Find the area of a segment of a circle with a central angle of 120 degrees and a radius of 8 cm

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Forget about particular values and name the things to be measured.

Call the radius of the circle $r$ and the angle subtended by the sector $t$.

The area of the part of the circle containing the sector and internal triangle is $tr^2/2$.

The internal triangle has altitude $r\cos(t/2)$ and base $2r\sin(t/2)$, so its area is

$\begin{array}\\ (1/2)(2r\sin(t/2))(r\cos(t/2)) &=r^2(\sin(t/2)\cos(t/2))\\ &=r^2\sin(t)/2\\ \end{array} $

since $\sin(t) =2\sin(t/2)\cos(t/2) $.

The area of the sector is the difference of these expressions, which is $tr^2/2-r^2\sin(t)/2 =r^2(t-\sin(t))/2 $.

Now you can substitute the values of the radius and angle of the sector.

As a check, for $t = \pi$, this gives $\pi r^2/2$, which is area of the semicircle.

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.

Find the area of the sector with a central angle of 120� and a radius of 8 inches. Leave in terms of π. ------------ 120/360 = 1/3 It's 1/3 of the area of the circle.

The sector of a circle is like a slice of a pizza. First you need to find the area of the circle, which is πr^2 With a radius of 8 inches, the area of the circle will be π*8^2 or 64π. Next you want to find the area of the 120 degree section of that circle. 120 degrees is 1/3 of a circle, so all you have to do next is multiply the are of the circle by 1/3 (because 360/120= 1/3). (1/3)(64π) --> All you really have to do here is multiply 64 by 1/3 or divide it by 3 (because to divide by a number is the same a multiplying by it's reciprocal) so you get (64/3)(π).

64/3 is 21 and 1/3 or 21.333..., so however you choose to write it, I will leave the answer as (64/3)π squared inches.