# Trigonometry Cube Equations

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## Introduction: Trigonometry Cube Equations

Learn how to make a Turner's Cube!

## Step 1: Get Your Calculations

Since the cube will have two internal loose cubes it is important to do these calculations so that the cubes stay contained within each other and do not fall out.

So we will make some calculations to find:

1. Size of each cube
2. Diameter of the holes
3. Depth of each hole

Finding the size of each cube

First determine what size cube you want. I wanted a 2x2x2 cube

Then determine what is the total number of internal cubes you want . I wanted the cube to contain 2 cubes so 2 internal cubes plus the cube that will hold them is a total of 3 cubes.

So to know what size to make each of the cubes we divide

`  Largest Cube/ Number of Layers -->          2/3 = .667 inches<br>`

so .667 will be the size difference of each cube so the sizes of each cube are as follows :

Large Cube : 2

Middle Cube : 2 - .667 = 1.333

Smallest cube: 1.333- .667 = .666

## Step 2: ​Finding the Diameter and Undercut of the Holes

To calculate the diameter and undercut we need the corner to corner distance

We will use the Pythagorean Theorem- which is a²+b² = c² modified to solve for c so , c= √(a²+b²)

now we will solve for the distances of each cube:

1. Smallest Cube measuring .666

c= √(a²+b²) --> √(.666²+.666²)= .9418

2. Middle Cube measuring 1.333

c= √(a²+b²) --> √(1.333²+1.333²)= .1.885

The corner to corner distance has to have the following relationship to the undercut and bore diameter.

undercut > corner to corner distance > bore diameter

The reason the undercut has to be larger than the corner to corner distance is because in order to brake the cubes free within each other you must cut away the attachment each cube has to each other on each corner. In order to brake the attachment you have to cut greater than the corner to corner distance. This is illustrated with the drawing labeled 2.

Similarly the reason why the diameter of the holes has to be smaller than the corner to corner distance is because if the hole was larger you would cut out the square and it would fall apart. This is demonstrated with drawing 1.

## Step 3: ​Finding the Bore Depths

To find the depth of the holes take the ((largest cube - (size of cube you are boring to)/2)

So the calculation is as follows:

1. Smallest cube:

((2-.666)/2) = .667

2. Middle Cube

((2- 1.333)/2) = .3335

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