how to calculate area of sector of a circle

Area of Sphere

The area of a sector of a circle is the amount of space enwrapped within the limit of the sector. A sphere e'er originates from the center of the circle. The sector of a circle is defined as the share of a band that is swallowed betwixt its two radii and the arc contiguous them. The articulated lorr-circle is the nearly common sphere of a circulate, which represents half a circle. Let us learn more about the area of sector, its convention, and how to work out the area of a sphere using radians and degrees.

1.	What is Arena of Sector of a Circle?
2.	Field of Sphere Formula
3.	Area of Sector in Radians
4.	FAQs along Area of Sector

What is Area of Sector of a Circle?

The space penned by the sphere of a circle is named the area of the sector. For instance, a pizza slice is an exemplar of a sector that represents a divide of a pizza. There are 2 types of sectors: insignificant and major sectors. A minor sector is a sphere that is inferior than a rig-circle, whereas, a major sector is a sector greater than a semi-circle.

The figure inclined below represents the sectors in a circle. The shaded region shows the area of the sector OAPB. Present, ∠AOB is the angle of the sector. It should personify noted that the unshadowed region is also a sector of the circle. So, the crosshatched region is the orbit of the small-scale sphere and the unshaded region is the area of the major sphere.

Now, let us learn about the area of a sector formula and its derivation.

Area of Sector Formula

In social club to find the unconditional space engulfed away the sector, we exercise the country of a sector rul. The area of a sphere can be measured using the following formulas,

• Area of a Sector of Circle = (θ/360º) × πr², where, θ is the sphere lean against subtended by the arc at the center, in degrees, and 'r' is the wheel spoke of the circle.
• Area of a Sphere of Circle = 1/2 × r²θ, where, θ is the sector angle subtended by the arc at the center, in radians, and 'r' is the radius of the circle.

Area of Sector Formula Derivation

Allow us apply the state method to derive the expression for the arena of the sector of a rotary. We know that a complete circle measures 360º. The area of a circle with an angle measuring 360º at the center is apt away πr², where 'r' is the radius of the circle.

If the angle at the center of the circle is 1º, the area of the sector is πr²/360º. So, if the angle at the eye is θ, the area of the sector is, Area of a Sector of Circle = (θ/360º) × πr², where,

• θ is the lean subtended at the center, given in degrees.
• r is the r of the traffic circle.

In other words, πr² represents the field of a full circle and θ/360º tells us how much of the circle is covered by the sector.

If the angle at the center is θ in radians, area of the sector of a encircle = (1/2) × r²θ, where,

• θ is the angle subtended at the center, given in radians.
• r is the radius of the circle.

It should beryllium noted that semitrailer-circles and quadrants are special types of sectors of a circle with angles of 180° and 90° severally.

Area of Sphere Using Degrees

Let us use these formulas and learn how to compute the area of the sphere of a circle when the subtended angle is given in degrees with the help of an example.

Example: A circle is bifid into 3 sectors and the focal angles made by the spoke are 160°, 100°, and 100° respectively. Find the domain of all the three sectors.

Solution:

The angle made away the showtime sector is θ = 160°. Therefore, the area of the ordinal sector = (θ/360°) × πr² = (160°/360°) × (22/7) × 6² = 4/9 × 22/7 × 36 = 352/7 = 50.28 square units.

The fish ready-made past the second sector is θ = 100°. Therefore, the area of the 2nd sphere is = (θ/360°) × πr² = (100°/360°) × (22/7) × 6² = 5/18 × 22/7 × 36 = 220/7 = 31.43 square units.

The angle made by the third sphere is the cookie-cutter as that of the second sector (θ = 100°). Thus, the area of the second sector is equal to the area of the third sector. Therefore, the area of the ordinal sector = 31.43 square units.

Area of Sector in Radians

If we motive to find the region of a sector when the angle is given in radians, we utilisation the formula, Area of sector = (1/2) × r²θ; where θ is the angle subtended at the center, granted in radians, and 'r' is the radius of the circle. So, let us understand where the formula comes from. We cognise that the normal for the area of a sector (in degrees) = (θ/360º) × πr² because it is a divide of a circle. The same concept is applied to the normal when we want to express IT in radians, but we just need to replace 360° with 2π because 2π (in radians) = 360°. This way, Area of sector in radians = (θ/2π) × πr². On further simplifying the normal, we get, area of sphere = (θ/2) × r² or (1/2) × r²θ. Let us understand how to find the area of a sector in radians with an example.

Example: Witness the sphere of a sphere if the r of the circle is 6 units, and the angle subtended at the center = 2π/3

Solution: Given, radius = 6 units; Slant measure out (θ)= 2π/3

The area of the given sector force out glucinium calculated with the normal, Area of sphere (in radians) = (θ/2) × r². On substituting the values in the formula, we get Expanse of sector (in radians) = [2π/(3×2)] × 6² = (π/3) × 36 = 12π.

Thus, the area of the given sphere in radians is expressed as 12π square units.

Real-Life history Example of Arena of Sector of Circle

One of the to the highest degree common real-life examples of the field of a sector is the piece of a pizza. The shape of the slices of a disclike pizza pie is like a sphere. Respect the figure given below that shows a pizza that is sectioned into 6 equal slices, where each slice is a sector, and the radius of the pizza is 7 inches. Now, let USA find the arena of the sector harp-shaped by each slice by victimisation the area of a sector formula. It should be noted that since the pizza pie is divided into 6 equal slices, the angle of sphere is 60°. Area of Pizza slice = (θ/360°) × πr² = (60°/360°) × (22/7) × 7² = 1/6 × 22 × 7 = 77/3 = 25.67 square units.

Tips on Field of Sector

Here is a list of a few meaningful points that are assistive in solving the area of sector problems.

• The area of a sphere of a R-2 is the fractional area of the circle.
• The area of a sphere of a circle with radius 'r' is calculated with the formula, Area of a sector = (θ/360º) × π r²
• The arc length of the sector of wheel spoke r fire be calculated with the formula, Arc Length of a Sector = r × θ

☛ Related Articles

• Arena of a R-2
• Arcs and Subtended Angles
• Segment of a Circle
• What is private eye?

Orbit of a Sector Examples

1. Example 1: If the fish of a sector of a rope is 60°, and the radius of the round is 7 inches, what is the surface area of the sector of this circle?

   Answer:

   The radius of the circle is 7 inches and the angle is 60°. Thus, rent us use the area of sphere normal. The country of sector = (θ/360°) × π r² = (60°/360°) × (22/7) × 7² = 77/3 = 25.67 square units. Therefore, the area of the minor sector is 25.67 aboveboard units.

2. Example 2: An umbrella has every bit spaced 8 ribs. If viewed arsenic a flatcar environ of radius 7 units, what would represent the orbit betwixt two consecutive ribs of the comprehensive? (Hint: The area 'tween two consecutive ribs would form a sphere of a circle)

   Solvent:

   The radius of the flat comprehensive = 7 units. There are 8 ribs in the umbrella. Since a complete Angle of a roach = 360°, the angle of each sector of the umbrella = 360/8 = 45° because the circulate is partitioned into 8 isothermal sectors. Thus, the area of sector = (θ/360°) × π r² = (45°/360°) × 22/7 × 7² = 77/4 = 19.25 square units. Therefore, the area between two consecutive ribs of the umbrella is 19.25 square units.

3. Example 3: A circle with a diameter of 2 units is mullioned into 10 equal sectors. Can you find the area of each sector of the circle?

   Solution:

   The diameter of the circle is 2 units, therefore, the radius of the circle is 1 unit. Since a complete angle of a circle = 360°, the angle of each sphere of the circle is 360/10 = 36° because the full-scale angle is divided into 10 equal parts. Surface area of Sector = (θ/360°) × πr² = 36°/360° × 22/7 × 1 = 11/35 = 0.314 square units. Therefore, the area of each sector of the circle is 0.314 square units.​

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Practice Questions on Area of a Sector

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FAQs on Area of Sector of Circle

What is the Orbit of a Sector of a Encircle?

The space enclosed by the sector of a roach is called the area of the sector of a circle. The split up of the circle that is enclosed by deuce radii and the corresponding arc is titled the sector of the circle.

What is the Formula for Area of Sector of Circle?

The two main formulas that are used to find the area of a sector are:

• Area of a Sphere of Circle = (θ/360º) × πr², where, θ is the Angle subtended at the substance, given in degrees, and 'r' is the r of the circle.
• Area of a Sphere of Surround = 1/2 × r²θ, where, θ is the angle subtended at the gist, precondition in radians, and 'r' is the spoke of the circle.

How to Calculate the Area of a Sector using Degrees?

When the angle subtended at the center is given in degrees, the area of a sector can Be premeditated using the following formula, area of a sector of circle = (θ/360º) × πr², where, θ is the angle subtended at the center, given in degrees, and r is the radius of the circle.

What do you Ignoble aside Sphere of a Circle?

A sphere is defined as the portion of a roach that is enclosed between its two radii and the arc adjoining them. The hemicycle is the well-nig popular sector of a circle, which represents half of a circle.

What do you Mean aside the Arc of a Circle?

A part of a curve or a part of a perimeter of a circle is called the arc. Many an objects have a curve ball in their shape. The curved portion of these objects is mathematically referred to as an arc.

How is the Area of Sector of Circle Formula Derivative?

The area of the sphere shows the sphere of a part of the circuit's area. We know that the area of a circle is calculated with the formula, πr². The formula for the area of a sector of a circle is derived in the following way:

• Apply the unitary method to derive the rul of the area of a sector of dress circle.
• We know, a complete dress circle measures 360º. The area of a lap with an angle measure 360º at the center is given by πr², where r is the radius of the circle.
• If the angle at the center of the circle is 1º, the country of the sphere is πr²/360º. Sol, if the angle at the meat is θ, the area of the sector is, Domain of a Sphere of a Circle = (θ/360º) × πr², where, θ is the angle subtended at the center, given in degrees, and r is the radius of the circle.
• In separate words, πr² represents the area of a full circle and θ/360º tells U.S.A how much of the encircle is covered by the sector.

How to Retrieve the Area of Sphere with Arc Length and Radius?

The area of a sector can be calculated if the arc distance and radius is tending. We number one calculate the angle (θ) subtended by the arc with the formula, Duration of Arch = (θ/360) × 2πr. Now, we already know the radius, and erst the tilt is known, the area of the sphere can be calculated with the formula, Area of a Sector of a Circuit = (θ/360º) × πr²

How to retrieve the Radius from Area of Sector?

If the area of a sector is known, and the lean on (θ) subtended by the arc is known, the r can be calculated by subbing the acknowledged values in the formula, Expanse of a Sphere of a Rophy = (θ/360º) × πr². For example, let us find the radius if the country of a sector is 36π, and the sector angle is conferred as 90°. We will substitute the given values in the formula, Expanse of a Sphere of a Circle = (θ/360º) × πr², that is, 36π = (90/360) × πr². So, the value of r² = 144, which means r = 12 units.

How to Incu the Area of Sector in Terms of Pi?

The area of sector butt also be denotative in terms of pi (π). For example, if the radius of a circle is given as 4 units, and the angle subtended by the arc for the sector is 90°, LET us find the field of the sector in price of pi. Area of sphere = (θ/360º) × πr². Substituting the values in the chemical formula, Surface area of sector = (90/360) × π × 4². After solving this, we get, the orbit as 4π.

How to Find the Area of a Sector in Radians?

In fiat to find the region of a sector with the central angle in radians, we use the formula, Area of sector = (θ/2) × r²; where θ is the angle subtended at the center, given in radians, and 'r' is the radius of the circle. For example, if the r of the circle is 12 units, and the sphere angle subtended by the spark at the center = 4π/3, let U.S.A find the area of the sector. Area of sphere (in radians) = (θ/2) × r². On subbing the values in the formula, we get Area of sphere (in radians) = [4π/(3×2)] × 12² = (2π/3) × 144 = 96π. Therefore, the area of the sector in radians is unequivocal A 96π satisfying units.

How to Find the Area of a Sector Without Angle?

If the sector angle is non given, but we know the discharge duration and the radius, the surface area of a sector can Be calculated. We first find the sphere angle by substituting the given values of the arc length and radius in the formula, Length of Arc = (θ/360) × 2πr. After calculative the angle, we can easily find the area of the sphere with the formula, Surface area of a Sector of a Circle = (θ/360º) × πr².

How to Find the Arc Distance of a Sector?

Arc length is the distance on the part of the circumference of a circle. The arc duration of a circuit can Be calculated using the tailing formulas:

• Arc Length = θ × r; where θ = Central weight subtended past the arc, and r = r of the circle. This convention is used when θ is in radian.
• Arc Length = θ × (π/180) × r; where θ = Centric angle subtended by the arc, and r = radius of the forget me drug. This formula is used when θ is in degrees.

how to calculate area of sector of a circle

Source: https://www.cuemath.com/measurement/area-of-a-sector/