how to find the area of a segment of a circle

Segment of a Circle

A segment of a circumvolve is the region that is bounded by an arc and a chord of the circumvolve. When something is divided into parts, each role is referred to as a segment. In the same way, a segment is a part of the circumvolve. But a segment is non whatsoever random part of a circumvolve, instead, it is a specific part of a circle that is cut past a chord of it. The segment of circle is the part that is formed by a chord of the circle (intersecting line) and an arc of the circle (role of the purlieus).

In this article, we will hash out the concept of segment of circumvolve, and understand its definition and properties. We will learn to detect the surface area and perimeter of the segment of a circumvolve and describe the theorems based on the segment along with some solved examples for a better agreement of the concept.

one.	What is the Segment of a Circumvolve?
2.	Properties of Segment of Circle
3.	Area of a Segment of Circumvolve
four.	Expanse of a Segment of Circumvolve Formula
five.	Perimeter of Segment of a Circle
6.	Theorems on Segment of a Circle
7.	FAQs on Segment of Circle

What is the Segment of a Circumvolve?

A segment of a circle is the region that is bounded by an arc and a chord of the circle. Let us recall what is meant by an arc and a chord of the circle.

• An arc is a portion of the circumvolve's circumference.
• A chord is a line segment that joins whatsoever two points on the circle'due south circumference.

In that location are two types of segments, i is a modest segment, and the other is a major segment. A minor segment is made past a minor arc and a major segment is made by a major arc of the circle.

Properties of Segment of Circumvolve

The properties of a segment of a circle are:

• Information technology is the area that is enclosed by a chord and an arc.
• The angle subtended by the segment at the center of the circle is the same as the angle subtended past the corresponding arc. This angle is ordinarily known every bit the central angle.
• A minor segment is obtained by removing the respective major segment from the total area of the circle.
• A major segment is obtained by removing the corresponding minor segment from the total area of the circle.
• A semicircle is the largest segment in whatsoever circle formed by the diameter and the respective arc.

Surface area of a Segment of Circle

An arc and two radii of a circle form a sector. These two radii and the chord of the segment together course a triangle. Thus, the expanse of a segment of a circle is obtained by subtracting the area of the triangle from the area of the sector. i.east., Area of a segment of circumvolve = area of the sector - area of the triangle

Let us use this logic to derive the formulas to discover the surface area of a segment of a circle. Annotation that this is the expanse of the modest segment. Usually, a segment of a circle refers to a small-scale segment.

Note: To find the area of the major segment of a circumvolve, nosotros just subtract the corresponding surface area of the minor segment from the total surface area of the circle.

Area of a Segment of Circumvolve Formula

Let us consider the modest segment of the above circle that is made by the chord PQ of a circle of radius 'r' that is centered at 'O'. We know that every arc of a circumvolve subtends an angle at the center which is referred to as the central angle of the arc. The angle made past the arc PQ is θ. We know from trigonometry that, the expanse of the triangle OPQ is (ane/two) rⁱⁱ sin θ. Also, we know that the surface area of the sector OPQ is:

• (θ / 360°) × πr², if 'θ' is in degrees
• (1/2) × r^twoθ, if θ' is in radians

Thus, the area of the modest segment of the circle is:

• (θ / 360°) × πr² - (1/ii) r² sin θ (OR) r^two [πθ/360° - sinθ / 2], if 'θ' is in degrees
• (1/2) × r²θ - (ane/2) r² sin θ (OR) (r^two / 2) [θ - sin θ], if 'θ' is in radians

Perimeter of Segment of a Circle

We know that the segment of a circumvolve is made up of an arc and a chord of the circle. Consider the same segment every bit in the to a higher place effigy.

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

We know 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 of the segment formula is:

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

Theorems on Segment of Circle

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

• Angles in the same segment theorem
• Alternating segment theorem

Angles in the Same Segment Theorem

It states that angles formed in the same segment of a circumvolve are always equal.

Alternate Segment Theorem

This theorem states that the bending formed by the tangent and the chord at the point of contact is equal to the angle formed in the alternate segment on the circumference of the circle through the endpoints of the chord.

Important Notes on Segment of Circle

• A segment of circle is the area enclosed past an arc and chord of the circumvolve.
• Nosotros have two types of segments of circle - minor and major segment.
• We tin can find the expanse of segment using, Area of Segment = Surface area of Sector - Area of Triangle

Related Articles

• Chord of a Circle
• Geometry
• Circle Formulas

Segment of a Circle Examples

1. Example 1: In a pizza piece, if the primal angle is 60 degrees and the length of its radius is four units, and then observe the area of the segment formed if we remove the triangle part out of the pizza slice. Utilize π = 3.142. Round your answer to two decimals. Ignore the edge office of the pizza.

   

   Solution:

   The radius of pizza is, r = 4 units.

   The central angle is, θ = 60 degrees.

   The area of the segment is,

   r² [πθ/360° - sin θ/2]

   = 4² [ (3.142 × sixty)/360 - sin60 / ii]

   ≈ 1.45 square units.

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

2. Case 2: If the surface area of a sector is 100 sq. ft and the area of the enclosed triangle is 78 sq. ft, what is the area of the segment?

   

   Solution:

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

   = 100 sq. ft. - 78 sq. ft.

   = 22 sq. ft.

   Therefore, the area of the segment is 22 sq. ft.

3. Instance 3: Find the area of the major segment of a circle if the area of the respective minor segment is 62 sq. units and the radius is 14 units. Utilise π = 22/vii.

   

   Solution:

   Area of the major segment = expanse of the circumvolve - area of the pocket-sized Segment

   = πr^two − 62

   = (22/vii) × 14 × 14 − 62

   = 554 sq. units

   Therefore, the area of the major segment is 554 sq. units.

get to slidego to slidego to slide

Breakup tough concepts through simple visuals.

Math will no longer be a tough discipline, especially when you understand the concepts through visualizations with Cuemath.

Volume a Free Trial Class

Practise Questions on Segment of a Circumvolve

go to slidego to slide

FAQs on Segment of a Circle

What Is a Segment of a Circle?

A segment of a circumvolve is the region that is bounded by an arc and a chord of the circumvolve. There are 2 types of segments, one is a minor segment (made by a minor arc) and the other is a major segment (made by a major arc).

What Is the Difference Between Chord and Segment of a Circle?

A chord of a circle is a line segment that joins whatsoever two points on its circumference whereas a segment is a region bounded by a chord and an arc of the circle.

What Is the Divergence Between Arc and Segment of a Circumvolve?

An arc is a portion of a circumvolve's circumference whereas a segment of a circumvolve is a region bounded past an arc and a chord of the circle.

What Is the Difference Betwixt a Sector of a Circle and a Segment of a Circle?

A sector of a circle is the region enclosed by 2 radii and the respective arc, while a segment of a circumvolve is the region enclosed by a chord and the corresponding arc.

What Is the Formula for Area of the Segment of a Circumvolve?

The area of the segment of the circle (or) small segment of a circle is:

• (θ / 360°) × πr² - (1/2) r² sin θ (OR) rⁱⁱ [πθ/360° - sin θ/2], if 'θ' is in degrees
• (1/2) × r^twoθ - (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 angle subtended by the arc of the segment.

How To Find the Area of a Segment of a Circle?

Here are the steps to find the area of a segment of a circumvolve.

• Identify the radius of the circle and label it 'r'.
• Identify the central angle made by the arc of the segment and characterization information technology 'θ'.
• Detect the surface area of the triangle using the formula (one/2) r^two sin θ.
• Find the expanse of the sector using the formula
  (θ / 360^o) × πr², if 'θ' is in degrees (or)
  (1/two) × r²θ, if θ' is in radians
• Subtract the area of the triangle from the area of the sector to find the area of the segment.

How To Find the Area of a Major Segment of a Circle?

The surface area of a major segment of a circumvolve is establish by subtracting the area of the respective small segment from the full area of the circle.

What Is the Alternating Segment Theorem of a Circle?

The alternate segment theorem states that the angle formed past the tangent and the chord at the point of contact is equal to the angle formed in the alternating segment on the circumference of the circle through the endpoints of the chord.

Is a Semicircle a Segment of the Circle?

We know that a diameter of a circle is also a chord of the circle (in fact, information technology is the longest chord of the circle). Too, we know that the semicircle's circumference is an arc of the circle. Thus, a semicircle is divisional past a chord and an arc and hence is a segment of the circumvolve.

Are the Angles in the Same Segment of a Circle Equal?

Yes, the angles formed by the aforementioned segment of a circle are equal. i.east., the angles on the circumference of the circle made by the same arc are equal.

How to Find the Perimeter of Segment of Circle?

The perimeter of a segment of circumvolve can be calculated by adding the length of the chord of circle and the length of the respective arc of the circle. The formula for the perimeter of segement is 2r sin (θ/2) + rθ.

Source: https://www.cuemath.com/geometry/segment-of-a-circle/#:~:text=What%20Is%20the%20Formula%20for,if%20'%CE%B8'%20is%20in%20radians

Posted by: devinemarisch.blogspot.com

Share this post