Measuring, constructing, and using angles

Measuring, constructing,

and using angles

ContentsWhy Measure Angles?How might we measure angles?

What is an “angle”?What does “measurement” mean?Our “measuring tool” for angles.

Contents (continued)Measuring an angle Drawing an angle of a given sizeThe “radian” unit of measure for angles.

Why measure angles?

Why measure angles?

So we can figure outhow big the Earth is!

If we choose two places that are north and south of each other,

A

B

then measure the distance between them,

Dist

ance

A

B

and also the highest angle the Sun reaches at each place on a given day,

Highest angle at Location A Highest angle at Location B

Sky simulations made with the free astronomy program Stellarium (http://stellarium.org/)

we can calculate the size of the Earth.

The first person to do this calculation was Eratosthenes, a librarian at the Great Library of Alexandria, Egypt, around 200 BC.

Actually, almost all of astronomy and geography depends upon being able to measure angles.

So do many jobs, such as surveying.

So, how might we measure angles?

First, let’s review what we mean by “angle”, and “measurement”.

One definition of “angle” is

“A pair of rays thathave the same endpoint”.The endpoint, V, is called the vertex.

You’ve probably learned that angles are related to “turns”, and that the same set of rays can be made by two different turns.

For example, here are …

the two “turns” that take us from ray VA to ray VB.

Counterclockwise turn Clockwise turn

Now that we’ve reviewed what “angle” means, what do we mean by “measurement”?

A good definition is found in The Archimedes Codex: How a Medieval Prayer Book is Revealing the True Genius of Antiquity’s Greatest Scientist:*

* Reviel Netz and William Noel, Da Capo Press, 2009, p. 41.

“To measure is to find a measuring tool and apply it successively to the object being measured. Suppose we want to measure a straight line.

According to the authors,

“For instance, suppose we want to measure your height, which is really saying that we want to measure the straight line from the floor to the top of your head.

“Then what we do is take a line the length of an inch [this is our measuring tool] and apply it successively, well over sixty times, but probably fewer than eighty times to measure your height.

“Since this is very tiresome, we have pre-marked measuring tapes that save us the trouble of actually applying the [one-inch line] successively,

“but, at the conceptual level, successive application is precisely what takes place.”

That definition of “measurement” needs some explanation. For example, when we read

“To measure is to find a measuring tool and apply it successively to the object being measured,”

we probably thought of “measuring tools” as rulers, etc.

Actually, the authors meant something quite different.

They meant that to measure a length (that’s the example they give), the “tool” we choose is a line segment with a convenient length.

The authors mentioned “an inch” as an example of a convenient length, but we could use a segment of any length we like.

For example, we could use this yellow segment:

Now, suppose that we wanted to “measure” this red segment.

We’d start at one end of the red segment, and “apply” the yellow segment to it repeatedly, until we got to the other end …

1

2

3

Here’s the summary of what we just did to “measure” the red segment.

1

2

3

Because we had to apply the yellow segment three times,

1

2

3

we say that the length of the red segment is “3 yellow segments”.

1

2

3

For example, if our yellow segment were an inch, the length of the red one would be “three inches”.

1

2

3

As the authors of The Archimedes Codex told us, this process would be tiresome if we had do many measurements.

For that reason, we make pre-marked rulers and measuring tapes,

1 2 3 4 5

so that we may “apply” several “yellows” at once.

1 2 3 4 5

If we wished, we could also mark off fractional parts of our yellow segment, just as we do with inches on a ruler.

Now that we’ve reviewed what “angle” and “measuring” mean,

let’s learn the common ways of measuring an angle.

As we’ve seen, the “measuring tool” that we use for lengths is some convenient segment.

However, when we measure angles, the “measuring tool” that we use is some sector of a circle.

For example, this one:

To measure an angle,

we align one side of our sector with one of the angle’s rays,

Note: “Point” of the sector must be aligned with vertex of the angle.


then keep “applying” our sector until we reach the other ray.


then keep “applying” our sector until we reach the other ray.

We use the same method for the other direction of “turn”:
















Although we could choose any sector as our “measuring tool”, the most-common is the one we get when we divide a circle into 360 equal sectors:

The angle enclosed by each of those sectors is called one degree.

We abbreviate that as “ 1° ”.

Important:No matter how big our circle,

a “ 1° ” sector always contains the same angle.

So, now we have our 1° measuring tool.

Let’s use it to measure an angle.

We had to apply our “ 1° ” tool eleven times,

so we say that the angle “measures 11°”.

Now that we have our angle-measurement tool, and know how to use it, we can look for ways to make a more-convenient version of it.

We already know that we can make a more-convenient version of a length-measurement tool by pre-marking “something” at chosen intervals, to make a tape or ruler.

For measuring angles, we can pre-mark 1° sectors on something:

However, this tool is “messy”, so we “clean it up” by …

leaving marks only at the center and edges of our circle.

Those are all we need for measuring an angle.

To make a “nicer” tool, we can number the degrees, and add a few other extras, to make the tool we call …

a protractor …

which comes in several versions.

To measure an angle with a protractor,

we align a “zero” line of the protractor with a ray of the angle …


Note where protractor aligns with vertex.




Note where protractor aligns with vertex.

then read how many degrees we’d “apply” to get to the other ray.

90°

Notice that we’ve measured the shorter of this angle’s two “turns”.

90°

Here’s the longer “turn” for this angle.

How many degrees should it measure?

We’ll discuss that later.

The 90° angle (also called a right angle) is an important one to know.

So is the angle that measures 180°. It’s called a “straight angle”.


180°180°



180°180°

90° 180°

Knowing these two angles…

helps us when we have to measure an angle like this one.

Does it measure 60°, or 120°?

By counting, we can see that there are 60 degrees between the rays.

60°

Also, 60° minus 0° is 60°. (That’s what we read on the inner scale).

60°

180° - 120° is 60°, too. (That’s what we read on the outer scale).

60°

Our angle 90°

Finally, our angle (the one we want to measure) is smaller than 90°.

All of these observations show that our angle measures 60°, not 120°.

60°

For comparison, here’s an angle that does measure 120°.

For comparison, here’s an angle that does measure 120°.

120°

Our angle 90°

We can see that it’s bigger than 90°,

Our angle 180°

but smaller than 180°.

Please remember that when we say that these angles measure 60° and 120°,

we’re talking about the measure of the smaller of their “turns”,

not the larger one.

These larger “turns” measure more than 180°, but less than the 360° that there are in a full circle.

They’re examples of what are called “reflex angles”.

So is the “larger turn” of a 90° angle, which we saw earlier.

We’ll now look, briefly, at how to find the measure of a reflex angle.

First, we can break the reflex angle into a straight angle (pink turn), …

and a 60° angle (orange turn).

The measure of the reflex angle is the sum of these: 180° + 60° = 240°.

60°

180°

240°

A second way is to remember that in a full circle, there are 360°.

Therefore, the number of degrees in the reflex angle has to be …

360° minus 120° (= 240°).

360°120°

240°

A third way to measure a reflex angle …

is with a full-circle protractor.

240°

Here are links to more information about reflex angles:

Reflex angleshttp://www.mathsisfun.com/reflex.html

Re-entrant angleshttp://www.thefreedictionary.com/re-entrant+angle

Besides wanting to measure angles, there are times when we want to (or must!) draw them.

Someone will tell us (or we will choose) the following things:

Where the vertex is,

the ray from which the angle is to start,

the direction of the “turn”, and

the measure of the angle that we are to draw.

105°

Now, just as we did for measuring angles, …

105°

we align a protractor with the vertex and ray.

105°

Next, we go in the direction of the given turn,

105°

for the required number of degrees.

105°

Here’s the mark for 105°.

We make a little dot there on our paper, …

105°

Here’s the mark for 105°.

then remove our protractor, ...

105°

and draw a ray from the vertex through our dot.

105°

We’ve now constructed the required angle.

105°

We’ll finish this discussion of angles by mentioning another common “tool” for measuring them.

We imagine making that tool by drawing an circle, and one of its radii, …

then making a segment as long as the radius, …

sticking one end of the segment to the circle, …

and bending the segment to fit the circle.

Now, we draw rays from the center of the circle through the endpoints of the segment.

The angle we’ve “cut out” of the circle is defined as “1 radian”.

This “measuring tool” has a property that makes it useful in subjects that you’ll study later.

That property is that

In a full circle, there are 2π radians.

Can you see why this is so?

Hint: What is the relationship between the length of a circle’s radius, and the length of its circumference?

Summary

Rather than review the details of how to measure angles, I’d like to emphasize that …

Summary (cont’d)

• Many types of work depend upon being able to measure angles. For example, astronomy and surveying.

Summary (cont’d)

• There are different ways of measuring angles.

Summary (cont’d)

• When we need to find a way to measure something, we can invent our own “tools”.

Summary (cont’d)

• It’s often helpful to imagine some simple version of a tool, or of solving any problem, then look for a way of making that simple tool or technique more convenient.

END

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Measuring, constructing, and using angles