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Circles, Sectors, Segments
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The following table gives the formulas for the area of sector and area of segment for angles in degrees or radians. Scroll down the page for more explanations, examples and worksheets for the area of sectors and segments.
A sector is like a “pizza slice” of the circle. It consists of a region bounded by two radii and an arc lying between the radii.
The area of a sector is a fraction of the area of the circle. This area is proportional to the central angle. In other words, the bigger the central angle, the larger is the area of the sector.
The following diagrams give the formulas for the area of circle and the area of sector. Scroll down the page for more examples and solutions.
We will now look at the formula for the area of a sector where the central angle is measured in degrees.
Recall that the angle of a full circle is 360˚ and that the formula for the area of a circle is πr2.
Comparing the area of sector and area of circle, we derive the formula for the area of sector when the central angle is given in degrees.
where r is the radius of the circle.
This formula allows us to calculate any one of the values given the other two values.
Worksheet to calculate arc length and area of a sector (degrees)
Calculate The Area Of A Sector (Using Formula In Degrees)
We can calculate the area of the sector, given the central angle and radius of circle.
Example:
Given that the radius of the circle is 5 cm, calculate the area of the shaded sector.
(Take π = 3.142).
Solution:
Area of sector = 60°/360° × 25π
= 13.09 cm2
Calculate Central Angle Of A Sector
We can calculate the central angle subtended by a sector, given the area of the sector and area of circle.
Example:
The area of a sector with a radius of 6 cm is 35.4 cm2. Calculate the angle of the
sector. (Take π = 3.142).
Solution:
How To Derive The Formula To Calculate The Area Of A Sector In A Circle?
It explains how to find the area of a sector of a circle. The formula for the area of a circle is given and the formula for the area of a sector of a circle is derived.
Example:
Janice needs to find the area of the red section of the circular table top in order to buy the
right amount of paint. What is the area of the red section of the circular table top?
Solution:
Step 1: Find the area of the entire circle using the area formula A = πr2.
Step 2: Find the fraction of the circle by putting the angle measurement of the sector over 360°, the total number of degrees in a circle.
Step 3: Multiply the fraction by the area of the circle. Leave your answer in terms of π.
- Show Video Lesson
How To Calculate The Area Of A Sector Using The Formula In Degrees And The Missing Radius Given The Sector Area And The Size Of The Central Angle?
Example 1: Find the area of the shaded region.
Example 2: Find the radius of the circle if the area of the shaded region is 50π
- Show Video Lesson
Formula For Area Of Sector (In Radians)
Next, we will look at the formula for the area of a sector where the central angle is measured in radians. Recall that the angle of a full circle in radians is 2π.
Comparing the area of sector and area of circle, we get the formula for the area of sector when the central angle is given in radians.
where r is the radius of the circle.
This formula allows us to calculate any one of the values given the other two values.
Worksheet to calculate arc length and area of sector (radians)
The following video shows how we can calculate the area of a sector using the formula in radians.
Example:
A lawn sprinkler located at the corner of a yard rotates through 90° and sprays water 30ft.
What is the area of the sector watered?
- Show Video Lesson
How To Determine The Area Of A Sector?
The formula is given in radians.
How to determine the area of a segment? (the area bounded by a chord and an arc).
Example 1: Find the area of the sector of a circle with radius 8 feet formed by a central angle of 110°
Example 2: Find the area of the shaded region in the circle with radius 12cm and a central angle of 80°.
- Show Video Lesson
Area Of Segment (Angle In Degrees)
The segment of a circle is a region bounded by the arc of the circle and a chord.
The area of segment in a circle is equal to the area of sector minus the area of the triangle.
How To Derive The Area Of A Segment Formula?
How do you find the area of a segment of a circle?
- Show Video Lesson
How To Calculate The Area Of Segments Of Circles?
It uses half the product of the base and the height to calculate the area of the triangle.
- Show Video Lesson
How To Calculate The Area Of Sector And The Area Of Segment?
It uses the sine rule to calculate the area of triangle.
- Show Video Lesson
Area Of Segment (Angle In Radians)
Finding the area of a segment (angle given in radians)
- Show Video Lesson
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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!):
- 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
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.