# Angles  (Page 3/29)

 Page 3 / 29

This brings us to our new angle measure. One radian    is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ times the radius, a full circular rotation is $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ radians. So

See [link] . Note that when an angle is described without a specific unit, it refers to radian measure. For example, an angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length (circumference) divided by a length (radius) and the length units cancel out.

## Relating arc lengths to radius

An arc length     $\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ is the length of the curve along the arc. Just as the full circumference of a circle always has a constant ratio to the radius, the arc length produced by any given angle also has a constant relation to the radius, regardless of the length of the radius.

This ratio, called the radian measure    , is the same regardless of the radius of the circle—it depends only on the angle. This property allows us to define a measure of any angle as the ratio of the arc length $\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ to the radius $\text{\hspace{0.17em}}r.\text{\hspace{0.17em}}$ See [link] .

$\begin{array}{l}s=r\theta \\ \theta =\frac{s}{r}\end{array}$

If $\text{\hspace{0.17em}}s=r,$ then

To elaborate on this idea, consider two circles, one with radius 2 and the other with radius 3. Recall the circumference of a circle is $C=2\pi r,$ where $r$ is the radius. The smaller circle then has circumference $2\pi \left(2\right)=4\pi$ and the larger has circumference $2\pi \left(3\right)=6\pi .$ Now we draw a 45° angle on the two circles, as in [link] .

Notice what happens if we find the ratio of the arc length divided by the radius of the circle.

Since both ratios are $\text{\hspace{0.17em}}\frac{1}{4}\pi ,$ the angle measures of both circles are the same, even though the arc length and radius differ.

One radian    is the measure of the central angle of a circle such that the length of the arc between the initial side and the terminal side is equal to the radius of the circle. A full revolution (360°) equals $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ radians. A half revolution (180°) is equivalent to $\text{\hspace{0.17em}}\pi \text{\hspace{0.17em}}$ radians.

The radian measure    of an angle is the ratio of the length of the arc subtended by the angle to the radius of the circle. In other words, if $\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ is the length of an arc of a circle, and $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ is the radius of the circle, then the central angle containing that arc measures $\text{\hspace{0.17em}}\frac{s}{r}\text{\hspace{0.17em}}$ radians. In a circle of radius 1, the radian measure corresponds to the length of the arc.

A measure of 1 radian looks to be about 60°. Is that correct?

Yes. It is approximately 57.3°. Because $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ radians equals 360°, $1\text{\hspace{0.17em}}$ radian equals $\text{\hspace{0.17em}}\frac{360°}{2\pi }\approx 57.3°.$

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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