Arc Length (Formula)

Learn how to find the arc length in a circle using radian angle measures in this free math video tutorial by Mario's Math Tutoring.

0:26 Formula for Finding Arc Length Using Radius and Central Angle in Radians
0:51 Example 1 Find Arc Length Given Radius 10 cm and Central Angle Theta pi/4
1:41 Example 2 Find Central Angle Given Radius 12 cm and Arc Length 8 cm

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[HominemsAdvert]

Found this 1 minute video more helpful than the entire class I had on the lesson, great job!

[yourlocalcoward]

THANK YOU all the other videos were like 15 minutes long

[JoséAntonioBottino]

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

that is, half of the circumference 2 • 𝜋 • r
s = 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

that is, half of the circumference 2 • 𝜋 • r
s = 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

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 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).

[InfotainmentVidyarthi]

so in the final answer (5*pi)/2, what will pi represent pi = 3.14 or pi =180° . In simple terms what is the answer

[JoséAntonioBottino]

At minute 0:15 you say “theta has to be in radians”. That is not so. The variable θ is the number of radians and not the measure of the angle.

The reference to if “they give you the angle, the central angle, in degrees just convert it to radians then you can use this relationship s = r θ”, confirms the error.

I will make another comment where I show how the formula is obtained.

[egangray]

Why is white megamind out here teachin math on youtube?