Sector Area Calculator
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
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:
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!):
- 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
💡 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!
- 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
- 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).