If is the angle (in degrees) of a sector of a circle of radius r, then area of the sector is

Updated November 03, 2020

By Mariecor Agravante

Circles are everywhere in the real world, which is why their radii, diameters and circumference are significant in real life applications. But there are other parts of circles – sectors and angles, for instance – that also have importance in everyday applications as well. Examples include sector sizes of circular food like cakes and pies, the angle traveled in a Ferris wheel, the sizing of a tire to a particular vehicle and especially the sizing of a ring for an engagement or wedding. For these reasons and more, geometry also has equations and problem calculations dealing with central angles, arcs and sectors of a circle.

The central angle is defined as the angle created by two rays or radii radiating from the center of a circle, with the circle’s center being the vertex of the central angle. Central angles are particularly relevant when it comes to evenly dividing up pizza, or any other circular-based food, among a set number of people. Say there are five people at a soiree where a large pizza and a large cake are to be shared. What is the angle that both the pizza and the cake have to be divided at to ensure an equal slice for everyone? Since there are 360 degrees in a circle, the calculation becomes 360 degrees divided by 5 to arrive at 72 degrees, so that each slice, whether of the pizza or the cake, will have a central angle, or theta (θ), measuring 72 degrees.

An arc of the circle refers to a “portion” of the circle’s circumference. The arc length therefore is the length of that “portion.” If you imagine a pizza slice, the sector area can be visualized as the entire slice of pizza, but the arc length is the length of the outer edge of the crust for that particular slice. From the arc length, the central angle can be calculated. Indeed, one formula that can help in determining the central angle states that the arc length (s) is equal to the radius times the central angle, or

s = r × θ

where the angle, theta, must be measured in radians. So to solve for the central angle, theta, one need only divide the arc length by the radius, or

\frac{s}{r} = θ

To illustrate, if the arc length is 5.9 and the radius is 3.5329, then the central angle becomes 1.67 radians. Another example is if the arc length is 2 and the radius is 2, the central angle becomes 1 radian. If you want to convert radians to degrees, remember that 1 radian equals 180 degrees divided by π, or 57.2958 degrees. Conversely, if an equation asks to convert degrees back into radians, then first multiply by π, and then divide by 180 degrees.

Another useful formula to determine central angle is provided by the sector area, which again can be visualized as a slice of pizza. This particular formula can be seen in two ways. The first has the central angle measured in degrees so that the sector area equals π times the radius-squared and then multiplied by the quantity of the central angle in degrees divided by 360 degrees. In other words:

πr^2 × \frac{\text{central angle in degrees}}{360 \text{ degrees}} = \text{sector area}

If the central angle is measured in radians, the formula instead becomes:

\text{sector area} = r^2 × \frac{\text{central angle in radians}}{2}

Rearranging the formulas will help to solve for the value of the central angle, or theta. Consider a sector area of 52.3 square centimeters with a radius of 10 centimeters. What would its central angle be in degrees? The calculations would begin with a sector area of 52.3 square centimeters being equal to:

\frac{θ}{360 \text{ degrees}} × πr^2

Since the radius (​r​) equals 10, the entire equation can be written as:

\frac{52.3}{100π} × 360

so that theta can be written as:

\frac{52.3}{314} × 360

Thus the final answer becomes a central angle of 60 degrees.

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.

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:

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.

The formula for sector area is simple - multiply the central angle by the radius squared, and divide by 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.

💡 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

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!

• 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

• 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

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.