What is the area of a sector with radius equal to 8 and measure of arc equal to 300°?

Area of Sector

The surface area of a sector of a circle is the amount of space enclosed inside the boundary of the sector. A sector always originates from the eye of the circle. The sector of a circle is defined as the portion of a circumvolve that is enclosed between its two radii and the arc bordering them. The semi-circle is the most common sector of a circle, which represents half a circumvolve. Allow u.s. learn more about the area of sector, its formula, and how to calculate the area of a sector using radians and degrees.

ane.	What is Area of Sector of a Circle?
2.	Expanse of Sector Formula
3.	Area of Sector in Radians
iv.	FAQs on Area of Sector

What is Area of Sector of a Circumvolve?

The infinite enclosed past the sector of a circle is called the area of the sector. For example, a pizza piece is an example of a sector that represents a fraction of a pizza. There are ii types of sectors: minor and major sectors. A pocket-size sector is a sector that is less than a semi-circle, whereas, a major sector is a sector greater than a semi-circumvolve.

The figure given below represents the sectors in a circle. The shaded region shows the expanse of the sector OAPB. Here, ∠AOB is the angle of the sector. Information technology should be noted that the unshaded region is also a sector of the circle. So, the shaded region is the area of the small-scale sector and the unshaded region is the area of the major sector.

Now, let us learn near the expanse of a sector formula and its derivation.

Area of Sector Formula

In order to find the full space enclosed by the sector, we employ the area of a sector formula. The area of a sector can be calculated using the following formulas,

• Area of a Sector of Circle = (θ/360º) × πr², where, θ is the sector bending subtended by the arc at the center, in degrees, and 'r' is the radius of the circle.
• Area of a Sector of Circle = one/2 × r²θ, where, θ is the sector angle subtended by the arc at the middle, in radians, and 'r' is the radius of the circle.

Surface area of Sector Formula Derivation

Let us apply the unitary method to derive the formula for the area of the sector of a circumvolve. We know that a consummate circle measures 360º. The area of a circumvolve with an angle measuring 360º at the centre is given by πr², where 'r' is the radius of the circle.

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

• θ is the angle subtended at the centre, given in degrees.
• r is the radius of the circumvolve.

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

If the angle at the center is θ in radians, expanse of the sector of a circle = (1/two) × r²θ, where,

• θ is the bending subtended at the middle, given in radians.
• r is the radius of the circle.

It should exist noted that semi-circles and quadrants are special types of sectors of a circle with angles of 180° and 90° respectively.

Area of Sector Using Degrees

Let united states use these formulas and acquire how to calculate the expanse of the sector of a circumvolve when the subtended angle is given in degrees with the help of an example.

Example: A circle is divided into 3 sectors and the primal angles made by the radius are 160°, 100°, and 100° respectively. Find the area of all the three sectors.

Solution:

The angle made by the first sector is θ = 160°. Therefore, the area of the first sector = (θ/360°) × πr² = (160°/360°) × (22/7) × 6ⁱⁱ = four/9 × 22/7 × 36 = 352/7 = l.28 square units.

The angle fabricated by the second sector is θ = 100°. Therefore, the expanse of the second sector is = (θ/360°) × πr² = (100°/360°) × (22/7) × six² = 5/eighteen × 22/seven × 36 = 220/7 = 31.43 square units.

The angle made by the third sector is the same as that of the second sector (θ = 100°). Thus, the area of the 2nd sector is equal to the expanse of the tertiary sector. Therefore, the area of the third sector = 31.43 square units.

Expanse of Sector in Radians

If nosotros demand to find the area of a sector when the angle is given in radians, we utilize the formula, Area of sector = (1/2) × r²θ; where θ is the angle subtended at the center, given in radians, and 'r' is the radius of the circle. So, let us understand where the formula comes from. Nosotros know that the formula for the surface area of a sector (in degrees) = (θ/360º) × πr² because information technology is a fraction of a circle. The same concept is applied to the formula when we desire to express it in radians, merely we just demand to replace 360° with 2π because 2π (in radians) = 360°. This means, Surface area of sector in radians = (θ/2π) × πr². On further simplifying the formula, we go, area of sector = (θ/ii) × r^two or (1/two) × r²θ. Let united states of america understand how to find the area of a sector in radians with an example.

Example: Find the area of a sector if the radius of the circumvolve is 6 units, and the angle subtended at the center = 2π/3

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

The area of the given sector can be calculated with the formula, Surface area of sector (in radians) = (θ/2) × r². On substituting the values in the formula, nosotros get Area of sector (in radians) = [2π/(3×2)] × 6ⁱⁱ = (π/3) × 36 = 12π.

Therefore, the area of the given sector in radians is expressed as 12π square units.

Real-Life Example of Expanse of Sector of Circle

I of the virtually mutual existent-life examples of the area of a sector is the slice of a pizza. The shape of the slices of a circular pizza is like a sector. Observe the figure given below that shows a pizza that is sectioned into 6 equal slices, where each piece is a sector, and the radius of the pizza is 7 inches. Now, let us discover the surface area of the sector formed by each slice by using the expanse of a sector formula. It should exist noted that since the pizza is divided into half dozen equal slices, the bending of sector is threescore°. Area of Pizza slice = (θ/360°) × πr² = (60°/360°) × (22/vii) × 7ⁱⁱ = i/vi × 22 × 7 = 77/3 = 25.67 square units.

Tips on Surface area of Sector

Hither is a listing of a few important points that are helpful in solving the expanse of sector problems.

• The expanse of a sector of a circumvolve is the fractional expanse of the circle.
• The surface area of a sector of a circumvolve with radius 'r' is calculated with the formula, Area of a sector = (θ/360º) × π r²
• The arc length of the sector of radius r can be calculated with the formula, Arc Length of a Sector = r × θ

☛ Related Articles

• Area of a Circle
• Arcs and Subtended Angles
• Segment of a Circle
• What is pi?

Area of a Sector Examples

1. Instance 1: If the angle of a sector of a circle is 60°, and the radius of the circle is 7 inches, what is the surface area of the sector of this circle?

   Solution:

   The radius of the circle is 7 inches and the angle is 60°. And so, allow u.s. employ the area of sector formula. The surface area of sector = (θ/360°) × π r² = (sixty°/360°) × (22/7) × 7ⁱⁱ = 77/three = 25.67 square units. Therefore, the area of the modest sector is 25.67 foursquare units.

2. Example two: An umbrella has equally spaced 8 ribs. If viewed every bit a flat circle of radius 7 units, what would be the expanse between two consecutive ribs of the umbrella? (Hint: The area betwixt two consecutive ribs would form a sector of a circle)

   Solution:

   The radius of the apartment umbrella = 7 units. There are 8 ribs in the umbrella. Since a consummate angle of a circumvolve = 360°, the angle of each sector of the umbrella = 360/8 = 45° considering the circumvolve is divided into eight equal sectors. Thus, the surface area of sector = (θ/360°) × π r² = (45°/360°) × 22/7 × 7² = 77/iv = 19.25 foursquare units. Therefore, the area between 2 consecutive ribs of the umbrella is xix.25 square units.

3. Example three: A circumvolve with a diameter of two units is divided into 10 equal sectors. Can you lot discover the expanse of each sector of the circle?

   Solution:

   The bore of the circumvolve is 2 units, therefore, the radius of the circumvolve is 1 unit. Since a complete angle of a circumvolve = 360°, the angle of each sector of the circle is 360/10 = 36° because the complete angle is divided into 10 equal parts. Surface area of Sector = (θ/360°) × πr^two = 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 Area of a Sector of a Circumvolve?

The infinite enclosed by the sector of a circle is called the area of the sector of a circle. The part of the circle that is enclosed by ii radii and the corresponding arc is chosen the sector of the circumvolve.

What is the Formula for Expanse of Sector of Circle?

The two chief formulas that are used to notice the expanse of a sector are:

• Area of a Sector of Circle = (θ/360º) × πrⁱⁱ, where, θ is the angle subtended at the center, given in degrees, and 'r' is the radius of the circle.
• Area of a Sector of Circle = 1/2 × r^twoθ, where, θ is the angle subtended at the centre, given in radians, and 'r' is the radius of the circumvolve.

How to Summate the Surface area of a Sector using Degrees?

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

What practise yous Mean by Sector of a Circle?

A sector is divers as the portion of a circle that is enclosed between its 2 radii and the arc bordering them. The semicircle is the most common sector of a circumvolve, which represents half of a circle.

What do yous Mean by the Arc of a Circle?

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

How is the Area of Sector of Circumvolve Formula Derived?

The area of the sector shows the area of a office of the circle'due south area. We know that the area of a circumvolve is calculated with the formula, πr^two. The formula for the area of a sector of a circle is derived in the following way:

• Apply the unitary method to derive the formula of the area of a sector of circle.
• We know, a complete circle measures 360º. The area of a circle with an angle measuring 360º at the center is given past πr², where r is the radius of the circle.
• If the bending at the eye of the circle is 1º, the expanse of the sector is πr²/360º. So, if the bending at the centre is θ, the area of the sector is, Area of a Sector of a Circumvolve = (θ/360º) × πrⁱⁱ, where, θ is the angle subtended at the center, given in degrees, and r is the radius of the circle.
• In other words, πr² represents the surface area of a full circumvolve and θ/360º tells us how much of the circumvolve is covered by the sector.

How to Find the Area of Sector with Arc Length and Radius?

The expanse of a sector can be calculated if the arc length and radius is given. We first calculate the bending (θ) subtended by the arc with the formula, Length of Arc = (θ/360) × 2πr. Now, we already know the radius, and once the bending is known, the expanse of the sector tin be calculated with the formula, Area of a Sector of a Circle = (θ/360º) × πr^two

How to find the Radius from Expanse of Sector?

If the area of a sector is known, and the bending (θ) subtended by the arc is known, the radius tin can be calculated by substituting the given values in the formula, Expanse of a Sector of a Circle = (θ/360º) × πr². For example, let u.s. find the radius if the area of a sector is 36π, and the sector bending is given as ninety°. We volition substitute the given values in the formula, Area of a Sector of a Circle = (θ/360º) × πr², that is, 36π = (90/360) × πr². And so, the value of r² = 144, which means r = 12 units.

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

The area of sector tin also be expressed 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 ninety°, permit united states of america find the expanse of the sector in terms of pi. Area of sector = (θ/360º) × πr². Substituting the values in the formula, Area of sector = (ninety/360) × π × 4². After solving this, we get, the expanse as 4π.

How to Find the Area of a Sector in Radians?

In society to observe the area of a sector with the central bending in radians, we apply the formula, Area of sector = (θ/two) × 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 radius of the circle is 12 units, and the sector angle subtended by the arc at the center = 4π/3, allow united states detect the surface area of the sector. Expanse of sector (in radians) = (θ/ii) × r². On substituting the values in the formula, we go Area of sector (in radians) = [4π/(3×2)] × 12^two = (2π/iii) × 144 = 96π. Therefore, the area of the sector in radians is expressed every bit 96π square units.

How to Find the Area of a Sector Without Angle?

If the sector angle is not given, but we know the arc length and the radius, the surface area of a sector tin be calculated. We start find the sector angle by substituting the given values of the arc length and radius in the formula, Length of Arc = (θ/360) × 2πr. After computing the angle, nosotros tin hands find the surface area of the sector with the formula, Area of a Sector of a Circumvolve = (θ/360º) × πr^two.

How to Find the Arc Length of a Sector?

Arc length is the distance along the role of the circumference of a circle. The arc length of a circle can be calculated using the post-obit formulas:

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

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

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