Arc Length of a Circle Formula - Sector Area, Examples, Radians, In Terms of Pi, Trigonometry
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- เผยแพร่เมื่อ 14 ต.ค. 2024
- This geometry and trigonometry video tutorial explains how to calculate the arc length of a circle using a formula given the angle in radians the and the length of the radius. It also explains how to calculate the sector area in terms of pi or in radians. In addition, it also explains how to determine the area of a sector of a circle using an equation where the angle is given in degrees instead of radians. This video contains plenty of examples and practice problems.
Trigonometry - Basic Introduction:
• Trigonometry
Introduction to Angles:
• Introduction to Angles
Radians and Degrees:
• Radians and Degrees
What Exactly Is a Radian?
• Trigonometry - What Ex...
Coterminal Angles:
• Coterminal Angles In R...
_________________________
Decimal Degrees to DMS:
• Decimal Degrees to DMS...
Arc Length of a Circle and Sector Area:
• Arc Length of a Circle...
Linear Speed and Angular Velocity:
• Linear Speed and Angul...
Clock to Angle Problems:
• Clock Aptitude Reasoni...
The Unit Circle:
• The Unit Circle, Basic...
_______________________________________
Final Exams and Video Playlists:
www.video-tuto...
Full-Length Videos and Worksheets:
/ collections
Trigonometry Formula Sheet:
bit.ly/47diggI
![Radians, Arc Length & Sector Area of a Circle - [2-21-1]](http://i.ytimg.com/vi/lcuxPNBXhlY/mqdefault.jpg)
![Radians, Arc Length & Sector Area of a Circle - [2-21-1]](/img/tr.png)
Trigonometry Formula Sheet: bit.ly/47diggI
Final Exams and Video Playlists: www.video-tutor.net/
Full-Length Videos & Worksheets: www.patreon.com/MathScienceTutor/collections
Trigonometry - Basic Introduction: th-cam.com/video/g8VCHoSk5_o/w-d-xo.html
How To Graph Trig Functions: th-cam.com/video/fQPOBOFb2sY/w-d-xo.html
Evaluating Inverse Trig Functions: th-cam.com/video/so9Tjk6Ga2U/w-d-xo.html
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For calculating in degrees, my teacher gave me this formula:
S=rθπ/180°
But I find yours much more easier to remember:
S=(θ/360°) 2πR
I guess it's just a method of dividing, but this formula cleared my concept, so thanks!😊
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In the formula
Acs = (1 / 2) • θ • r^2
the θ denotes the number of radians (it does not have the unit "rad").
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God bless your heart fr! i was really struggling with the basics of circles because my professor has this long, extra, and difficult way of getting the arc lengths, or radians or degrees and you just made it so simple for me! i really appreciate it
Many people wonder why radians do not appear when we have radians*meters (rad • m).
Here is an attempt at an explanation:
Let s denote the length of an arc of a circle whose radius measures r.
If the arc subtends an angle measuring β = n°, we can pose a rule of three:
360° _______ 2 • 𝜋 • r
n° _______ s
Then
s = (n° / 360°) • 2 • 𝜋 • r
If β = 180° (which means that n = 180, the number of degrees), then
s = (180° / 360°) • 2 • 𝜋 • r
The units "degrees" cancel out and the result is
s = (1 / 2) • 2 • 𝜋 • r
s = 𝜋 • r
that is, half of the circumference 2 • 𝜋 • r.
If the arc subtends an angle measuring β = θ rad, we can pose a rule of three:
2 • 𝜋 rad _______ 2 • 𝜋 • r
θ rad _______ s
Then
s = (θ rad / 2 • 𝜋 rad) • 2 • 𝜋 • r
If β = 𝜋 rad (which means that θ = 𝜋, the number of radians), then
s = (𝜋 rad / 2 • 𝜋 rad) • 2 • 𝜋 • r
The units "radians" cancel out and the result is
s = (1 / 2) • 2 • 𝜋 • r
s = 𝜋 • r
that is, half of the circumference 2 • 𝜋 • r.
If we take the formula with the angles measured in radians, we can simplify
s = (θ rad / 2 • 𝜋 rad) • 2 • 𝜋 • r
s = θ • r
where θ denotes the "number of radians" (it does not have the unit "rad").
θ = β / (1 rad)
and θ is a dimensionless variable [rad/rad = 1].
However, many consider θ to denote the measure of the angle and for the example believe that
θ = 𝜋 rad
and radians*meter results in meters
rad • m = m
since, according to them, the radian is a dimensionless unit. This solves the problem of units for them and, as it has served them for a long time, they see no need to change it. But the truth is that the solution is simpler, what they have to take into account is the meaning of the variables that appear in the formulas, i.e. θ is just the number of radians without the unit rad.
Mathematics and Physics textbooks state that
s = θ • r
and then
θ = s / r
It seems that this formula led to the error of believing that
1 rad = 1 m/m = 1
and that the radian is a dimensionless derived unit as it appears in the International System of Units (SI), when in reality
θ = 1 m/m = 1
and knowing θ = 1, the angle measures β = 1 rad.
In the formula
s = θ • r
the variable θ is a dimensionless variable, it is a number without units, it is the number of radians.
When confusing what θ represents in the formula, some mistakes are made in Physics in the units of certain quantities, such as angular speed.
My guess is that actually the angular speed ω is not measured in rad/s but in
(rad/rad)/s = 1/s = s^(-1).
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@organic chemistry I think you got your formula wrong, you wrote (theta/360)*πr^2 but shouldn't it be (π/360)*theta*r^2 at 10:27
@@jelly_blue oh sorry I didn't see that, Thank you
thank you for these videos! they help us out so much
Hello, just a question, why do you use radian form always instead of degree form?
Idk how to explain it but the formula only works when the arc degree measure is in radians.
thank you math daddy
This is a step to break down the basics..... other times it can boggle the mind.
hello sir,
is there a video for area of irregular shape in a circle or area between two circles
3:49 why is this method way easier to understand then the one my teacher taught us.
Like I prefer this over measure of arc/360 • 2 (pi)r
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The conversion to degrees at 10:10 - how come you didn't multiply by 360/2π ? I'm confused because I can see how the twos then cancel to get the formula. Definitely being thick here
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The Organic Chemistry Teacher, how can I find the radius if I am only given the perimeter and theta?
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hi there, if I have the measurement from A to B and one more measurement from the center of a and b to the outer edge of circle "s"
Am I able to at least get the length of the arc? And then therefore will I be able to figure out how large my circle will be? I do not have a radius.
A+B= 10-10/16"
D+ "F"= 2"
Any help would be amazing. I'm building a large wheel for a project my grandson and I are making. A large hamster wheel, large enough for cat 😂 thank you in advance. Be well
Can you make a video finding area and arc length of polar coordinates , thanks
I wish you showed how to get the radius if given the area of a sector and theta.
Plz plz teach about calculating major segment and minor segment plz. I tried but I failed to calculate it
Hi There, to the 10:00mn position, why do we use 1/2 and what is the meaning of 1/2? thanks in advance
I understood it using the formula of the area of a triangle.
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Hello, I am a geometry math teacher who is now needing to create videos for my students while doing distance learning through remote teaching. I am wanting to create videos similar to this for them however I do not know what software to use. What do you use or recommend?
I don't think he would see a comment on a video this old, you might have more luck asking on a newer video.
Bleach Smoothie thank you:)
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Thank you
This sounds like Jonah Marais from Why Don’t We!!
In India we learn this in 10th grade [15yrs] and we do not call it trigonometry🤣😂
BUT TO BE HONEST GREAT WORK
ur not that guy pal I learned it at 9th grade
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in 10:27 why is the pi not equal to 180 degress ?????????
What's the formula to find R? Is it S/θ or Sθ?
r is the radius
r= the diameter divided by 2 @Demesio Frederick
Demesio Frederick Arc length over theta (S/0) to find R if theta is in radians
SAT TOMORROW THANK YOU
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Thx tomorrow i have the 0606 mock