A segment the a circle is the region that is bounded by one arc and also a chord of the circle. As soon as something is separated into parts, each component is described as a segment. In the same way, a segment is a part of the circle. However a segment is not any type of random component of a circle, instead, it is a specific part of a circle that is cut by a chord that it. Let us learn around the meaning of a segment of a circle and the formula to discover the area of a segment that a one in detail here.
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|1.||What is the Segment the a Circle?|
|2.||Properties the Segment of Circle|
|3.||Area of a Segment of Circle|
|4.||Area of a Segment of circle Formula|
|5.||Perimeter of Segment of a Circle|
|6.||Theorems top top Segment that a Circle|
|7.||FAQs ~ above Segment of Circle|
What is the Segment of a Circle?
A segment the a one is the region that is bounded by one arc and also a chord of the circle. Let us recall what is supposed by one arc and also a chord that the circle.
There space two species of segments, one is a young segment, and also the other is a significant segment. A minor segment is make by a boy arc and also a major segment is do by a major arc that the circle.
Properties the Segment that Circle
The nature of a segment that a circle are:A minor segment is derived by removed the corresponding significant segment indigenous the complete area the the circle.A significant segment is acquired by remove the equivalent minor segment native the complete area that the circle.
Area that a Segment the Circle
An arc and two radii that a circle type a sector. These 2 radii and also the chord the the segment together kind a triangle. Thus, the area the a segment that a one is derived by individually the area of the triangle from the area the the sector. I.e.,
Area the a segment of one = area the the ar - area that the triangle
Let united state use this reasonable to have the recipe to find the area of a segment the a circle. Keep in mind that this is the area the the minor segment. Usually, a segment of a circle refers to a minor segment.
Note: To find the area that the major segment of a circle, we just subtract the equivalent area that the boy segment indigenous the full area of the circle.
Area that a Segment of circle Formula
Let us take into consideration the minor segment that the over circle that is made by the chord PQ the a circle of radius 'r' that is centered at 'O'. We recognize that every arc of a one subtends an edge at the facility which is referred to as the main angle the the arc. The angle made through the arc PQ is θ. We understand from trigonometry that, the area of the triangle OPQ is (1/2) r2 sin θ. Also, we know that the area the the sector OPQ is:(θ / 360o) × πr2, if 'θ' is in degrees(1/2) × r2θ, if θ' is in radians
Thus, the area that the minor segment that the one 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
Perimeter that Segment the a Circle
We understand that the segment of a one is made up of one arc and a chord the the circle. Think about the exact same segment together in the over figure.
Perimeter of the segment = length of the arc + length of the chord
We understand that,the length of the arc is rθ, if 'θ' is in radians and πrθ/180, if 'θ' is in degrees.the length of the chord = 2r sin (θ/2)
Thus, the perimeter the the segment formula is:The perimeter that the segment of a one = rθ + 2r sin (θ/2), if 'θ' is in radians.The perimeter that the segment of a one = πrθ/180 + 2r sin (θ/2), if 'θ' is in radians.
Theorems top top Segment that a Circle
Mainly, there space two theorems based on the segment that a Circle.Angles in the same segment theoremAlternate segment theorem
Angles in the exact same Segment Theorem
It claims that angles developed in the very same segment the a one are constantly equal.
Alternate Segment Theorem
This theorem states that the angle developed by the tangent and also the chord at the point of call is same to the angle created in the alternative segment on the circumference of the circle v the endpoints of the chord.
Here room a few related topics to the segment that a circle, take it a look.
Examples ~ above Segment that a Circle
Example 1: In a pizza slice, if the central angle is 60 degrees and the size of its radius is 4 units, then discover the area the the segment created if we eliminate the triangle part out that the pizza slice. Use π = 3.142. Round her answer to two decimals.
The radius the pizza is, r = 4 units.
The main angle is, θ = 60 degrees.
The area that the segment is,
r2 <πθ/360o - sin θ/2>
= 42 < (3.142 × 60)/360 - sin 60/2>
≈ 1.45 square units.
Therefore,the area that the segment the the pizza = 1.45 square units.
Example 2: If the area the a sector is 100 sq. Ft and also the area the the enclosed triangle is 78 sq. Ft, what is the area the the segment?
Area the the segment = area that the sector- area the 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 that the major segment that a one if the area of the equivalent minor segment is 62 sq. Units and also the radius is 14 units. Usage π = 22/7.
Area the the significant segment = area that the circle - area of the boy Segment
= πr2 − 62
= (22/7) × 14 × 14 − 62
= 554 sq. Units
Therefore, the area of the major segment 554 sq. Units.
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FAQs top top Segment that a Circle
What Is a Segment of a Circle?
A segment that a one is the an ar that is bounded by an arc and a chord that the circle. There room two varieties of segments, one is a minor segment (made by a minor arc) and also the various other is a major segment (made by a significant arc).
What Is the Difference in between Chord and also Segment the a Circle?
A chord the a circle is a heat segment the joins any two clues on the circumference whereas a segment is a region bounded by a chord and an arc of the circle.
What Is the Difference in between Arc and also Segment that a Circle?
An arc is a section of a circle's circumference whereas a segment of a one is a an ar bounded by an arc and also a chord that the circle.
What Is the Difference between a ar of a Circle and also a Segment that a Circle?
A ar of a circle is the region enclosed by 2 radii and also the matching arc, if a segment of a circle is the an ar enclosed by a chord and the matching arc.
What Is the Formula for Area of the Segment the a Circle?
The area that the segment the the circle (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 that the circle and 'θ' is the angle subtended by the arc the the segment.
How To uncover the Area that a Segment of a Circle?
Here space the measures to discover the area of a segment the a circle.Identify the radius of the circle and label it 'r'.Identify the main angle made by the arc the the segment and also label the 'θ'.Find the area of the triangle using the formula (1/2) r2 sin θ.Find the area that the sector using the formula(θ / 360o) × πr2, if 'θ' is in degrees (or)(1/2) × r2θ, if θ' is in radiansSubtract the area of the triangle native the area that the ar to discover the area the the segment.
How To find the Area the a significant Segment that a Circle?
The area the a significant segment the a one is found by individually the area that the matching minor segment indigenous the total area that the circle.
Are the angles in the same Segment the a circle Equal?
Yes, the angles developed by the same segment that a circle space equal. I.e., the angle on the one of the circle made by the same arc room equal.
What Is the alternative Segment to organize of a Circle?
The alternative segment theorem says that the angle developed by the tangent and the chord at the allude of contact is same to the angle created in the alternate segment on the circumference of the circle with the endpoints the the chord.
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Is a Semicircle a Segment that the Circle?
We know that a diameter that a circle is also a chord that the circle (in fact, the is the longest chord of the circle). Also, we recognize that the semicircle's one is an arc of the circle. Thus, a semicircle is bounded by a chord and also an arc and hence is a segment that the circle.