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**Area of A Sector of A Circle Formula**

**Introduction to Area Of A Sector Of A Circle Formula**

The area of sector of a circle is the space enclosed within the boundary of the sector. A sector always originates from the center of the circle. Let us learn more about the area of sector formula, and how to find the area of a sector in radians and degrees.

**What is Area of Sector of a Circle?**

The space enclosed by the sector of a circle is called the area of the sector. For example, a pizza slice is an example of a sector that represents a fraction of a pizza. There are two types of sectors: minor and major sectors. A minor sector is a sector that is less than a semi-circle, whereas, a major sector is a sector greater than a semi-circle.

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

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

**Sector Definition**

The sector of a circle is defined as the portion of a circle that is enclosed between its two radii and the arc adjoining them. It is considered as a portion of a circle with two radii and an arc. A circle is divided into two sectors, the minor sector and the major sector where the minor sector is the smaller portion in the circle and the larger sector is the major portion. The semi-circle is the most common sector of a circle, which represents half a circle.

**Area of Sector Formula**

In order to find the total space enclosed by the sector, we use 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 angle subtended by the arc at the center, in degrees, and ‘r’ is the radius of the circle.
- Area of a Sector 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**

Let us apply the unitary method to derive the formula for the area of the sector of a circle. We know that a complete circle measures 360º. The area of a circle with an angle measuring 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 area of the sector is πr2/360º. So, if the angle at the center is θ, the area of the sector is,

Area of a Sector of Circle = (θ/360º) × πr2, where,

- θ is the angle subtended at the center, given in degrees.
- r is the radius of the circle.

In other words, πr2 represents the area 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, then the formula for the area of the sector of a circle = (1/2) × r2θ, where,

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

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

**Real-Life Example of Area of Sector of Circle**

One of the most common real-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 slice is a sector, and the radius of the pizza is 7 inches. Now, let us find the area of the sector formed by each slice by using the sector area formula. It should be noted that since the pizza is divided into 6 equal slices, the angle of sector is 60°.

Area of Pizza slice = (θ/360°) × πr2

= (60°/360°) × (22/7) × 72

= 1/6 × 22 × 7

= 77/3

= 25.67 square units.

**Solved Examples on Area of a Sector of a Circle**

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

Solution:

The radius of the circle is 7 inches and the angle is 60°. So, let us use the area of sector formula.

The area of sector = (θ/360°) × πr2

= (60°/360°) × (22/7) × 72

= 77/3 = 25.67 square units.

Therefore, the area of the minor sector is 25.67 square units.

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

Solution:

The radius of the flat umbrella = 7 units.

There are 8 ribs in the umbrella.

Since a complete angle of a circle = 360°,

the angle of each sector of the umbrella = 360/8 = 45° because the circle is divided into 8 equal sectors.

Thus, the area of secor = (θ/360°) × πr2

= (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.

**Example 3: **A circle with a diameter of 2 units is divided 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 sector of the circle is 360/10 = 36° because the complete angle is divided into 10 equal parts.

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.

**Frequently Asked Questions on Area of a Sector of a Circle**

1: What is the Area of a Sector of a Circle?

Answer: The space 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 two radii and the corresponding arc is called the sector of the circle.

2: What is the Formula for Area of Sector of Circle?

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

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.

Area of a Sector of Circle = 1/2 × r2θ, where, θ is the angle subtended at the center, given in radians, and ‘r’ is the radius of the circle.

3: How to Calculate Area of Sector using Degrees?

Answer: When the angle subtended at the center is given in degrees, the area of a sector can be calculated 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.

4: What do you Mean by Sector of a Circle?

Answer: A sector is defined as the portion of a circle that is enclosed between its two radii and the arc adjoining them. The semicircle is the most common sector of a circle, which represents half of a circle.

5: What do you Mean by the Arc of a Circle?

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

6: How is the Area of Sector of Circle Formula Derived?

Answer: The area of the sector shows the area of a part of the circle’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:

7: Apply the unitary method to derive the formula of the area of a sector of circle.

Answer: We know, a complete circle measures 360º. The area of a circle with an angle measuring 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 area of the sector is πr2/360º.

So, if the angle at the center is θ, the area of the sector is,

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

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

Answer: The area of a sector can be calculated if the arc length and radius is given. We first calculate the angle (θ) subtended by the arc with the formula,

Length of Arc = (θ/360) × 2πr. Now, we already know the radius, and once the angle is known, the area of the sector can be calculated with the formula,

Area of a Sector of a Circle = (θ/360º) × πr2

9:How to Find the Radius from Area of Sector?

Answer: If the area of a sector is known, and the angle (θ) subtended by the arc is known, the radius can be calculated by substituting the given values in the formula,

Area of a Sector of a Circle = (θ/360º) × πr2.

For example, let us find the radius if the area of a sector is 36π, and the sector angle is given as 90°. We will substitute the given values in the formula,

Area of a Sector of a Circle = (θ/360º) × πr2,

that is, 36π = (90/360) × πr2.

So, the value of r2 = 144, which means r = 12 units.

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

Answer: The area of sector can 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 90°, let us find the area of the sector in terms of pi. Area of sector = (θ/360º) × πr2. Substituting the values in the formula,

Area of sector = (90/360) × π × 42.

After solving this, we get, the area as 4π.

11: How to Find the Area of a Sector Without Angle?

Answer: If the sector angle is not given, but we know the arc length and the radius, the area of a sector can be calculated. We first 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 calculating the angle, we can easily find the area of the sector with the formula,

Area of a Sector of a Circle = (θ/360º) × πr2.