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Sector Area Calculator

Table of contents

What is a sector of a circle? Sector definitionSector area formulaSpecial cases: area of semicircle, area of quadrantSector area calculator – when it may be useful?FAQs

With this sector area calculator, you'll quickly find any circle sector area, e.g., the area of a 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.

What is a sector of a circle? Sector definition

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:

Sector with acute central angle.Sector with right central angle - quadrantSector with obtuse central angleSector with straight central angle - semicircle.Sector with reflex central angle

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.

Sector area formula

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

Special cases: area of semicircle, area of quadrant

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

Sector with straight central angle - semicircle.
  • 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

Sector with right central angle - quadrant.
  • 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

Sector area calculator – when it may be useful?

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.

FAQs

What is the sector of a circle?

The sector of a circle is a slice of a circle, bound by two radiuses and an arc of the circumference. We identify sectors of a circle using their central angle. The central angle is the angle between the two radiuses. Sectors with a central angle equal to 90° are called quadrants.

How do I calculate the area of the sector of a circle?

To calculate the area of the sector of a circle, you can use two methods.

  • If you know the radius and central angle:

    1. Convert the central angle into radians:

      α [rad] = α [deg] · π/180°

    2. Multiply the radius squared by the angle in radians.

    3. Divide the result by 2.

  • If you know the area of the circle and central angle:

    1. Calculate the ratio between the full angle and the central angle.

    2. Multiply the result by the area of the circle.

What is the area of the 90° sector of a circle with r = 1?

The area of a sector with a central angle α = 90° of a circle with radius r = 1 is π/4. To calculate this result, you can use the following formula:

A = r² · α/2,

substituting:

  • r = 1; and
  • α = 90° · π/180° = π/2.

Thus:

A = (1² · π/2)/2 = π/4.

Notice that this is also a quarter of the area of the whole circle.

How do I find the central angle of a sector?

To find the central angle of a sector of a circle, you can invert the formula for its area:

A = r² · α/2,

where:

  • r — The radius; and
  • α — The central angle in radians.

The formula for α is then:

α = 2 · A/r²

To find the angle in degrees, multiply the result by 180°/π.

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