What is the formula for volume of a rectangular prism

Rectangular Prism Calculator

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Units

Significant Figures

Answer:

total surface area

Stot =

lateral surface area

Slat =

bottom surface area

Sbot =

Rectangular Prism Shape

l = length
w = width
h = height
d = diagonal
Stot = total surface area
Slat = lateral surface area
Stop = top surface area
Sbot = bottom 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, ft2 or ft3. 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 mm2 and V in mm3.

Formulas for a rectangular prism:

• Volume of Rectangular Prism:
  • V = lwh
• Surface Area of Rectangular Prism:
  • S = 2(lw + lh + wh)
• Space Diagonal of Rectangular Prism: (similar to the distance between 2 points)
  • d = √(l2 + w2 + h2)

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 = √(l2 + w2 + h2)

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 = √(l2 + w2 + h2)

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 = √(l2 + w2 + h2)

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 = √(d2 - l2 - w2)
• 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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Created by Hanna Pamuła, PhD candidate

Reviewed by Bogna Szyk and Jack Bowater

Last updated: Sep 03, 2022

With this volume of a rectangular prism calculator - a.k.a. a box volume calculator - you'll find the volume of any box-shaped container in a blink of an eye. No fuss required. You need to enter only three values, and we'll calculate the volume for you (though it's not so tricky, you could figure it yourself😊). If you're searching for the definition, check the section What is a rectangular prism. However, if you're still wondering how to find the volume of a rectangular prism, we'll show you step-by-step how to use our calculator on an (almost) real-life example - the volume of a cat 🐈.

What is a rectangular prism?

A rectangular prism is a 3D shape with 6 faces, all of which are rectangles. Other names for a rectangular prism are a cuboid, or simply a box.

Every rectangular prism has:

• 8 vertices;
• 12 sides; and
• 6 sides.

What is the formula for the volume of a rectangular prism?

Finding the volume of a rectangular prism is a straightforward task - all you need to do is to multiply the length, width, and height together:

Rectangular prism volume = length × width × height

Where can you use this formula in real life? Let's imagine three possible scenarios:

1. You bought a fish tank for your golden fish🐠. It's in a regular box shape, nothing fancy, like a corner bow-front aquarium. If you're wondering how much water you need to fill it, simply use the volume of a rectangular prism formula. It is a similar story for other pets kept in tanks and cages, like turtles or rats - if you want a happy pet, then you should guarantee them enough living space.

2. The time has come - you've decided that this year you'd like to grow your own carrots🥕 and salad🥗. For that, you need to construct a raised bed and fill it with the potting soil. But how much dirt should you buy? Well, that's the same question as how to find the volume of a rectangular prism: measure your raised bed, use the formula, and run to the gardening center.

3. You are going on the vacations of your dreams 🌴. You have to pack your stuff for the three weeks, and you're wondering which suitcase🧳 will fit more in:

• your good old large suitcase, 30 x 19 x 11 inches,
• or the new fancy one, 28 x 21 x 12 cm

That's again the problem solved by volume of a rectangular prism formula. Solve it manually, or find it using our calculator.

If you're searching for a calculator for other 3D shapes - like e.g. a cube, which is a special case of a rectangular prism - you may want to check out our comprehensive volume tool. It has a gazillion different shapes! (Fourteen, to be exact.)

How to find the volume of a rectangular prism?

Well, now that you know what a rectangular prism is and its volume formula, all the calculations should be a piece of cake! Just measure the three dimensions of your rectangular prism, and use the method from the previous paragraph. Alternatively, you can simply use our box volume calculator.

So, let's have a look at the example – let's calculate the volume of a cat🐈. Yes, a cat, as cats are almost like liquids (they take on the shape of whatever container they are in). Assuming that the cat completely fills a plastic container with dimensions 12 inches x 10 inches x 8 inches:

1. Input the container's length into the first field of our volume of a rectangular prism calculator. It's 12 inches in our case.
2. Enter the width of the box. Put 10 inches into the proper field.
3. Finally, input the height of your container, 8 inches.
4. And there it is: the volume of a rectangular prism calculator did the job. Now we know that our cat's volume is 960 cubic inches. Isn't that pawesome?🐾

(Of course, it's a really rough estimate. Take it with a pinch of salt 🧂 or even two pinches.)

So, now that you know your cat's volume, you can go and search for a perfect cardboard box📦 for your pet!

FAQ

How do I calculate the volume of a rectangular prism?

To find the volume of a rectangular prism, you need to:

1. Determine the lengths of the sides: width, length, height.
2. Multiply together the three values from Step 1.
3. The result you've got is the volume of your solid.

Don't forget to include the units if it is given! Since it's volume, you need cubic units.

What is the volume of a rectangular prism with sides 2, 5, and 7?

The answer is 70. To see how to get this result, recall the formula for the volume of a rectangular prism: volume = length × height × width. Hence, we compute the volume as 2 × 5 × 7 = 70.
Remember to include the units: for instance, if your measurements are in inches (in), the volume will be in cubic inches (in³).

How do I calculate the volume of a rectangular prism given diagonals?

To determine the volume of a rectangular prism when you know the diagonals of its three faces, you need to apply the formula volume = 1/8 × √(a² - b² + c²)(a² + b² - c²)(-a² + b² + c²), where a, b, and c are the diagonals you're given. This formula can be easily derived by using the Pythagorean theorem.

Hanna Pamuła, PhD candidate

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