \times h = 5 \times 2 \times 3 = 30$ m$^3$

The volume of this prism is $30$ cubic meters.

Use the formula for the volume of a rectangular prism, and solve these examples on your own!

Calculate the volume of a rectangular prism with a length of 10 inches, a width of 4 inches, and a height of 6 inches.

Determine the volume of the rectangular prism pictured above.

Find the volume of a rectangular prism pictured above.
What is the volume of a rectangular storage container that is 6 meters long, 3 meters wide, and 2.5 meters tall?

### Finding the Volume of a Prism with $V=Bh$

Another formula to find the volume of a rectangular prism is using $V = B \times h$. Here, $B$represents the area of the base of the shape. For a rectangular prism, this base is the area of the rectangle at the bottom. This formula is really helpful, not just for rectangular prisms, but also for other 3D shapes like cylinders which have a circular base (volume of a cylinder). It helps us understand how much space these shapes occupy by considering their base area and height.

Let's calculate the volume of a rectangular prism with a length of 4 feet, a width of 3 feet, and a height of 6 feet.

Determine the dimensions of the prism:

• Length ($l$) = $4$ ft
• Width ($w$) = $3$ ft
• Height ($h$) = $6$ ft

Calculate the base area (B):

• The base area is found by multiplying the length and width.
• Base Area ($B$) = $l \times w = 4 \times 3 = 12$ square feet.

Calculate the volume using $V = B \times h$:

• Now, multiply the base area by the height.
• Volume ($V$) = $B \times h = 12 \times 6 = 72$ cubic feet.

The volume of this rectangular prism is $72$ cubic feet.

Practice using this formula on your own!

A rectangular prism has dimensions of length 10 cm, width 5 cm, and height 8 cm. Find its volume using the formula $V = B \times h$.

Find the volume of a rectangular prism pictured above, using $V = B \times h$.
A rectangular prism has a length of 3 meters, a width of 2 meters, and a height of 5 meters. Determine its volume with the formula $V = B \times h$.
Calculate the volume of a rectangular prism with dimensions of 12 ft (length), 7 ft (width), and 9 ft (height) using the formula $V = B \times h$.

## Formula for Volume of a Rectangular Prism – Application

Solving problems involving the volume of a rectangular prism enhances our understanding of space and capacity. It applies in diverse fields, from architecture and construction to everyday tasks like packing or storage.

Find the volume of a box with a length of 8 cm, width of 7 cm, and height of 10 cm.

A rectangular garden bed measures 30 feet in length, 4 feet in width, and 1.5 feet in depth. What is its volume in cubic feet?
Calculate the volume of a toy box with dimensions 24 inches in length, 18 inches in width, and 15 inches in height using the formula V = Bh.

A fish tank has dimensions of 75 cm in length, 30 cm in width, and 40 cm in height. What is the volume of the tank in cubic centimeters?
Find the volume of a storage unit that is 2 yards long, 1 yard wide, and 1.5 yards high using $V = Bh$.

A rectangular prism-shaped shipping box has a height of $8$ inches and a volume of $1,536$ cubic inches. What is the area of the base, $B$ of the box?

## Formula for Volume of a Rectangular Prism – Summary

Key Learnings from this Text:

• Understanding the volume of a rectangular prism helps in calculating space in practical situations.
• The volume formula, $V = l \times w \times h$, is straightforward and widely applicable.
• Alternative method: $V = B \times h$, where $B$ is the base area, deepens conceptual understanding.
• Real-world application of this knowledge spans from storage organization to construction planning.
Formula Explanation
$V = l \times w \times h$ Multiply length ($l$), width ($w$), and height ($h$) to find the volume.
$V = B \times h$ Calculate the base area ($B$ = length x width), then multiply by height ($h$).

## Formula for Volume of a Rectangular Prism – Frequently Asked Questions

What is the formula to find the volume of a rectangular prism?
Can the volume of a rectangular prism be the same as a cube?
How do you find the volume of a rectangular prism with missing dimensions?
What does the formula $V = B \times h$ mean for a rectangular prism?
Is it possible to find the volume of a rectangular prism using only its length and width?
How do units affect the calculation of a rectangular prism's volume?
What's the difference between volume and surface area of a rectangular prism?
Can the formula for volume be used for all rectangular prisms, regardless of size?
How does increasing one dimension of a rectangular prism affect its volume?
What is the significance of understanding the volume of a rectangular prism in real life?

### TranscriptFormula for Volume of a Rectangular Prism

Formulas for Volume of a Rectangular Prism Rectangular prisms are three-dimensional shapes that have a length, width, and height. When we measure the volume, we determine how much space is INSIDE the prism and what solid, (...) liquid, (...) or gas it has the ability to hold. We know that we can divide this shape into cubic units. But what happens when the prisms grow in size? We need a more efficient way to find the volume. There are two mathematical formulas to find the cubic measurement of a rectangular prism. Remember, a formula is a mathmatical expression that helps us find a specific value. It's like a set of instructions that guide in in performing calculations. One formula for finding volume looks like this. is the length, is the width, and stands for height. If we multiply the measurement of these three sides, we will find the measurement for the space inside, known as the volume. Suppose you are at the pet store trying to find the perfect aquarium size for your fish. You can use this formula to determine the size of the tank quickly. Based on these dimensions, what is the volume of this aquarium? When we multiply the length, six inches, by the width, two inches, by the height, ten inches, we get the product one hundred twenty. Always remember to include the unit of measurement in your answer, which in this case is one hundred twenty inches cubed. Now suppose that you've bought too many fish and you need to upgrade a pool that's the perfect size to accommodate all your new family members. To find the find the cubic measurement of this rectangular prism, we will use the mathematical formula volume equals base times height, or equals times . The base of this pool is fifteen feet, and the height is four feet. Pause the video to solve and press play when you're ready to see the solution. Fifteen times four is sixty, so the volume is sixty feet cubed. But what if we didn't know the base? Use length times width times height to solve; either way we get the same answer. We can also solve using equals base times height, but first we need to find the area of the base by calculating length times width. Five times three is fifteen. From there, we can solve for volume using base times height. To summarize, there are two formulas used to find the volume of rectangualr prisms: length times width times height or, base times height. And it's a good thing we know two ways to solve because volume is EVERYWHERE and EVERYTHING.

## Formula for Volume of a Rectangular Prism exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Formula for Volume of a Rectangular Prism .
• ### Find the volume of the rectangular prism.

Hints

To calculate the volume, multiply the length, width, and height.

What is 10 x 5 x 5?

Solution

The length is 10 inches, the width is 5 inches and the height is five inches.

Multiply these together: 10 x 5 x 5 = 250.

Therefore, the correct answer is $250 in^3$.

• ### Label the length, width, height, and the volume.

Hints

The height is the perpendicular distance from the base to the top, the length is how long it is, and the width is perpendicular to the length.

The formula to calculate the volume is V= l x w x h.

Solution

The length is 2in, the width is 2in, and the height is 10in.

When we multiply 10 x 2 x 2, we get the total volume of the prism, $40 in^3$.

• ### Calculate the volume.

Hints

The length has 4 cubes, so the length is 4. Count the cubes to find the width and height. Then multiply.

Count the cubes to identify the length, width, and height.

Solution

The length is 4, the width is 2, and the height is 2. Multiply 4 x 2 x 2 to get the volume, $16 in^3$.

• ### Fill in the missing side to find the volume of the rectangular prism.

Hints

To find the missing side, compare the length with other side., A square has 4 equal sides.

Find the volume using the formula V= l x w x h.

Solution

The width is 5 inches. There is a square at the end of the prism which is why the width is 5 inches. 12 x 5 x 5 = $300 in^3$.

• ### Which formulas are used to find the volume of a rectangular prism?

Hints

There are two formulas used to calculate the volume of a rectangular prism.

One way to find the volume is to multiply the base times the height.

V stands for volume, b stands for base, h stands for height, and w stands for width.

Solution

To find the volume of a rectangular prism, use V= b x h or V= l x w x h.

V stands for volume, b stands for base, h stands for height, and w stands for width.

• ### Find the volume of each rectangular prism and sort from smallest to largest.

Hints

Use the formula V= l x w x h to find the volume of each. Then sort from smallest to largest.

Start by calculating the volume of the purple rectangular prism. Multiply 7 x 7 x 7.

Solution

Use the formula V= l x w x h to find the volume of each. Then sort from smallest to largest.

• The volume of the red prism is $72 in^3$.
• The volume of the blue prism is $80 in^3$.
• The volume of the orange prism is $110 in^3$.
• The volume of the purple prism is $343 in^3$.

So, the shapes from smallest to largest are red, blue, orange, purple.