# Cube Calc: find V, a, d

Created by Álvaro Díez
Reviewed by Dominik Czernia, PhD and Jack Bowater
Last updated: Oct 14, 2022

Welcome to cube calc: find V, a, d, a simple cube calculator that shows you how to calculate the volume of a cube (V), the side of a cube (a), or any other property you want.

While we are at it, we will take a look at the definition and equations for all of these parameters, and show you how you can find anything and everything about a cube.

## The cube, more than a simple dice

The cube is one of the simplest 3D shapes in existence. It's a regular polyhedron made of regular polygons. In particular, it consists of six identical squares, and it can roll (albeit not perfectly). It's this regularity that makes it a great candidate for dice, but also a simple shape which you can use to learn about the 3D properties of objects.

In this cube calc, we will focus on finding the properties of a cube. How do you find the volume of a cube? What is the volume of a cube? How to find the surface area of a cube? All the answers to those questions, and their relevant definitions, are here, but first we need to introduce some terminology so we're all on the same page:

• $a$ - represents the side of the cube;
• $V$ - denotes the volume of a cube;
• $d$ - is the letter we use for the diagonal of the cube;
• \$S$$- represents the surface area of the cube; and • $f$ - is the face diagonal, which is the diagonal of the square that makes up the sides of the cube. Now we are ready to go ahead and learn how to find the volume of a cube, and all of its other properties! ## How to use the cube calc: find V, a, d calculator? If all you are after is the values of all parameters of the cube, you're in luck. Our cube calc find V, a, d, S, and f with a single input. All you need to do is fill in the value you know, and it will auto-populate the rest of the fields with the correct answers. If you are interested in learning more about how to find the surface area of a cube, or how to compute any other values, read on! We will assume that the value you know is the side (a), except when calculating the side itself. You can also obtain the side by reversing any of the equations. ## How to find the volume of a cube? Cube calc: find V If you want to know how to find the volume of a cube from its side length, all you need to do is to multiply the side a together three times. The equation is, therefore: $V = a^3$ ## How to find the side of a cube? Cube calc: find a We have seen above how the cube calc finds $V$. But if you input $V$ hoping to get $a$, the cube calc finds $a$. It does so by turning the equation around like this:$$a = \sqrt[3]{V}

So $a$ is the cube root of $V$. Problem solved, let's move on to see what happens if you want to find the diagonal of a cube. How does the cube calc find $d$?

## How to find the diagonal of a cube? Cube calc: find d

The short answer to the question above is: using the Pythagorean theorem. The longer answer is that we need to look at the image of the cube first.

For the cube calc to find $d$, it first needs to calculate $f$, the face diagonal, which is the square diagonal of the faces of the cube. Since the diagonal in a square divides it into two right triangles, you can simply use the right triangle calculator to find $f$ if you know a`.

Both legs are the same length (i.e., $f$), and therefore we can apply the Pythagorean theorem to find:

$d = a \times \sqrt{3}$

## How to find the surface area of a cube? Cube calc: find S

If you want to know how to calculate the surface area of a cube, it is similar to calculating the volume. While the cube calc finds $V$ by multiplying the side $a$ by itself three times (as it is in 3 dimensions), we find the surface area by finding the area of the squares (2 dimensions), and summing them all up.

$\begin{split} S &= a^2\!+\!a^2\!+\!a^2\!+\!a^2\!+\!a^2\!+\!a^2 \\ &= 6\times a^2 \end{split}$

## How to find the face diagonal of a cube? Cube calc: find f

We had already commented on how to find $f$ when we explained how the cube calc finds $d$, but let's do it again for completeness. Starting from $a$, we know that $f$ divides the squares into two right triangles with legs of size $a$. Applying Pythagoras, we get that:

$f = \sqrt{a^2+a^2}= a\times\sqrt{2}$

Finally, by combining all these equations and solving for whatever values you want, the calculator achieves its magic. That's why, from any value you give, the cube calc finds $V$, $a$, $d$, and $S$.

Álvaro Díez
Side (a)
in
Cube diagonal (d)
in
Face diagonal (f)
in
Volume (V)
cu in
Surface area (S)
in²
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