Main content

### Course: High school geometry > Unit 4

Lesson 7: Solving modeling problems with similar & congruent triangles# Geometry word problem: Earth & Moon radii

Similarity and the Golden ratio join forces again, this time to save the Earth! (Or maybe just finding its radius). Created by Sal Khan.

## Want to join the conversation?

- hello, why is it 6371 km square root of phi, but not square root of phi minus 6371 km. thanks.(30 votes)
- Yeah, that's tricky. I would look at it in three steps (where Sal combined it into one step). (1)The triangle we are given (the one drawn in the picture) is in units of earth-radii. Segment EL, for instance, is 1 earth radius - equal to the radius of earth. Segment ME, conveniently the length of both an earth and a moon radius, is given as sqrt(phi) earth radii (or aproximately 1.27... earth-radii).

(2) So then when you draw your triangle in kilometers, the base is 6371 km, and the height is 6371 km + r km (where r is the radius of the moon). (3) Since you have two congruent triangles, you can create a ratio: 6371/1 = (6371+r)/sqrt(phi). Cross multiply and you get: 6371(sqrt(phi)) = (6371 + r)(1) -- which catches you up to Sal.(35 votes)

- The problem says that the golden ratio is the
**only**number that has the property of its square being one more than itself. How did they prove this? or is it even proven?(13 votes)- It can be proven simply by solving for an unknown value x that satisfies the conditions. In other words, solving for x in x^2 = x + 1. This ends up just being a quadratic (x^2 - x - 1 = 0) that you can use the quadratic formula on. The only numbers that satisfy the equation are the golden ratio and one minus the golden ratio (1.618.... and -0.618....)(30 votes)

- How did he get 6371 times the square root of phi?(14 votes)
- The base was 1 as you could see on the picture and then it was proportioned so you got the actual length which is 6371 km as Sal drew it. This is the same as saying 6371 * 1. So logically the same should follow for the other side which is sqrt(phi) so you got 6371 * sqrt(phi).(9 votes)

- I'm failing to understand even the terms of this problem. How is the side the is the square root of phi also the square root of phi*one earth radii? I just can't see why Sal multiplied 6371km by the square root of phi in his new triangle. Please help!(7 votes)
- Let me try to help here.

The units were in earth radii which was one unit of Earth Radii, so Sal changed them to km by multiplying them by Earth's radii in terms of kilometers. He did this to all the numbers, in the equation of a^2+b^2=c^2, otherwise the equation would have been changed.

I hope I helped, kind of new to this myself.(9 votes)

- How can Phi go on forever and ever?(6 votes)
- Is it a non-repeating non terminating number. The definition of this is that the decimals in the number go on for an infinite amount. Other example of this is π and √2.(4 votes)

- Phi squared is
`phi + 1`

... Is the square root of phi anything special?(6 votes)- No matter how many times you take the square root of 1, you will always get 1. Similarly, if you recursively take the square root of a value x, it will approach 1. If x is positive it will diminish to 1, if 0 < x < 1 then x will increment up to the limit of 1. Not exclusive to phi, but a cool property of recursive square roots for this number range.(6 votes)

- If the radius of the moon and the radius of the earth are in proportion to φ, is the moon (in any way) and any other satellite (artificial or manmade) proportion to φ? Thanks in advance.(2 votes)
- It's possible this Moon:Earth ratio was luck. However, our best theory is that the Moon was formed via a planet-planet collision. So, it's also possible that when two planets collide, and are similar enough in size to create a moon at all, that it's biased to be in a ratio like ours is.(7 votes)

- Hi, at06:13please tell me how did he get (sqrt phi-1)?(3 votes)
- He factored out a 6371:

(6371√φ) - 6371 = 6371([√φ] - 1)

Note that when you distribute the 6371 on the right side of the equation, you get back to the left side of the equation. Therefore, both are equal.(3 votes)

- Where can I find the video on golden triangle(4 votes)
- This video is about the golden triangle :). If you want to learn about the golden ratio the previous video will do.(1 vote)

- I'm confused about how he got the first measurements. How does a^2 = the square root of phi?(2 votes)
- He set up the pythagorean theorem as the analogous equation.

a^2 + b^2 = c^2

Thus since phi + 1 = phi^2, one can conclude that

a^2 = phi <=> √(a^2) = √phi <=> a = √phi(4 votes)

## Video transcript

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.