Arc Length and Area of a Sector | Formulas | Sample Problems | Trigonometry | Pre-Calculus

Pre-Calculus
Arc Length and Area of a Sector | Formulas | Sample Problems | Trigonometry

Arc Length
In a circle of radius r, the length s of an arc intercepted by a central
angle with measure θ radians is given by s = rθ.

Area of a Sector
In a circle of radius r, the area A of a sector with a central angle
measuring θ radians is A=1/2 r^2 θ.

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[JoséAntonioBottino]

Many people wonder why radians do not appear when we have radians*meters.
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).

[cayabyabjanelleannev.]

Thank you po Prof D, learn po so much!

[AlineaPeñero]

salamat poo!! naiintindihan ko na po sya

[rasmineleid.yambao]

Very useful and helpful po, thankyouu Sir!❤️

[ivanaalawi]

It’s Maths time nmn always supporting ur channel po

[arandajohnkena.]

Good day to you, Professor Jeffrey Del Mundo. Thank you for the video lesson.
Watching from STEM - 1107

[domalantamarthamaeb.stem]

Thank you for this video lesson, Sir.

[JoséAntonioBottino]

In the formula
Acs = (1 / 2) • θ • r^2

the θ denotes the number of radians (it does not have the unit "rad").

[andreadullavin]

Done watching! Thank you for making this video lesson, Prof D. You made math fun and easy to learn. ♡♡
Student from Gas 1111 - Dullavin, Andrea N.

[asobu]

Thank you very much po Sir.It really help me a lot.

[_khionne]

THANK YOU SO MUCH, SIR!!
DONE WATCHING FROM STEM 1107.

[carmizamorro]

This really helps me ngayun sir thankyou so much! Permission to use ur example problems sir for academic purposes

[zamorairenee.]

Done watching from STEM 1107. Thank you Prof D!.

[YullieQ]

Hello sir..dito na nman ang estudyante mong may sakit sa limot pagdating sa numbers...team challenger

[alfonsojr.tvvlog]

Another interesting content, waiting for this from TEAM CHALLENGER, , , godbless