## Volume of a Rectangular Prism

Filling and wrapping is a great way to think about volume and surface area. When you fill something up, the space that is filled is the volume. When you wrap up a present or cover something you are now working with surface area.

Companies need to know the volume of a box in order to determine how much of their product will fit inside.

Here is an example:

The volume is equal to the number of 1 in x 1 in x 1 in cubes we can fit inside this box.

There could be 15 cubes across the length by 4 across the width, making a bottom layer of 15 x 4 = 60 cubes. You can also think of this as the area of the base.

Notice that the height of the box is 8 in. That means that we could fit 8 layers of 60 cubes.

In other words, the volume of this box is 480 cubic inches.

This could be simplified to the formulas **V = Bh** or **V = l w h** for all rectangular prisms.

Let’s Practice! Determine the volume of each prism.

1.)

V = l w h

V = (25 in.)(3 in.)(7 in.)

V = (75 in^{2})(7 in)

V = 525 in^{3}

2.)

V = l w h

V = (4.5 cm)(3.2 cm)(4.2 cm)

V = 60.48 cm^{3}

3.)

4.)

V = Bh

V = (112 in.^{2})(23 in.)

V = 2576 in.^{3}

**Let’s Review**

The volume of a prism is the space that fills it. You can determine the volume of a rectangular prism by either multiplying the area of the base times the height or by multiplying the length, width and height.

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## Rectangular Prism Calculator

### Rectangular Prism Shape

l = length

w = width

h = height

d = diagonal

S = surface area

V = volume

### Calculator Use

Enter any 3 variables for a rectangular prism into this online calculator to calculate the other 3 unknown variables. A cube is a special case where l = w = h for a rectangular prism.

* **Units:** Note that units are shown for convenience but do not affect the calculations. The units are in place to give an indication of the order of the results such as ft, ft^{2} or ft^{3}. For example, if you are starting with mm and you know h, l and w in mm, your calculations will result with d in mm, S in mm^{2} and V in mm^{3}.

### Formulas for a rectangular prism:

**Volume of Rectangular Prism**:**Surface Area of Rectangular Prism**:**Space Diagonal of Rectangular Prism**: (similar to the distance between 2 points)

A cube is a special case where l = w = h. So you can find the volume of a cube or surface area of a cube by setting these values equal to each other.

#### Calculations for a rectangular prism:

1. Given the length, width and height find the volume, surface area and diagonal of a rectangular prism

- h, l and w are known; find V, S and d
- V = lwh
- S = 2(lw + lh + wh)
- d = √(l
^{2}+ w^{2}+ h^{2})

2. Given the surface area, length and width find the height, volume and diagonal of a rectangular prism

- S, l and w are known; find h, V and d
- h = (S — 2lw) / (2l + 2w)
- V = lwh
- d = √(l
^{2}+ w^{2}+ h^{2})

3. Given the volume, length and width find the height, surface area, and diagonal of a rectangular prism

- V, l and w are known; find h, S and d
- h = V / lw
- S = 2(lw + lh + wh)
- d = √(l
^{2}+ w^{2}+ h^{2})

4. Given the diagonal, length and width find the height, volume and surface area of a rectangular prism

- d, l and w are known; find h, V and S
- h = √(d
^{2}— l^{2}— w^{2}) - V = lwh
- S = 2(lw + lh + wh)

For more information on cuboids see: Weisstein, Eric W. «Cuboid.» From *MathWorld*—A

Wolfram Web Resource, Cuboid.

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## Volume of Rectangular Prism in F#

open System

open System.Windows.Forms

open System.ComponentModel

open System.Drawing

let RecPrismForm=new Form(Text=»Rectangular prism volume»)

RecPrismForm.BackColor<-Color.Bisque

//enter the length value

let lengthlbl=new Label(Top=10,Left=0,Width=130)

lengthlbl.Text<-«Enter Length value:»

let lenghttxtbox=new TextBox(Top=10,Left=140,Width=80)

//enter the width value

let widthlbl=new Label(Top=40,Left=0,Width=130)

widthlbl.Text<-«Enter Width value:»

let widthtxtbox=new TextBox(Top=40,Left=140,Width=80)

//enter the height value

let heightlbl=new Label(Top=70,Left=0,Width=130)

heightlbl.Text<-«Enter Height Value:»

let heighttxtbox=new TextBox(Top=70,Left=140,Width=80)

//after compute prism volume displayed here

let pvolumelbl=new Label(Top=120,Left=80,Width=80)

pvolumelbl.Text<-«Prism Volume:»

let prismvolumelabel=new Label(Top=120,Left=180,BorderStyle=BorderStyle.FixedSingle)

//create buttons

let manipulatebtn=new Button(Top=180,Left=120,Width=80)

manipulatebtn.BackColor<-Color.Ivory

manipulatebtn.Text<-«Compute»

let exitbutton=new Button(Top=180,Left=220,Width=80)

exitbutton.BackColor<-Color.Ivory

exitbutton.Text<-«Exit»

//adding the controls in form

RecPrismForm.Controls.Add(lengthlbl)

RecPrismForm.Controls.Add(lenghttxtbox)

RecPrismForm.Controls.Add(widthlbl)

RecPrismForm.Controls.Add(widthtxtbox)

RecPrismForm.Controls.Add(heightlbl)

RecPrismForm.Controls.Add(heighttxtbox)

RecPrismForm.Controls.Add(pvolumelbl)

RecPrismForm.Controls.Add(prismvolumelabel)

RecPrismForm.Controls.Add(manipulatebtn)

RecPrismForm.Controls.Add(exitbutton)

//Manipulate the prism valume

manipulatebtn.Click.Add(fun compute->

let lenghtvalue=Convert.ToDouble(lenghttxtbox.Text)

let widthvalue=Convert.ToDouble(widthtxtbox.Text)

let heightvalue=Convert.ToDouble(heighttxtbox.Text)

let prismvol=lenghtvalue*widthvalue*heightvalue

prismvolumelabel.Text<-Convert.ToString(prismvol))

exitbutton.Click.Add(fun exit -> RecPrismForm.Close())

Application.Run(RecPrismForm)

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## Units for the Volume of a Rectangular Prism

by A Visitor

**Question**

Find the volume of a rectangular prism that is 5 inches in height, 4 inches in width, and 3 inches in length. Is the answer 60 inches or 60 cubic inches?

**Answer**

The calculated number **60** (W×L×H = 4×3×5) is correct. As for the units, you can find it by following the logic below:

Imagine that you are given a number, let’s say 1. In your mind, you know that it is just a number and it does not have much meaning.

Now, if I tell you the number is in inches (1 inch), you will know that the number represents length.

Next, if we can represent *length* with the unit inch, how do we represent the unit for *area*?

To answer this, let’s say we have a square that the width of 2 inches and the length of 3 inches, we can calculate its area as follows:

Area = W × L

= 2 inch × 3 inch

= 6 inch^{2}

To calculate the numbers (2 × 3 = 6) is the easy part. As for the unit, since we are multiplying two lengths with the unit inches together, logically, we can write the resulting unit as *inch ^{2}* (square inches). The

^{2}on the unit represents area.

Similarly, for rectangular prism with 5 inches in height, 4 inches in width, and 3 inches in depth, we can calculate its volume as follows:

Area = W × L × H

= 4 inch × 3 inch × 5 inch

= 60 inch^{3}

Applying the same logic, since we are multiplying three lengths with unit the inches together, logically, we can write the resulting unit as *inch ^{3}* (cubic inches). The

^{3}on the unit represents volume.

Also, this site have free math lessons on volume of solids that may be of interest to you (scroll down page the see the list of lessons). The link is given below:

- Lessons on Volume of Solids

www.mathexpression.com

## Volume of a Rectangular Prism (with videos, worksheets, solutions & activities)

Videos, worksheets, stories and songs to help Grade 6 students learn how to find the volume of a rectangular prism or cuboid.

How to find the volume of a rectangular prism

Volume of a rectangular prism or box examples

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

www.onlinemathlearning.com

## Volume of a Rectangular Prism Online Quiz

Following quiz provides Multiple Choice Questions (MCQs) related to **Volume of a Rectangular Prism**. You will have to read all the given answers and click over the correct answer. If you are not sure about the answer then you can check the answer using **Show Answer** button. You can use **Next Quiz** button to check new set of questions in the quiz.

**Q 1 — Find the volume of the following rectangular prism in cubic cm.**

#### Answer : A

#### Explanation

**Step 1:**

Volume of rectangular prism = l × w × h cubic units

**Step 2:**

Volume of given prism = 9 × 3 × 9 = 243 cubic cm.

* Show Answer*

**Q 2 — Find the volume of the following rectangular prism in cubic cm.**

*Answer : D*

*Explanation*

**Step 1:**

*Volume of rectangular prism = l × w × h cubic units*

**Step 2:**

*Volume of given prism = 4 × 9 × 2 = 72 cubic cm*

* Show Answer*

**Q 3 — Find the volume of the following rectangular prism in cubic cm.**

*Answer : B*

*Answer : B*

*Explanation*

*Explanation*

**Step 1:**

*Volume of rectangular prism = l × w × h cubic units*

**Step 2:**

*Volume of given prism = 5 × 6 × 4 = 120 cubic cm.*

* Show Answer*

**Q 4 — Find the volume of the following rectangular prism in cubic cm.**

*Answer : C*

*Answer : C**Explanation*

*Explanation***Step 1:**

*Volume of rectangular prism = l × w × h cubic units*

**Step 2:**

*Volume of given prism = 5 × 3 × 3 = 45 cubic cm.*

* Show Answer*

**Q 5 — Find the volume of the following rectangular prism in cubic cm.**

*Answer : C*

*Answer : C**Explanation*

*Explanation***Step 1:**

*Volume of rectangular prism = l × w × h cubic units*

**Step 2:**

*Volume of given prism = 9 × 5 × 8 = 360 cubic cm.*

* Show Answer*

**Q 6 — Find the volume of the following rectangular prism in cubic cm.**

*Answer : A*

*Answer : A**Explanation*

*Explanation***Step 1:**

*Volume of rectangular prism = l × w × h cubic units*

**Step 2:**

*Volume of given prism = 2 × 6 × 8 = 96 cubic cm.*

* Show Answer*

**Q 7 — Find the volume of the following rectangular prism in cubic cm.**

*Answer : D*

*Answer : D**Explanation*

*Explanation***Step 1:**

*Volume of rectangular prism = l × w × h cubic units*

**Step 2:**

*Volume of given prism = 9 × 7 × 9 = 567 cubic cm.*

* Show Answer*

**Q 8 — Find the volume of the following rectangular prism in cubic cm.**

*Answer : B*

*Answer : B**Explanation*

*Explanation***Step 1:**

*Volume of rectangular prism = l × w × h cubic units*

**Step 2:**

*Volume of given prism = 4 × 2 × 9 = 72 cubic cm.*

* Show Answer*

**Q 9 — Find the volume of the following rectangular prism in cubic cm.**

*Answer : A*

*Answer : A**Explanation*

*Explanation***Step 1:**

*Volume of rectangular prism = l × w × h cubic units*

**Step 2:**

*Volume of given prism = 3 × 5 × 9 = 135 cubic cm.*

* Show Answer*

**Q 10 — Find the volume of the following rectangular prism in cubic cm.**

*Answer : C*

*Answer : C**Explanation*

*Explanation***Step 1:**

*Volume of rectangular prism = l × w × h cubic units*

**Step 2:**

*Volume of given prism = 3 × 6 × 4 = 72 cubic cm.*

* Show Answer*

*volume_of_rectangular_prism.htm*

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