Area of Sector

The part of the circle which is enclosed by an arc and two radii drawn to the extremities of the arc is called a sector.
area of a sector

sector and area of sector

The total space inside the boundary of the sector is called as the area of the sector. Area is measured in terms of square unit.

Let O be the center, r be the radius, AB be the arc. The shaded portion AOB is the sector; l is the length of the arc. θ is the angle of the sector at the center.

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C is the circumference of the circle O, and C = 2πr units

Angle at the center of the circle = 360º

l/C = theta^o/360^o

Plug the value for C

l/(2 pi r) = theta^o/360^o --- (1)

Length of the sector , l = theta/360 xx 2 pi r units

Area of a Sector Formula

Arcs are proportional to the angle subtended at the center.

Since sector is bounded by two radius and arc, perimeter of the sector = length of arc + radius + radius

= l + r + r = l + 2r units

Area of the sector is proportional to the angles subtended at the center.

Area of the sector    =  θ                    --- (2)

Area of the circle         360

Area of the sector = (theta xx pi r^2)/360

From (1) and (2) , we have

Area of the sector    =   l

Area of the circle         2πr

Therefore, area of the sector = l/(2pir) xx pir^2 = l xx r/2

Note: When the angle of the sector is not given while the length of the arc and radius are given, the formula for area is (lr)/2 sq. units

Examples

Below are the examples on area of sector -

Example 1: Calculate the area of the sector whose perimeter is 110 cm and radius is 20 cm.

Solution:

Step 1:

Write the given details.

Perimeter = 110 cm

Radius = 20 cm

Step 2:

Write the formula for perimeter and find "l"

P = l + 2r

l = P - 2r

Step 3:

Plug the values of P and r

l = 110 - 2(20)

= 70 cm

Step 4:

Since we know the length of the arc and radius of the sector, we could use the formula A = (lr)/2 to calculate area

Step 5:

Plug the values in the formula

A = (70 xx20)/2 = 700 sq. cm

Step 6:

Write the solution

Area of the sector = 700 sq. cm

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Example 2:

Calculate the area of the sector whose perimeter is 202 cm and the angle subtended at the center is 216°

Solution:

Step 1:

Write the given details

Perimeter , P = 202 cm

Angle, θ = 216°

Step 2:

Write the perimeter formula and plug the values

Perimeter of the sector = l + 2r = 202 cm       --- (1)

Step 3:

Write the arc length formula and plug the known values

l = theta/360^o xx 2pir = 216/360 xx 2 pi r = (6 pir)/5 --- (2)

Step 4: Plug (2) in (1) and find r

(6pir)/5 + 2r = 202

(6 xx 22)/(5xx7)r + 2r = 202

132/35r + 2r = 202

132r + 70r = 202 x 35

r = 202/202 xx 35 = 35 cm

Step 5:

Calculate the area of the sector using the formula A = (lr)/2

A = (6pir xx r)/(2 xx 5)

= 2310 sq. cm