Imagine a cube balancing perfectly on its vertex as in the picture. When looking down from above...

Imagine a cube balancing perfectly on its vertex as in the picture. When looking down from above, the upper most vertex lines up with the bottom most vertex.
This cube has a side length of 1 unit.
Find the distance from the circled vertex to the floor as shown in the picture.
Show your work.

the distance is 1.524 in OR 38.327 mm

Show your work fam.
Also, exact values only. Balancing and measuring cube like objects dont count.

pythagorians theorem. fucking autist

Form a triangle, hypotenuse is 1 unit, we know its a cube so the angle is 90 on the inside and 45 on the outside to the plane it sits on to form 180, then solve for X or the red line. 45-45-90 triangle. 1/square root(2) = ~0.7071

This would be true if the cube was resting on an edge, not a vertex as shown in the picture.

This ffs

nigger, do your own goddamn homework.

If you think it works, try it lol. Doubt you'll get the answer that way tho.

it would boil down to the pythagorean theorem,
where a would be the distance from the corner to the center of any given face of the entire cube
b would be the same since the point on the center of the face would have the same distance from the ground. (imagine we turn the cube in OPs image CCW on the point its standing on)

here we get into some interesting stuff about the square root of 2 yadda yadda

but assuming this 1 unit stuff we're looking at a diagonal of 1.414 (.707 units from the corner to the center)...


anyway i opened it up in Unity and with a little scripting got this result:
0.4964234

Nigga dis easy. The distance is basically the height of the tetrahedron with an edge length of 1 unit.

h=sqrt(2/3)*a where a is the length of the edge.

0.82

the Y position shown here is consistent with your 1-unit rule, it is ultimately the distance of that "corner" object from the ground on its y (up and down) axis, ie it is .4964234 units above the "ground"

It's still partially true. Hypotenuse is still 1, but the angle between the vertex and the tabletop isn't 45. Instead, it's the same angle as any regular hexagon, which is 60 degrees. Like all 30-60-90 triangles, the shorter side is exactly half of the hypotenuse. Answer is .5 units. Use whatever pythogorean math you want to get there, but the answer always remains the same.

Now fuck off and do something less autistic.

your formula is for a regular tetrahedron. This is not a regular tetrahedron.

It's true anyway, the edge of the cube doesn't get shorter because you rotated the cube

I came here to fap, but did this instead, so fuck you.

Ok, this guy () had a near miss, but as this guy () pointed out, we are not guaranteed that the cube is resting at a 45 degree angle to the plane.

First, let's assume that when OP says "balancing perfectly on its vertex" he means that a line passing between the vertex on the table and the vertex directly across the cube sits perpendicular to the plane. Any other orientation would probably require the cube to be unevenly weighted, so it makes sense.

Any triangle containing that line that crosses through the cube would have to contain one that runs along an edge and one side that bisects a face.

So, pick a triangle. The side that is a cube edge is given to be length 1. The side that bisects a face must have length SQRT(2) because it bisects each face into two right triangles with two sides length 1. By the Pythagorean Theorem, the middle side must be SQRT(2).

So the line through the middle of the cube is the hypotenuse of a right triangle with sides 1, SQRT(2) and... SQRT(3), again by the Pythagorean Theorem. (If anyone doubts that it is a right triangle, remember that each edge meets the face at a 90 degree angle).

OK! That was already a lot of math to get to this point with no explanation of why we care. It's important to know the length of the core line because we need to know the angle between the edge and the table in the triangle with the red line we want to find.

Consider the triangle with the core line, one line bisecting the yellow face, and one line along the edge of the blue face suspended above the ground. The angle between the core line and the blue edge line can be found using the Law of Cosines. It turns out that this angle is... 60 degrees.

Because that angle and the angle between the blue edge and the table must add up to 90 degrees, the blue edge is at a 30 degree angle from the table.

cont...

For fuck sake. Pythagorean can be used for 3d. Take the cubed root of the total area of the 3 projected sides. So take the cubed root of 3 which =1.4422 then divide by 3 because it's an exact 1/3
The answer is 0.4807 just about

HA it's a trap, anons, the rubix cube is on a desk. You all fail, Dunder Mifflin needs better than this

1/root(3). third the way up the cube. top vertex is root3 by pythagorean theorem. divide by 3 and you get root3/3 which is 1/root3 fuck your retarded easy math

cont...

Since the blue edge is 30 degrees from the table and the red line drops down perpendicular to the table (by definition as the distance from the circled vertex to the table), the angle between the blue edge and the red line is 60 degrees.

In a 30-60-90 degree right triangle, the shorter side is always opposite the 30 degree angle and, in fact, is always half the hypotenuse. Here, the hypotenuse is the blue edge, given as length 1.

The red line is 1/2 units long.

You can also verify that last bit with the Law of Cosines, if you care.

So this is what STEMlords do for fun...

I think I'll stick to saving the world, please.

>Or should I say: your welcome.

So close! SQRT(3)/3 is the length of the side along the table.

Sorry, that last bit should say Law of Sines

Sorry I meant total volume. Here's the equation tho
[[(1^3)+(1^3)+(1^3)]cube root]/3=0.4807

what

This. Cube resting perfectly on one vertex forms a 2D hexagon if you're viewing it from the side. The math is super easy from there. No need to complicate this shit boys.

Hey where's my gold star, I got it first

you silly fucking niggers need to get a grip. 3d pythag thrm is root sum of distances squared, in this case root 3. the red circle is halfway up to the 2nd layer of vertices, and by symmetry a third the way up the cube. therefore red dot is 1/sqrt(3) or .577 that is the answer

It's still not the right length. It's the length of the side between the bottom of the red line and the vertex on the table.

No gold star for you, only silver.

You're wrong just like

Good job, you're viewing it from about a 30 degree angle...

>>/sci/

Or from directly above

Why are you all assuming that the side is not at a 45 degree angle?
OP said:
>the upper most vertex lines up with the bottom most vertex.
The only way this is the case this is true and the cube is "balancing perfectly" is if it makes a 45 degree angle.

Put the cube on its edge. The angle is 45 degrees. Now slowly raise the cube from balancing on its edge to balancing on its vertex.
The angle shrinks.

Sorry, but I'm right. The length I gave is what the question asked, exactly how the picture shows too.
Here's some more decimal places so it more right
0.480749857
I am curious though, what do you *think is the answer

Sqrt (3)/2

For some reason I keep getting 0.5774

because that’s the answer.
see , proof is correct.

We cannot be sure that the vertex is a third of the way up

I done goofed, been too long since math class.forgot cube root is for something else. I shall commit sudoku wearing my silver medal

Sure you can. Divide one of the three downwards-oriented sides into an upper and lower triangle. their altitudes are equal, so from the bottom vertex to the 3 next vertices is the same as the distance from those 3 to the next 3. from there it is symmetrical and the last third is equal to the first.

Yes we fuckin can, picture just one of those sides. The distance from the bottom point to one of the side points to its top point is equal, same thing applies when going vertically, therefore since that's true, another one at proper angle is also true, making the answer 1/3 of the total height

define one goober to be equal to the distance from the circled vertex to the ground when a cube with sides = to 1 unit is balanced on a vertex with the upper most vertex directly over the bottom vertex.

therefore the distance is one goober

simple

sqrt(3)/3=0.577

the distance from the 'lower' side point to the 'upper' side point might not be the same as the distance outlined in red though

I agree with you

LOOK HERE CAUSE I HAVE THE ANSWER

Everyone saying that the angle is 30 degrees is wrong, this isn't a hexagon, even if it looks like one from a certain angle. People in this thread have already got it right answer is 0.577

Here's my working,
Point a = the point touching the ground
Point b = the point circled in red in the image
Point c = the topmost point of the cube

We know edge ab = 1
We can work out the distance from b to c applying pythag to the other edges of that face. I just plugged this through my calculator so I'll tell you
edge bc = 1.41

From there we can use edge ab and edge bc to form another triangle (which will go through the middle of the cube) and calculate edge ac
edge ac = 1.73
and now that we have all three sides of that triangle, we also know all three angles.
a= 54.7
b= 90
c= 35.3

We also know that line ac is a completely vertical line, which means it is 90degrees up from the ground.
90 - 54.7 = 35.3

NEW TRIANGLE
now we can work out the actual answer to the problem.
x = our edge we want to find the length of
a = the point on the ground
b = red circle
c = the right angle

from here it's simple triangle maths
hypoteneuse is 1
angle a = 35.3
plug it all through the calculator

x = 0.578

someone else has probably worked this out by the time I finished typing

X*Cos(30)=Vertex height

Sorry
X*Tan(30)=Vertex Height

I always mix Sin, Cos, and Tan

What are you making in unity?

Assumptions:
3. Cube is perpendicular to the ground.
2. Ground is flat.
1. Normal force extends through both the resting vertex and through the opposite-most vertex ("top").

Assuming the above is true, then:
The marked perimeter vertex would be 45deg off normal and 45deg off axis, thus forming a 45-45-90 triangle.

You don't even need Pythagorean Theorem for this: It's a right isosceles, and is thusly a derivative of the Golden Triangle.

Ratios are 1:1:SQR(2)

Figure it out.