# Sector Area Calculator

By Hanna Pamuła, PhD candidate
Last updated: Feb 16, 2021

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.

## 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

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.

## 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 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.

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

`Semicircle area = α * r² / 2 = πr² / 2`

1. Quadrant area: `πr² / 4` • 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`

## 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 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.

Hanna Pamuła, PhD candidate
Central angle
deg
in
Diameter
in Sector area (A)
in²
Arc length (L)
in
Chord length (c)
in
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