Area of Sector

The area covered by a segment of a circle is identified as the area of a sector. A circle can be split into two sections, the major sector and the minor sector. The sector that occupies the larger area is known as the major sector, while the one that covers a smaller area is referred to as the minor sector.

The fractional part of the total area of a circle is the sector’s area. The amount of area that a sector occupies can be calculated using the formulas mentioned below:

Formula 1: Area of Sector = θ/360  πr² (Here, θ is expressed in degrees)

Formula 2: Area of Sector = ½ r² θ (θ is in radians)

Let’s now go through the step-by-step derivation of the formula 1,

The unitary method is used in deriving the formula for the area of a sector of a circle. 

• The area of a full circle with a central angle of 360^o is represented by πr² (where r is the radius).
   
• Let the central angle be 1^o, and the area of the sector is πr²/ 360^o. 
   
• To calculate the area of a sector, multiply (/360^o) by πr².
   
• The angle subtended at the center, θ, is given in degrees.
   
• The symbol r represents the circle's radius.

In simple terms, 

• πr² is the area of a full circle.
   
• θ/360º is the area of a circle covered by the sector.

Therefore, if θ is in degrees, the formula we use for the area of a sector:

Area of Sector = θ/360º × (πr²)

Now we will derive the formula 2 :

We know that θ is the central angle in radians

Area of a circle = πr²

The total angle of a circle is 2 π radians.

Consider a sector with a central angle θ takes up the following fraction of the circle:

The fraction of a circle equals 2 π θ 

We can calculate the area of the sector by multiplying the fraction by the total area of the circle,

Area of Sector =  θ × 2 π × r²

The sector area is calculated as 2π θ ⋅ πr ².

Area of Sector = θ r ²/ 2

⇒ ½ r2θ 

The area of the sector can be calculated using the formulas mentioned below:

If the angle is in radians, use the formula:

Area of Sector =  θ / 360^o. πr2 (where  θ = central angle in degrees, r = radius)

If the angle is in radians:

Area of Sector = ½ r² θ (where θ = central angle in radians, r = radius)

Let’s look at an example:

Calculate the area of the circle.

Given: Radius = 7cm and the central angle of the sector = 90^o

Substitute the values into the equation of Area of Sector 
=  θ / 360^o. πr²   90/360. π(7)²

Substituting the value of π= 3.14,

= ¼ π(49) = 38.48 cm²

The area of the sector can be denoted using the square units mentioned below:

• (cm²): Unit used to denote the square centimeters.

• (m²): Unit used to denote square meters.

• (in²): Unit used to denote square inches.
   

• At θ = 360^o
  The sector will be a full circle when the central angle = 360^o or 2π radians.
  Area = πr² 

• At θ = 180^o
  The sector will be a semi-circle when the central angle θ = 180^o.
  Area = ½ πr²

• At θ = 90^o
  The sector will be a quarter-circle when the central angle  θ = 90^o.
  Area = 1/4 πr²

• Before applying the formula to calculate the sector's area, confirm whether the angle is expressed in degrees or radians.

• Students should recall the formulas, such as θ / 360^o. πr² (for degrees) and ½ r² θ(for radians) while finding the area of the sector.

• They can simplify the calculations by directly applying fractions for angles. For example, For angles 180^o and 360^o, the fractions of the circular area such as ½ and 1 are used respectively.

• To make the calculations easier, the values such as π= 3.14 or 22/7 (fractional form) can be learned.

• They should ensure the standard unit in the final answer is expressed in square units.

• Major Sector: The sector that occupies the major portion of a circle.

• Minor Sector: The sector that covers the smaller portion of a circle.

• Central angle: The angle at the center of a circle, which is used to calculate the area of the sector.

• Radians: It is a unit of measurement for angle and is represented using a symbol (θ).

• Degrees: It is a unit of measurement for angles. The total area of a circle in degrees is 360^o.