A segment of a one is the region that is bounded by an arc and also a chord the the circle. As soon as something is split into parts, each part is referred to as a segment. In the very same way, a segment is a part of the circle. Yet a segment is not any random part of a circle, instead, that is a specific component of a circle that is cut by a chord that it. Let us learn about the meaning of a segment of a circle and the formula to find the area that a segment the a circle in information here.

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1.What is the Segment of a Circle?
2.Properties of Segment that Circle
3.Area that a Segment that Circle
4.Area of a Segment of one Formula
5.Perimeter that Segment the a Circle
6.Theorems ~ above Segment that a Circle
7.FAQs top top Segment that Circle

What is the Segment of a Circle?


A segment of a one is the region that is bounded by an arc and a chord that the circle. Let united state recall what is expected by one arc and a chord the the circle.

There space two types of segments, one is a minor segment, and the other is a significant segment. A boy segment is made by a young arc and also a major segment is made by a significant arc that the circle.

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Properties of Segment of Circle


The nature of a segment that a one are:

A boy segment is acquired by removing the corresponding major segment indigenous the full area of the circle.A major segment is derived by removed the equivalent minor segment native the complete area of the circle.

Area of a Segment of Circle


An arc and also two radii the a circle form a sector. These 2 radii and the chord the the segment together kind a triangle. Thus, the area that a segment that a circle is derived by individually the area of the triangle from the area the the sector. I.e.,

Area the a segment of one = area that the sector - area the the triangle

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Let us use this reasonable to have the formulas to find the area that a segment the a circle. Keep in mind that this is the area the the boy segment. Usually, a segment the a circle describes a boy segment.

Note: To discover the area the the significant segment the a circle, we simply subtract the equivalent area that the young segment from the total area that the circle.


Area of a Segment of circle Formula


Let us think about the young segment of the over circle the is make by the chord PQ that a one of radius 'r' the is focused at 'O'. We recognize that every arc of a circle subtends an angle at the facility which is described as the central angle the the arc. The angle made by the arc PQ is θ. We recognize from trigonometry that, the area the the triangle OPQ is (1/2) r2 sin θ. Also, we understand that the area of the ar OPQ is:

(θ / 360o) × πr2, if 'θ' is in degrees(1/2) × r2θ, if θ' is in radians

Thus, the area the the boy segment that the circle is:

(θ / 360o) × πr2 - (1/2) r2 sin θ (OR) r2 <πθ/360o - sin θ/2>, if 'θ' is in degrees(1/2) × r2θ - (1/2) r2 sin θ (OR) (r2 / 2) <θ - sin θ>, if 'θ' is in radians

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Perimeter the Segment of a Circle


We understand that the segment of a circle is comprised of an arc and also a chord of the circle. Take into consideration the exact same segment together in the over figure.

Perimeter that the segment = length of the arc + size of the chord

We understand that,

the size of the arc is rθ, if 'θ' is in radians and also πrθ/180, if 'θ' is in degrees.the size of the chord = 2r sin (θ/2)

Thus, the perimeter that the segment formula is:

The perimeter of the segment that a circle = rθ + 2r sin (θ/2), if 'θ' is in radians.The perimeter that the segment the a one = πrθ/180 + 2r sin (θ/2), if 'θ' is in radians.

Theorems ~ above Segment the a Circle


Mainly, there room two theorems based on the segment the a Circle.

Angles in the very same segment theoremAlternate segment theorem

Angles in the very same Segment Theorem

It states that angles created in the very same segment that a one are always equal.

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Alternate Segment Theorem

This theorem claims that the angle created by the tangent and the chord at the point of contact is same to the angle created in the alternate segment top top the one of the circle v the endpoints of the chord.

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Related Topics

Here are a few related topics to the segment of a circle, take it a look.


Examples on Segment of a Circle


Example 1: In a pizza slice, if the main angle is 60 degrees and also the size of the radius is 4 units, then discover the area of the segment formed if we remove the triangle component out that the pizza slice. Use π = 3.142. Round her answer to 2 decimals.

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Solution:

The radius that pizza is, r = 4 units.

The central angle is, θ = 60 degrees.

The area of the segment is,

r2 <πθ/360o - sin θ/2>

= 42 < (3.142 × 60)/360 - sin 60/2>

≈ 1.45 square units.

Therefore,the area the the segment of the pizza = 1.45 square units.


Example 2: If the area of a sector is 100 sq. Ft and the area of the attached triangle is 78 sq. Ft, what is the area of the segment?

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Solution:

Area that the segment = area that the sector- area of the triangle

= 100 sq. Ft. - 78 sq. Ft.

= 22 sq. Ft.

Therefore, the area the the segment is 22 sq. Ft.


Example 3: Find the area of the major segment that a one if the area the the corresponding minor segment is 62 sq. Units and the radius is 14 units. Use π = 22/7.

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Solution:

Area that the major segment = area the the circle - area that the minor Segment

= πr2 − 62

= (22/7) × 14 × 14 − 62

= 554 sq. Units

Therefore, the area that the significant segment 554 sq. Units.


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Practice questions on Segment of a Circle


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FAQs on Segment that a Circle


What Is a Segment that a Circle?

A segment the a one is the an ar that is bounded by one arc and also a chord that the circle. There space two types of segments, one is a minor segment (made by a young arc) and also the other is a significant segment (made by a major arc).

What Is the Difference between Chord and Segment the a Circle?

A chord of a circle is a heat segment the joins any kind of two points on the circumference vice versa, a segment is a an ar bounded by a chord and an arc the the circle.

What Is the Difference between Arc and also Segment that a Circle?

An arc is a section of a circle's circumference vice versa, a segment the a one is a region bounded by an arc and a chord that the circle.

What Is the Difference in between a sector of a Circle and also a Segment that a Circle?

A sector of a circle is the an ar enclosed by 2 radii and also the corresponding arc, while a segment the a circle is the an ar enclosed by a chord and the equivalent arc.

What Is the Formula because that Area that the Segment the a Circle?

The area of the segment that the one (or) boy segment the a circle is:

(θ / 360o) × πr2 - (1/2) r2 sin θ (OR) r2 <πθ/360o - sin θ/2>, if 'θ' is in degrees(1/2) × r2θ - (1/2) r2 sin θ (OR) (r2 / 2) <θ - sin θ>, if 'θ' is in radians

Here, 'r' is the radius of the circle and 'θ' is the edge subtended by the arc the the segment.

How To discover the Area the a Segment that a Circle?

Here space the steps to find the area that a segment of a circle.

Identify the radius the the circle and label that 'r'.Identify the main angle do by the arc the the segment and also label that 'θ'.Find the area of the triangle making use of the formula (1/2) r2 sin θ.Find the area that the sector making use of the formula(θ / 360o) × πr2, if 'θ' is in degrees (or)(1/2) × r2θ, if θ' is in radiansSubtract the area the the triangle native the area the the sector to find the area that the segment.

How To find the Area of a major Segment the a Circle?

The area that a significant segment of a circle is uncovered by individually the area the the corresponding minor segment indigenous the total area that the circle.

Are the angle in the very same Segment the a one Equal?

Yes, the angles created by the very same segment the a circle are equal. I.e., the angles on the circumference of the circle made by the same arc are equal.

What Is the alternate Segment organize of a Circle?

The alternative segment theorem claims that the angle formed by the tangent and also the chord at the suggest of contact is equal to the angle formed in the alternative segment top top the one of the circle with the endpoints of the chord.

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Is a Semicircle a Segment of the Circle?

We understand that a diameter that a circle is likewise a chord of the circle (in fact, the is the longest chord of the circle). Also, we recognize that the semicircle's circumference is an arc of the circle. Thus, a semicircle is bounded by a chord and an arc and hence is a segment the the circle.