What is a Sector?
A sector is much the same as a “pie” or a "pizza slice". It typically involves a province (sector) encompassed by 2 Radii and 1 Arc lying between the radii. To better understand what a sector of a circle is, here is a specimen from the real-life examples of food. A variety of foods that we eat resembles the sector of a circle, typically because they replicate circles.
Find the Sector in the Circle
A circle consists of an interior, its area. Divide the circle into pieces and you will get a sector. Similarly, cut the cake into Slices and you will get the sector of a circle.
Crack the Area of a Sector in Simple Few Steps
To ace over the calculation of area of sector, understanding the anatomy of the circle is the key.
What are the parts of a circle?
A Midpoint - It is the most crucial one as it is the connecting dot positioned right in the circle's centre.
Radius - A line that runs through the midpoint to the exterior of the circle
Diameter - A line extending across the circle and passes through its midpoint.
Take Notice of Diameter - it is actually two radii combined. Therefore, a circle's radius is invariably half the length of its diameter and its diameter is always double the length of its radius.
Types of Sectors
A semicircle (half circle) and a Quadrant (Quarter of a circle) are two main types of Sector:
Besides, There are primarily two "portions" of a circle: Of which, the "pizza" slice is known as the Sector. Whereas, a cut carried off from between two points on the circle by a "chord" is called a Segment. This means Chunks cut off by joining any two points on the circle are segments.
Seeing that, both sectors and segments are part of a circle's interior, both have areas. We can measure their area using formulas.
Easy Formulas to Calculate Area of a Sector of a Circle
Calculating the Area of Sector of a Circle Using Degrees
A whole of a circle surrounds 360°, thus the ratio of the sector's angle calculation to 360° is directly proportional to the fraction of the circle's area is computed. Hence, for the area of the sector of a circle, it's the median angle of 360°, times the area of the circle. The sector of a circle that subtends an angle of 180 degrees at the centre is called a semicircle. For instance, if the sector’s angle happens to measure 180°, and the two radiuses creating it are 10 inches, you would divide 180 by 360 (180/360) or 1/2 to obtain the portion of the circle.
In numeric terms, the area of a sector of a circle formula will be = \[\frac {\theta}{360} \times \pi r^{2}\]
Hence, the area of the semicircle will be = \[\frac {1}{2} \times \pi r^{2}\]
This is because the angle is 180 degrees.
Solved Example
Question: Given is a circle with a radius of 2 cm and ∠ A = 30°. Find the area of the sector of the circle below?
Solution: Area of circle = πr2 = π 22 = 4π
Total degrees in a circle = 360°
Given that the central angle is 30 degrees and the radius is 2cm,
Therefore, 30° slice = \[\frac {30}{360}\] fraction of circle.
= \[\frac {30}{360} \times \pi r^{2}\].
Area of sector = \[\frac {\theta}{360}\]* Total Area
= \[\frac {\theta}{360} \times \pi r^{2}\]
= \[\frac {1}{12} \times \frac {22}{7} \times 4\]
= 1.047 square cm
Calculating the Area of Sector of a Circle Using Radians
If the question is available with radians as a replacement for degrees to calculate the area of sector angle, the usual method of finding the sector's area remains the same. Radians are units that are utilized for computing angles. A circle on a whole circumscribes 2 π radians, so the portion of 2 π is the fraction of the circle's area that is being measured.
To put it simply, if the sector's angle is \[\frac {\pi}{2}\] radians, the portion of the circle to be measured will be \[\frac {\pi}{2}\] divided by 2π, or \[\frac {1}{4}\] of the circle.
So, to find the area, multiply the circle's area by the fraction of the circle that is being dealt with. Just make use of radians instead of degrees. The sector of a circle formula in radians is:
A = \[\frac{\text{angle in radian}}{\pi} \times \pi r^{2}\]
For example, if a sector contains an angle of \[\frac {\pi}{3}\] at the centre. Then the area of the sector would be \[\frac {\pi}{3}\]
= \[\frac{\frac{\pi}{3}}{\pi} \times \pi r^{2}\]
= \[\frac {1}{3} \times \pi r^{2}\]
Calculating the Area of Sector Using the Known Portions of a Circle
In cases where the portion of a circle is known, don't divide degrees or radians by any value.
For example, if the known sector is 1/4 of a circle, then just multiply the formula for the area of a circle by ¼, and you are good to go to find the area of the sector.
Thankfully, this sector of a circle formula would work with any stated fraction of a circle. Even if the known portion is 1/120 of a circle, simply multiply the formula for the area of a circle by 1/120.
So, when given the portion of the circle, the sector of a circle formula is:
A = (portion of the circle) (π x r2)
For example, if we have to find the area of the quadrant of a circle, then it is already known to us that the angle subtended by the quadrant at the centre of the circle is always 90 degrees and it divides the circle into four parts of equal area.
Hence, we can also say that the area of a single quadrant of a circle is ¼ of the area of the circle.
Area of Quadrant =\[\frac {90}{360} \times \pi r^{2}\]
= \[\frac {1}{4} \times \pi r^{2}\]
So if the radius of a circle is 10 cm then the area of its one of the quadrant would be :
= \[\frac {1}{4} \times \pi \times 100\]
= 78.57 square cm.
If the angle at the centre that is θ would be in radians, area of the sector of a circle = \[\frac {1}{2} \times r^{2} \theta\], where,
Here, θ will be the angle subtended at the centre, given in radians and r is the radius of the circle.
It should be noted that semicircles and quadrants are special types of sectors of a circle. They made an angle of 180° and 90° respectively with the centre.
Area of Sector Using Degrees
Now let us discuss the area of the sector of a circle whose angle is given to you in degrees with the help of the example given here.
Example: Determine the area of a sector if the radius of the circle is 8 units, and the angle subtended at the centre = \[\frac {\pi}{3}\].
Solution: Given, radius = 8 units; Angle measure (θ)= \[\frac {\pi}{3}\]
The area of the given sector can be calculated with the given formula,
\[A = (\frac {\text{sector angle}}{360}) \times (\pi \times r^{2}) \]
Hence, Area of sector would be = \[(\frac {\theta}{360}) \times \pi r^{2}\]
When we substitute the given values we get the Area of the sector
= \[\frac{\frac{\pi}{3}}{360} \times \frac {22}{7} \times 64\]
=12π
Hence, the area of the given sector in radians is expressed as 12π square units
FAQs on Area of Sector
1. How to Measure the Area of a Sector of a Circle?
A sector is simply a pie, portion, or wedge of a circle. A sector always has its origin at the midpoint of the circle. The area of a sector is thus a fraction of the area of the circle. This area is equivalent to the median angle.
Typically, the formula to calculate the area of a sector of a circle is A = pi *r2. But, to find a part of the circle’s area—that part can be identified using the sector's angle.
2. Mention a real-life problem related to the area of sectors.
The slice of pizza is the most common example. The shape of the slices of a circular pizza is in the shape of a sector. Let a pizza be sectioned into 6 equal slices, where each slice is a sector. Let the radius of the pizza be 14 inches. Now let us find the area of a single slice of pizza using the formula of area of the sector. Now, if a single pizza is divided into 6 pieces, then the angle subtended by a single sector would be 60°.
Area of Pizza slice = \[(\frac {\theta}{360^{0}}) \times \pi r^{2}\] = \[(\frac {60^{0}}{360^{0}}) \times (\frac{22}{7}) \times 14 \times 14 \] = \[\frac {2156}{3}\] = 718.67 square units. Hence the area of a slice of pizza is 718.67 square inches.
Let Ram have eaten one slice of pizza. Then the area of the rest of the slices will be 5 × 718.67= 3593.35 square inches.
So, with the help of sectors, we can solve our daily life problems.
3. A circle of radius 6 units is divided into 3 sectors and the central angles made by the radius are 130°, 130°, and 100°, respectively. Find the area of all three sectors.
The angle made by the first sector is θ = 130°.
Therefore, the area of the first sector = \[(\frac {\theta}{360^{0}}) \times \pi r^{2} = (\frac{130^{0}}{360^{0}}) \times (\frac {22}{7}) \times 6^{2} \] = \[\frac {13}{36} \times \frac {22}{7} \times 36 = \frac {352}{7}\] = 40.82 square units.
The angle made by the second sector is θ = 130°.Therefore, the area of the second sector = \[(\frac {\theta}{360^{0}}) \times \pi r^{2} = (\frac{130^{0}}{360^{0}}) \times (\frac {22}{7}) \times 6^{2} \] = \[\frac {13}{36} \times \frac {22}{7} \times 36 = \frac {352}{7}\] = 40.82 square units.
Therefore, the area of the third sector = \[(\frac {\theta}{360^{0}}) \times \pi r^{2} = (\frac{100^{0}}{360^{0}}) \times (\frac {22}{7}) \times 6^{2} = \frac {22}{7} \times 36\] = 31.4 square units.
4. How to calculate the area of a quadrant of a circle using the sector formula?
To find the area of the quadrant of a circle, we know that the angle subtended by the quadrant at the center of the circle is always 90 degrees. A quadrant divides the circle into four parts of equal area.
Hence, we can also say that the area of a single quadrant of a circle is ¼ of the area of the circle.
Area of Quadrant = \[\frac {90}{360} \times \pi r^{2}\]
= \[\frac {1}{4} \times \pi r^{2}\]
So, if the radius of a circle is 10 cm, then the area of its one of the quadrant would be:
= \[\frac {1}{4} \times \pi \times 100\]
= 78.57 square cm
5. What is the difference between sector and segment in any circle?
The sector of a circle can be called the arc of the circle which subtends an angle at the centre of the circle.
The segment is the part inside a circle divided by a chord. A chord is any of the regular lines on the circle which meet two points on the circle. A chord divides a circle into two parts. These parts are called the segment in the circle. The bigger area of the circle divided by a chord is called the major segment, while the smaller one is called the minor segment. A sector always subtends an angle at the centre of the circle while a segment has not any relation with the centre.