Golden ratio to find radius of moon

The dimensions of
the Earth and moon

are in relationship
to each other

forming a golden triangle.

Represented by phi,
the golden ratio

is the only number which has
the mathematical property

of its square being
one more than itself.

And there's a whole video
on phi on the Khan Academy,

and I suggest you watch it.

It will give you chills.

And if you watch that and
you do this problem here,

you'll have even more chills.

But let's just try to
tackle this problem.

So they say that phi plus
1 is phi squared, which

is neat by itself.

And this is where they just
write out what phi looks like.

So phi is approximately 1.61803.

It keeps going on and on and on.

Plus 1 is going to be 1.61803
on and on and on squared,

which is 2.61803.

So it's just another
way of expressing this.

They tell us by applying
the Pythagorean equation

to this equation, a
right triangle with sides

phi square root of phi
and 1 is constructed.

So what are they saying?

Let's say, well look, this looks
kind of like the Pythagorean

equation, or the
Pythagorean theorem.

If we make this a squared,
we make this b squared,

and we make this c
squared, you could

think about that as
expressing the relationship

between the sides
of a right triangle

where our hypotenuse
c is equal to phi,

that this side, the shorter
side b, is equal to 1--

the square root of 1 is just
1-- and the longer side, but not

the longest, the longer of
the two non hypotenuse sides,

is going to be the
square root of phi.

That's all they're saying
with that first sentence.

As shown below, and this is
a little bit mind blowing.

As shown below, the radii
of the Earth and moon

are in proportion to phi.

So this is exciting.

Let me do this in a
color you can see.

If you were to take
the Earth's radius-- So

that's this radius
right over here.

I keep wanting to change
colors, and I'm having trouble.

So if you were to take
the Earth's radius

and add it to the
moon's radius, the sum

of those two radii, the ratio
of that sum to Earth's radii

is the square root of phi.

And that just makes you
think about the universe

a little bit.

You should just pause
that video and ponder.

Who cares about this
actual question?

Well, we should answer
the question as well,

but that's kind of
eerie because this

isn't the only
place this shows up.

This shows up throughout nature
and throughout mathematics.

It's just a fascinating number
for a whole set of reasons,

and this is just a
little bit eerie.

But anyway, we have
a problem to do.

If the radius of the earth
is 6,371 kilometers, then

what is the radius of the moon?

So let's actually redraw
this triangle here,

but let's draw it in
terms of kilometers.

Right over here,
the measurements

are in terms of Earth radii.

This right over here
is 1 Earth radii.

If this is 1 earth radii,
then the entire distance,

the combined radii of
the moon and the Earth,

is square root of
phi Earth radii,

and the hypotenuse of this
triangle is phi Earth radii.

So this is in terms
of Earth radii,

but let's redraw the
triangle, and let's

draw it in terms of kilometers.

So let's draw it in
terms of kilometers.

I'll try to draw it
pretty similar to that.

So if we draw it in
terms of kilometers,

and I'm going to do a
very rough drawing of it.

So that's the Earth
right over here.

I'll just draw a
part of the Earth.

I don't have to draw
the whole thing.

I think you get the idea.

And then this is the
moon right over here.

I think you get the idea.

They told us that the radius of
the Earth is 6,371 kilometers.

Well, they also told us
that this height, the height

of this right triangle, is
square root of phi Earth radii.

So if we write it in
terms of kilometers

it's going to be 6,371 times the
square root of phi kilometers.

Square root of phi Earth radii.

So that's this entire
distance right over here.

Now what they want us to figure
out is the radius of the moon.

They want to figure out this
distance right over here.

So let's call that r for
the radius of the moon.

So how can we figure out r?

Well, we also know
what this segment is.

This segment, which
I will do in green,

is also the radius of the Earth.

The Earth is roughly a sphere.

So we could say that this
distance right over here

is also 6,371 kilometers.

So this actually breaks down
to a fairly simple problem.

The combined radii, we can
write it in two different ways.

We could write it as the radius
of the moon plus the radius

of the Earth, which
is 6,371 kilometers.

And we'll just assume that
everything we're doing

is in kilometers now.

And we can write it as, the
combined radii, as 6,371 times

the square root of phi.

Once again, the combined
radii is square root of phi

times the length of
the Earth's radius.

So here we just have it
in terms of Earth radii.

Here we have it in
terms of kilometers.

This is the Earth's radius.

You multiply that times
square root of phi,

you get the combined radius.

Well, now we just
have to solve for r.

We can subtract 6,371
from both sides,

and we get r is equal to 6,371
times the square root of phi,

minus 6,371.

And if we want, we
could factor a 6,371

from both of those
terms, and we would

get r is equal to 6,371 times
square root of phi minus 1.

And we are done.

And this is still pretty
neat and pretty cool.