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?FAQsWith 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:

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!):

The area of a circle is calculated as

`A = πr²`

. This is a great starting point.The full angle is 2π in radians, or 360° in degrees, the latter of which is the more common angle unit.

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`

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**

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`

## 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.

### 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:

Convert the central angle into

**radians**:`α [rad] = α [deg] · π/180°`

**Multiply**the radius squared by the angle in radians.**Divide**the result by 2.

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

Calculate the

**ratio between the full angle and the central angle**.**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`

.