Last modified on August 3rd, 2023

chapter outline

 

Sector of a Circle

You must have noticed a slice of a circular pizza. Its shape looks what is called sector of a circle in mathematics.  For example, when a pizza of radius 10 inches is sliced into 6 pieces, each piece is a sector.

What is a Sector of a Circle

The sector of a circle is the part enclosed by two radii and the included arc of a circle. It is represented by the symbol ⌔.

Sector of a Circle

Types of Sector in a Circle

There are 2 types of sector: minor and major sector. A minor arc is smaller than a semi-circle sector and thus has a central angle less than 180, while a major sector is greater than a semicircle and thus has a central angle more than 180.

Types of Sector in a Circle

Formulas

Area of a Sector of a Circle

There are three formulas for calculating the area of a sector of a circle. All the three formulas give the same result but are used based on the information given about the sector.

How to Find the Area of a Sector Given the Central Angle in Degrees

If the radius is known and the central angle of the sector is given in degrees, the formula to find the area of a sector is given below.

Area of a Sector of a Circle

Let us solve some examples to understand the concept better.

Find the area of a sector with a central angle of 60° and a radius of 16 cm. Express your answer to the nearest tenth.

Solution:

As we know,
Area (A) of a sector = (θ/360°) × πr2, here θ = 60°, r = 16 cm, π = 3.141
= (60°/360°) × 3.141 × (16)2
= 3.141 × 16 × 16/60
= (50.25 × 0.26) cm2
= 13.06 cm2

Calculate the area of a sector with a radius of 20 yards and an angle of 90 degrees.

Solution:

As we know,
Area (A) of a sector = (θ/360°) × πr2, here θ = 90°, r = 20 yards, π = 3.141
= (90°/360°) × 3.141 × (20)2
= (1256.4/4) yards2
= 314.1yards2

Find the radius of a semi-circle with an area of 24 square inches.

Solution:

A semi-circle being the half a circle; therefore, the angle θ = 180 degrees.
As we know,
Area (A) of a sector = (θ/360°) × πr2, A = 24 square inches, here θ = 180°, π = 3.141
24 = (180°/360°) × 3.141 × r2
24 = 3.141 × r2/2
r2 = 48/3.141 = 15.28 inches

How to Find the Area of a Sector Given the Central Angle in Radians

If the radius is known and the central angle of the sector is given in radians, the formula to find the area of a sector is given below.

Area of a Sector of a Circle Formula

Let us solve some examples to understand the concept better.

Find the area of a sector with a radius of 8 m and a central angle of 0.62 radians.

Solution:

As we know,
Area (A) of a sector = (θr2)/2, here θ = 0.62 radians, r = 8 m
= 0.62 × 8 × 8/2
= 19.84 m2

The area of a sector is 225 m2. If the sector’s radius is 8 m, find the central angle of the sector in radians.

Solution:

As we know,
Area (A) of a sector = (θr2)/2, here A = 225,r = 8 m
225 = θ × 8 × 8/2
θ =225 × 2/64
θ = 7.031 radians

How to Find the Area of a Sector Given the Arc Length

If the radius and the length of the arc are known, the formula to find the area of a sector is given below.

Sector of a Circle Formula

Let us solve some examples to understand the concept better.

The length of an arc is 64 mm. Find the area of the sector formed by the arc if the circle’s radius is 15 mm.

Solution:

As we know,
Area (A) of a sector = r × L/2, here r = 15 mm, L = 64 mm
= (15 × 64/2) mm2
= 480 mm2

Find the area of a sector whose arc length is 10 inches and radius is 6 inches.

Solution:

As we know,
Area (A) of a sector = r × L/2, here r = 6 inches, L = 10 inches
= (6 × 10/2) inches2
= 30 inches2

Perimeter of a Sector of a Circle

The perimeter of a circle sector is the combined length of two radii and the arc that forms the sector. The formula to calculate the perimeter of a sector is derived below.

Derivation

Perimeter (P) of a sector = radius (r) + radius (r) + arc length (L)

= 2 radius (r) + arc length (L)

= 2r + L

Perimeter of a Circle Sector

As arc length is calculated using the relation

Arc length (L) = (θ/360) × 2πr

Thus, the above formula can also be written as:

Perimeter (P) of a sector = 2r + [(θ/360) × 2πr], here r = radius, θ = central angle in radians, π = 3.141 = 22/7

Let us solve some examples to understand the concept better.

A circular arc whose radius is 14 cm, forms an arc length of 6 cm. Find the perimeter of the sector formed.

Solution:

As we know,
Perimeter (P) of a sector = 2r + L, here r = 14 cm, L = 6 cm
= (2 × 14) + 6
= 34 cm

Find the perimeter of the sector of the circle with radius 22 cm and a making an angle of 30° at the center.

Solution:

As we know,
Perimeter (P) of a sector = 2r + [(θ/360) × 2πr], here r = 22 cm, θ =30°, π = 3.141
= (2 × 22) + [(30/360) × 2 × 3.141 × 22]
= (44 + 11.51) cm
= 55.51 cm

Last modified on August 3rd, 2023

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