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## NASA

### Course: NASA > Unit 2

Lesson 2: Measuring the solar system- A flat earth
- Arc length
- Circumference of Earth
- Occultations
- Occultation vs. transit vs. eclipse
- Size of the moon
- Angular measure 1
- Angular measure 1
- Trigonometric ratios in right triangles
- Angular Measure 2
- Angular Measure 2
- Intro to parallax
- Parallax: distance
- Parallax method
- Solar distance
- Solve similar triangles (advanced)
- Size of the sun
- Scale of solar system

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# Angular measure 1

## A common mistake

A simple yet fundamental concept in astronomy is angular measure. It addresses a common error made by non-astronomers. To illustrate this error, try describing the size of the Sun from your point of view.

It’s tempting to say it’s about an inch wide, or the size of a quarter. There is a problem with this description however. Do you know why?

Perhaps my arm is much shorter than yours, and therefore my understanding of “a quarter” is different. The measurement depends on the exact distance to the quarter. In order to use this type of measurement we’d have to say it’s the size of a quarter observed x inches away.

## Angular measure

Astronomers use a simpler method based on how many degrees you would tilt your telescope (or head) to scan across an object. This is known as angular measure.

**Click and drag the circle below to see how the angular measure changes**:This method leads to the conclusion that both the Sun and Moon are about a half of a degree in size. This means if we put 720 Moons side-by-side in a circle they would complete a ring around the sky! Convince yourself of this, it's very important:

What about measuring really tiny objects such as planets? Just as we do with microscopic objects, we simply increase the resolution of our measurement. We can divide one degree of arc into 60 arcminutes. We further divide each arcminute into 60 arcseconds:

Therefore

**one degree**is equal to 60 x 60 =**3,600 arcseconds**## Triangulation

When using angular measure we define an isosceles triangle between the observer and the sides of the object we are measuring. As follows:

Notice we can cut this triangle (and angle) in half to form a right triangle. We love right triangles because it allows us to use trigonometry!

**tan (angular measure/2) = radius / distance**

## Quick review (trig in action)

Imagine a pole is 12 meters high and we have to tilt our head 36.8 degrees from the horizon to see the top.

**How far away are we from the pole?****tan (ABC) = opposite / adjacent**

tan (36.8) = 12 / distance

distance = 12 / tan(36.8)

distance = 16 meters

Next let's review basic trigonometry & angular measure.

## Want to join the conversation?

- In the example of the Quick review (trig in action) Imagine a pole is 12 meters high and we have to tilt our head 36.8 degrees from the horizon to see the top. How far away are we from the pole? Would we have to take in to consideration the height of the observer eye's from the ground at (point B) the place observed from? The angle at his feet would not be 36.8 degrees to the top of the 12 feet high pole. So would the distance from the pole be wrong? Instead of tan (ABC) = opposite / adjacent; tan (36.8) = 12 / distance; distance = 12 / tan(36.8); distance = 16 meters. Would we have to know the height of eyes from ground? Would we use tan(AED) = opposite / adjacent; tan(36.8) = (12 - height of eyes from ground) / distance ; distance = (12 - height of eyes from ground) / tan(36.8); 1.3 meters high gives a distance of about 14.26 meters, 2 meters high gives a of about 13.36 meters.(15 votes)
- Yes, sometimes the height of the observer is taken into consideration but in most cases the observer is considered as point mass if the distance being calculated is larger than the dimensions of the observer(8 votes)

- I believe that this should be explained in a video not just as an article as there are concepts here which require a more detailed explanation rather then the fine print.(11 votes)
- Isn't the angle depicting arcminutes and arcseconds a little wide for one degree?(7 votes)
- Yes it is, but I challenge you to draw it correctly ;)(7 votes)

- What is an arcminute and arc second? Also, what is tan in tan(angular measure/2)=radius/distance?(6 votes)
- Arcminute and arcsecond are explained in the text. Tan means tangent, the trigonometric function.(7 votes)

- I am still not clear about what angular measure is . Can you please make video to explain it since i am a beginner and i am starting from scratch.(9 votes)
- if 360 degrees = 24 hours why 1 degree = 60 arcminutes?

if it have nothing to do with Earth rotation period why they call it arcminute?

what lasts 60*360 minutes?(3 votes)- I think the "minute" and "second" parts in arcminute and arcsecond are there solely because an arcminute has 60 arcseconds, as a minute has 60 seconds.

Does this help?(2 votes)

- This method leads to the conclusion that both the Sun and Moon are about a half of a degree in size. This means if we put 720 Moons side-by-side in a circle they would complete a ring around the sky!

what is meant by the statement "half of a degree in size"?

how does it lad to the conclusion about the 720 moons in 360'(3 votes) - How do we calculate the value of tan(36.8)? Moreover, the example's last step seems theoretically improbable, because tan(36.8)= -1.25786223766!!(3 votes)
- "This method leads to the conclusion that both the Sun and Moon are about a half of a degree in size."

I could not understand this conclusion.

My question is from what distance are they measuring the size?(3 votes)- They are measuring the arc of sky that they cover. Assuming flat ground, the horizon to directly overhead is 90 degrees. The Sun and Moon cover a half degree of that distance, or you could put 180 Moons/Suns end to end in a line from the horizon to directly overhead.(0 votes)

- Is there a website/calculator that would give me the angular size of the moon (as viewed from Earth of course) for different dates and if possible, to be more precise for different places in the world?(2 votes)