
Find the area of the minor segment of a circle of radius 42 cm, if the length of the corresponding arc is 44 cm.
Answer
498.9k+ views
Hint: In this sum, we will first draw figure for better understanding. After that, we will find angles subtended by an arc of the given length. For finding the area of the segment, we need to find areas of sector and area of a triangle formed. We can find the area of the sector by using angles found. For finding the area of the triangle we will use basic formula but for finding base and height we will use congruency, right-angled triangle properties.
Complete step-by-step solution
Formulas which we will use are:
I) where r is radius of circle.
II) Area of segment = Area of the sector - Area of a triangle.
III) where r is radius of circle.
IV)
Let us draw the figure first:
Now, let us find angle subtended by arc with help of length of arc 44 cm, radius 42 cm. As we know,
Putting the values we get:
As we can see from diagram, area of segment APB = Area of sector OAPB - Area of triangle OAB . . . . . . . . . (2) so, let us first find area of sector OAPB and area of triangle OAB. Given radius r = 42 cm.
Putting value of from (1), we get:
But we don't know values of base or height. So let us find that,
Draw
Therefore,
In we can see that
OA = OB as both is the radius of the same circle.
OM = OM which is common in both triangles.
Therefore, by RHS congruency criterion,
By using CPCT (corresponding parts of congruent triangle) we can say,
Therefore,
As we have find
Therefore,
Also,
Now from
Also,
From (3) AB = 2AM
Therefore, putting value of AM, we get:
AB = 42 cm
Also,
So, we have found length of base and perpendicular for triangle AOB.
From (2) as we know, area of segment APB = area of sector OAPB - area of triangle OAB.
We have,
Note: Students should not get confused with length of arc or length of chord such that, by two points. Students should always draw diagrams for better understanding. Also, don't get confused with formulas of . Don't forget to put units after finding a value for area use squared units and length as linear units.
Complete step-by-step solution
Formulas which we will use are:
I)
II) Area of segment = Area of the sector - Area of a triangle.
III)
IV)
Let us draw the figure first:

Now, let us find angle subtended by arc with help of length of arc 44 cm, radius 42 cm. As we know,
Putting the values we get:
As we can see from diagram, area of segment APB = Area of sector OAPB - Area of triangle OAB . . . . . . . . . (2) so, let us first find area of sector OAPB and area of triangle OAB. Given radius r = 42 cm.
Putting value of
But we don't know values of base or height. So let us find that,
Draw
Therefore,

In
OA = OB as both is the radius of the same circle.
OM = OM which is common in both triangles.
Therefore, by RHS congruency criterion,
By using CPCT (corresponding parts of congruent triangle) we can say,
Therefore,
As we have find
Therefore,
Also,
Now from
Also,
From (3) AB = 2AM
Therefore, putting value of AM, we get:
AB = 42 cm
Also,
So, we have found length of base and perpendicular for triangle AOB.
From (2) as we know, area of segment APB = area of sector OAPB - area of triangle OAB.
We have,
Note: Students should not get confused with length of arc or length of chord such that, by two points. Students should always draw diagrams for better understanding. Also, don't get confused with formulas of
Latest Vedantu courses for you
Grade 8 | CBSE | SCHOOL | English
Vedantu 8 CBSE Pro Course - (2025-26)
School Full course for CBSE students
₹45,300 per year
Recently Updated Pages
Master Class 10 Computer Science: Engaging Questions & Answers for Success

Master Class 10 Maths: Engaging Questions & Answers for Success

Master Class 10 English: Engaging Questions & Answers for Success

Master Class 10 General Knowledge: Engaging Questions & Answers for Success

Master Class 10 Science: Engaging Questions & Answers for Success

Master Class 10 Social Science: Engaging Questions & Answers for Success

Trending doubts
What is the past participle of wear Is it worn or class 10 english CBSE

What is Whales collective noun class 10 english CBSE

What is potential and actual resources

For what value of k is 3 a zero of the polynomial class 10 maths CBSE

What is the full form of POSCO class 10 social science CBSE

Which three causes led to the subsistence crisis in class 10 social science CBSE
