What will be the length of the arc if the angle of sector is 60 degree and radius is 3.5 cm?

With this sector area calculator, you'll quickly find any circle sector area, e.g., the area of semicircle or quadrant. In this short article we'll:

  • provide a sector definition and explain what a sector of a circle is.
  • show the sector area formula and explain how to derive the equation yourself without much effort.
  • reveal some real-life examples where the sector area calculator may come in handy.

So let's start with the sector definition - what is a sector in geometry?

A sector is a geometric figure bounded by two radii and the included arc of a circle

Sectors of a circle are most commonly visualized in pie charts, where a circle is divided into several sectors to show the weightage of each segment. The pictures below show a few examples of circle sectors - it doesn't necessarily mean that they will look like a pie slice, sometimes it looks like the rest of the pie after you've taken a slice:

You may, very rarely, hear about the sector of an ellipse, but the formulas are way, way more difficult to use than the circle sector area equations.

The formula for sector area is simple - multiply the central angle by the radius squared, and divide by 2:

But where does it come from? You can find it by using proportions, all you need to remember is circle area formula (and we bet you do!):

  1. The area of a circle is calculated as A = πr². This is a great starting point.
  2. The full angle is 2π in radians, or 360° in degrees, the latter of which is the more common angle unit.
  3. Then, we want to calculate the area of a part of a circle, expressed by the central angle.
  • For angles of 2π (full circle), the area is equal to πr²: 2π → πr²
  • So, what's the area for the sector of a circle: α → Sector Area
  1. From the proportion we can easily find the final sector area formula:

Sector Area = α × πr² / 2π = α × r² / 2

The same method may be used to find arc length - all you need to remember is the formula for a circle's circumference.

💡 Note that α should be in radians when using the given formula. If you know your sector's central angle in degrees, multiply it first by π/180° to find its equivalent value in radians. Or you can use this formula instead, where θ is the central angle in degrees:

Sector Area = r² × θ × π / 360

Finding the area of a semicircle or quadrant should be a piece of cake now, just think about what part of a circle they are!

  • Knowing that it's half of the circle, divide the area by 2:

    Semicircle area = Circle area / 2 = πr² / 2

  • Of course, you'll get the same result when using sector area formula. Just remember that straight angle is π (180°):

    Semicircle area = α × r² / 2 = πr² / 2

  • As quadrant is a quarter of a circle, we can write the formula as:

    Quadrant area = Circle area / 4 = πr² / 4

  • Quadrant's central angle is a right angle (π/2 or 90°), so you'll quickly come to the same equation:

    Quadrant area = α × r² / 2 = πr² / 4

We know, we know: "why do we need to learn that, we're never ever gonna use it". Well, we'd like to show you that geometry is all around us:

  • If you're wondering how big cake you should order for your awesome birthday party - bingo, that's it! Use sector area formula to estimate the size of a slice 🍰 for your guests so that nobody will starve to death. Check out how we've implemented it in our cake serving calculator.
  • It's a similar story with pizza - have you noticed that every slice is a sector of a circle 🍕? For example, if you're not a big fan of the crust, you can calculate which pizza size will give you the best deal (don't forget about the tip afterwards).
  • Any sewing enthusiasts here?👗 Sector area calculations may be useful in preparing a circle skirt (as it's not always a full circle but, you know, a sector of a circle instead).

Apart those simple, real-life examples, the sector area formula may be handy in geometry, e.g. for finding surface area of a cone.

What will be the length of the arc if the angle of sector is 60 degree and radius is 3.5 cm?

Go back to Calculators page

To use the arc length calculator, simply enter the central angle and the radius into the top two boxes. If we are only given the diameter and not the radius we can enter that instead, though the radius is always half the diameter so it’s not too difficult to calculate. 

The calculator will then determine the length of the arc. It will also calculate the area of the sector with that same central angle.

How to Calculate the Area of a Sector and the Length of an Arc

Our calculators are very handy, but we can find the arc length and the sector area manually. It’s good practice to make sure you know how to calculate these measurements on your own.

What will be the length of the arc if the angle of sector is 60 degree and radius is 3.5 cm?

How to Find the Arc Length

An arc length is just a fraction of the circumference of the entire circle. So we need to find the fraction of the circle made by the central angle we know, then find the circumference of the total circle made by the radius we know. Then we just multiply them together. Let’s try an example where our central angle is 72° and our radius is 3 meters. 

First, let’s find the fraction of the circle’s circumference our arc length is. The whole circle is 360°. Let’s say our part is 72°. We make a fraction by placing the part over the whole and we get \(\frac{72}{360}\), which reduces to \(\frac{1}{5}\). So, our arc length will be one fifth of the total circumference. Now we just need to find that circumference.

The circumference can be found by the formula C = πd when we know the diameter and C = 2πr when we know the radius, as we do here. Plugging our radius of 3 into the formula, we get C = 6π meters or approximately 18.8495559 m.

Now we multiply that by \(\frac{1}{5}\) (or its decimal equivalent 0.2) to find our arc length, which is 3.769911 meters. Note that our units will always be a length.

How to Find the Sector Area

Just as every arc length is a fraction of the circumference of the whole circle, the sector area is simply a fraction of the area of the circle. So to find the sector area, we need to find the fraction of the circle made by the central angle we know, then find the area of the total circle made by the radius we know. Then we just multiply them together. Let’s try an example where our central angle is 72° and our radius is 3 meters.

First, let’s find the fraction of the circle’s area our sector takes up. The whole circle is 360°. Our part is 72°. We make a fraction by placing the part over the whole and we get \(\frac{72}{360}\), which reduces to \(\frac{1}{5}\). So, our sector area will be one fifth of the total area of the circle. Now we just need to find that area.

The area can be found by the formula A = πr2. Plugging our radius of 3 into the formula we get A = 9π meters squared or approximately 28.27433388 m2

Now we multiply that by \(\frac{1}{5}\) (or its decimal equivalent 0.2) to find our sector area, which is 5.654867 meters squared.  Note that our answer will always be an area so the units will always be squared.