Sector Area and Arc Length

In summary, the formula for the area of a sector can be expressed in either radians or degrees, but the important thing is to use the correct formula based on the given angle.
  • #1
frozonecom
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I'm just wondering because I'm really confused right now.

My teacher gave us the formula:

[itex]K= \frac{1}{2}sr[/itex]

for area of a given sector where "s" is the arc length and "r" is the given radius.

the formula for the arc length is:

[itex]s=\Theta r[/itex]

Though, I can't seem to understand how he came up with the formula for the area of the sector, because searching the internet always came with the result that the formula for area of the sector is:

[itex] A= \frac{\Theta}{360} \pi r^2[/itex]

I hope someone can help me. :)
 
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  • #2
Your first formula is measured in "radians". A full revolution is 2*pi radians.

Your last formula is measured in degrees. A full revolution is 360 degrees.

Multiply your last formula by 1 in a clever way: 1 = (360 degrees)/(2*pi)
 
  • #3
Oh! So both formulas are actually the same.

The difference is just that I'll use the "right formula" based on the given theta!

Thanks! :)
 
  • #4
Here's an easy way to think of it. You know that the area of a circle is pi r^2, right? So, think of the circle as a "sector" of angle 360 degrees, or 2pi radians. You need to multiply the angle by whatever factor will give you the result pi r^2. And then that factor is the same for any other angle measured in the same units.
 
  • #5


Hello,

I understand your confusion and I am happy to provide some clarification. The formula your teacher gave you, K= \frac{1}{2}sr, is actually a simplified version of the more commonly used formula for the area of a sector, A= \frac{\Theta}{360} \pi r^2. Let me explain how this formula is derived.

A sector is a portion of a circle that is bounded by two radii and an arc. The area of this sector is essentially a fraction of the total area of the circle. This fraction is determined by the central angle of the sector, denoted by \Theta. The central angle is measured in degrees, with a full circle being 360 degrees. So if the central angle of the sector is 90 degrees, the sector will be one-fourth of the total area of the circle.

Now, to find the area of the sector, we need to find the fraction of the total area that the sector occupies. This is where the formula A= \frac{\Theta}{360} \pi r^2 comes in. This formula takes into account the central angle \Theta and the radius r, which are both necessary for determining the area of the sector. The \frac{\Theta}{360} part of the formula represents the fraction of the total area that the sector occupies. And multiplying it by \pi r^2 gives us the actual area of the sector.

Now, let's look at the formula your teacher gave you, K= \frac{1}{2}sr. This formula is essentially the same as A= \frac{\Theta}{360} \pi r^2, but it has been simplified by combining the \frac{1}{2} and the s. This is because the arc length s is equal to \frac{\Theta}{360} times the circumference of the circle, which is represented by 2\pi r. So when we combine the \frac{1}{2} and the s, we get \frac{\Theta}{360} \times 2\pi r, which simplifies to \frac{\Theta}{180} \pi r. This is the same as \frac{\Theta}{360} \pi r^2, just written in a different way.

I hope this explanation helps clear up your confusion. Remember, both formulas are correct and can be used interchangeably depending on what information you have available. If you have any further questions,
 

What is sector area?

Sector area is the measurement of the surface enclosed by an arc and two radii in a circle.

How is sector area calculated?

To calculate sector area, you can use the formula A = (θ/360) x πr², where θ is the central angle in degrees and r is the radius of the circle.

What is arc length?

Arc length is the distance along the circumference of an arc, from one endpoint to the other.

How is arc length calculated?

To calculate arc length, you can use the formula L = (θ/360) x 2πr, where θ is the central angle in degrees and r is the radius of the circle.

How are sector area and arc length related?

Sector area and arc length are both measurements of sections of a circle. They are related through the central angle, as both formulas for calculating them involve the central angle in degrees.

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