Area of a Sector How to Find (Formula Radians)

Learn how to find the Area of a Sector using radian angle measures in this free math video tutorial by Mario's Math Tutoring. We discuss what a sector is as well as the formula for finding the area of a sector with a central angle in radians.

0:06 What is a Sector
0:38 Formula for Area of a Sector
1:10 Example 1 Find Area of Sector with Radius 4 in and Central Angle of pi/6.
2:23 Example 2 Find Area of Sector with Radius 10 cm and Central Angle 150 Degrees.
2:53 Convert from Degrees to Radians

Related Videos:

Converting Degrees to Radians

Finding Area of a Sector Using Proportions

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[explosionyonko]

How did 8 turned into 4 when you where simplifying it?

[ireto]

What if, you have different length of r?
What is the area then?

[chriswilliams]

To me the degrees formula makes more sense.

The fraction of degrees divided by the whole (theta/360) times the area of a circle pi r^2.
As a formula area in degrees A=(theta/360)*(pir^2)
Can someone explain the area rationale of area of a section of a circle in radians? Why is it half of the radians and half the area of a section of a circle?

[JoséAntonioBottino]

At minute 0:23 you say that “theta has to be in radians”. That is not so. The correct thing is that, in that formula, θ is the number of radians.

@chriswilliams5291:

Next I am going to show how the formula is obtained:

Let Acs denote the area of a circular sector whose radius measures r.

If the arc subtends an angle measuring β = n°, we can pose a rule of three:
360° 𝜋 • r^2
n° Acs

Then
Acs = (n° / 360°) • 𝜋 • r^2

If β = 180° (which means that n = 180, the number of degrees), then
Acs = (180° / 360°) • 𝜋 • r^2

The units “degrees” cancel out and then
Acs = (1 / 2) • 𝜋 • r^2

i.e. half the area of the circle Ac = 𝜋 • r^2

If the arc subtends an angle measuring β = θ rad, we can pose a rule of three:
2 • 𝜋 rad 𝜋 • r^2
θ rad Acs

Then
Acs = (θ rad / 2 • 𝜋 rad) • 𝜋 • r^2

If β = 𝜋 rad (meaning that θ = 𝜋, the number of radians), then
Acs = (𝜋 rad / 2 • 𝜋 rad) • 𝜋 • r^2

The units “radians” cancel out and then
Acs = (1 / 2) • 𝜋 • r^2

or half the area of the circle Ac = 𝜋 • r^2

If we take the formula with the angles measured in radians, we can simplify
Acs = (θ rad / 2 • 𝜋 rad) • 𝜋 • r^2
Acs = (1 / 2) • θ • r^2

where θ is the number of radians (does not have the unit symbol “rad”)
θ = β / (1 rad)

and θ is a dimensionless variable [rad/rad = 1].

However, many consider θ to be the measure of the angle and for the example they believe that
θ = 𝜋 rad

and radian*square meter results in square meter
rad • m^2 = m^2

since, according to them, the radian is a dimensionless unit, as stated in the International System of Units (SI) brochure.
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.

[potogold]

What if the unit of the radius is different than inches? i.e. miles, cm, etc..