Table of Contents

Last modified on August 3rd, 2023

A segment of a circle is the region bounded by a chord of the circle and its associated arc. It is represented by the symbol ‘⌓’. A semicircle is the biggest segment of a circle.

In the given circle, the segment is enclosed by the chord AB and its associated arc ACB.

There are two types of segments in a circle: minor and major segment. A segment with an intercepted arc less than the semicircle is called a minor segment, while a segment with an intercepted arc more than the semicircle is called a major segment. If nothing specifically is stated, a segment means the minor segment.

The formula to find the area of the segment of a circle can either be expressed in terms of degree or in terms of radians. The two formulas for calculating circle’s segment are given below.

As we know from our ‘Area of a Sector of a Circle’ an arc and two radii of a circle form a sector. These two radii and the chord of the segment together form a triangle. Thus the area of a segment of a circle can be obtained by subtracting the area of the triangle from the area of the sector.

Thus, mathematically

**Area _{Segment }= Area _{Sector} – Area _{Triangle}**

Let us use the above logic to derive the formulas to find the segment of a circle both in degree and in radians. As previously discussed, this is the area of the minor segment.

**Derivation**

In the given figure above,

If ∠AOB = θ is the central angle, then the area of the sector AOBC (A _{sector AOBC}) in degrees is given by the formula:

(A _{sector AOBC}) = θ/360° × πr^{2}

Let the area of ΔAOB be A_{ΔAOB}

Then the area of the segment ABC is written using the formula

Area of a Segment of a Circle = Area of the Sector – Area of the Triangle

(A _{segment ABC}) = (A _{sector AOBC}) – A_{ΔAOB}

** (A _{segment ABC}) = θ/360° × πr^{2 }– A_{ΔAOB} …… (1)**

Now, to find the area of ΔAOB, we know that the area of the triangle with side lengths a and b and the included angle C is given by ½ absinC

Appling the above formula in ΔAOB with side lengths r and r and the included angle O, we get,

**Area of ΔAOB = ****½ r ^{2}sin θ**

Thus, Equation (1) can now be written as,

Area of a Segment of a Circle = θ/360° × πr^{2} – ½ r^{2}sinC

Factoring by 1/2r^{2} we get,

** Area (A) of a Segment of a Circle = ½ × r ^{2} × (πθ /180 – sin θ)**

Thus, if the radius is known and the central angle of the segment is given in degrees, the formula to find the area of a segment is given below.

Let us solve some examples to understand the concept better.

**Find the area of a segment of a circle with a central angle of 60 degrees and a radius of 4 cm. Use π = 3.141.**

Solution:

As we know,

A = ½ × r^{2} × (πθ/180 – sin θ), here r = 4 cm, θ = 60°

= (½) × (4)^{2} × [(3.141× 60)/180- 0.866]

= (½) × 16 × (1.047 – 0.866)

= 1.448 cm^{2}

**Derivation**

Now let us consider the other variant of this formula.

Area of the sector AOBC (A _{sector AOBC}) when the central angle is measured in radians is given by the formula:

A _{sector AOBC} = (θr^{2})/2

Now using the same formula for area of ΔAOB **=** ½ r^{2}sin θ

Then the area of the segment ABC is written using the formula

**Area _{Segment }= Area _{Sector} – Area _{Triangle}**

(A _{segment ABC}) = (A _{sector AOBC}) – A_{ΔAOB}

(A _{segment ABC}) = (θr^{2})/2 – ½ r^{2}sin θ

Factoring by 1/2r^{2} we get,

**Area (A) of a Segment of a Circle = r ^{2}/2 (θ – Sin θ)**

Thus, if the radius is known and the central angle of the segment is given in deg, the formula to find the area of a segment is given below.

**Note**: To find the area of the major segment, we will subtract the corresponding area of the minor segment from the total area of the circle.

Mathematically,

**Area of the major segment = Area of the circle – Area of the minor Segment**

Let us solve some examples to understand the concept better.

**Find the area of the circular segment if the diameter of a circle is 12 cm and the central angle is 4.59 radians. Express your answer to two decimal places.**

Solution:

As we know,

Area (A) of a Segment of a Circle in radians = ½ × r^{2} (θ – Sin θ), here r = d/2 =12/2 = 6 cm, θ = 4.59 radians

= ½ × 6^{2} × (4.59 – Sin 4.59)

= ½ × 6^{2} × (4.59 – 0.080)

= (½ × 6^{2} × 4.51) cm^{2}

= 81.18 cm^{2}

**Given that a chord and radius of a circle are each 24 cm. Find the area of the minor circular segment.**

Solution:

From the diagram, it is clear that ΔOBC is an equilateral triangle. Hence the central θ is 60° = π/3 radians

As we know,

Area (A) of a Segment of a Circle in radians = ½ × r^{2} (θ – Sin θ), here r = 22 cm, θ = π/3

= ½ × (24)^{2} [(π/3 – Sin (π/3)]

= 52.18 cm^{2}

**Find the area of the major segment of a circle if the area of the corresponding minor segment is 88 m ^{2} and the radius is 22 m. Use π = 3.141.**

Solution:

As we know,

Area of the major segment = Area of the circle – Area of the minor Segment

= πr^{2} – 88

= [3.141× (22)^{2}] -88

= 1432.24 m^{2}

As we know, the segment of a circle is made of an arc and a chord of a circle. Thus mathematically,

Perimeter (P) of the segment = length of the arc + length of the chord

The formula to find the perimeter of the segment of a circle can either be expressed in terms of degree or in terms of radians. The two formulas for calculating the circle’s segment are given below.

As we know,

Length of the arc = rθ, and

Length of the chord = 2r sin(θ/2)

Thus,

**Perimeter (P) of the segment = πrθ/180 + 2r sin (θ/2)**, here r = radius, θ is in degrees

As we know,

Length of the arc = πrθ/180, and

Length of the chord = 2r sin(θ/2)

Thus,

**Perimeter (P) of the segment = rθ + 2r sin (θ/2), **here r = radius,θ is in radians

**Find the perimeter of the segment of a circle with a central angle of 30 degrees and a radius of 3 cm. Use π = 3.141.**

Solution:

Since the central angle is given in degrees,

Perimeter (P) of the segment = πrθ/180 + 2r sin (θ/2), here r = 3cm, θ = 30°

= (3.141 × 3 × 30)/180 + (2 × 3) sin (30/2)

= 1.5705 + (6 × 0.25)

= 3.07 cm

Last modified on August 3rd, 2023

Given that the height and chord length combination of any segment is unique, Is there a formula which links the height of the segment and length of the chord to the area of the segment.

There is no such direct formula, however, you can use the below given derived formula for your calculation:

Area of segment = ${\left( \dfrac{l^{2}-4h^{2}}{8hl}\right) ^{2}\cdot 2\tan ^{-1}\left( \dfrac{2h}{l}\right) -\dfrac{l^{2}-4h^{2}}{16h}}$, where ‘h’ is the height of the segment and ‘l’ is the length of the chord.