Sector Area Calculator

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, but 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.

🙋 Before you continue reading, you should already be familiarized with circles and ellipses. Read our circle calc: find c, d, a, r and our ellipse calculator to ensure you have a firm grasp of these subjects.

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

• Sector Area = r² × α / 2

But where does it come from? You can find it by using proportions. All you need to remember is the 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

4. 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. Read more about this in our circumference calculator and arc length calculator.

💡 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!

1. Semicircle area: πr² / 2

• 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 the sector area formula. Just remember that the straight angle is π (180°):

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

2. Quadrant area: πr² / 4

• As a 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 the sector area formula to estimate the size of a slice 🍰 for your guests so that nobody will starve to death.

• 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.

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