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 → πr²So, 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 / 2Of 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 / 4Quadrant'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; andα = 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; andα— 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.
