AREAS RELATED TO CIRCLE: NCERT SOLUTION

 AREA RELATED TO CIRCLE: CONCEPT AND SOLUTION

Minor Sector: The blue region in the circle is said to be a minor sector.

Major Sector:   The yellow region in the circle inside the circle is known as the major sector.

Area of the major and the minor sector:

The area of a circle is Πr²

Area of 360 degrees around a point in a circle = Πr²

If the angle made in the minor sector is ө,

Then, using the unitary method

360 degrees form an area = Πr²

^{Then Ө degree =} Πr²ϴ/360 

This is the area of the minor sector

The area of the major sector can be obtained by = Area of the circle - Area of the minor sector

                                                                              

                                                                            = Πr^{2  -} Πr²ϴ/360 

Segment of a circle: 

Minor segment:  The area of a circle between the chord and the arc is said to be the area of the segment.

The area in blue is the area of the minor segment, and the area in red is the area of the major segment. 

Finding the area of minor and major segments 

Exercise 12.1 

Question 1:  

The radii of two circles are 19 cm and 9 cm, respectively. Find the radius of the circle that has a circumference equal to the sum of the circumferences of these two circles. 

Solution: The circumference of the larger circle = the circumference of the first circle + the circumference of the second circle.

2πR = 2πr₁ + 2πr₂

2πR = 2π *19 +2π* 9

Cancel 2π from both sides

R = 19+ 9 = 28 cm

Question 2: The radii of two circles are 8 cm and 6 cm, respectively. Find the radius of the circle having an area equal to the sum of the areas of the two circles.

Solution:   Let  r1 = 8 cm, r2 = 6 cm.  

Let the radius of the larger circle be R.

then

The area of two smaller circles equals the area of the larger circle

Question 3: Fig. 12.3 depicts an archery target marked with its five scoring regions from the center outwards as Gold, Red, Blue, Black, and White. The diameter of the region representing the Gold score is 21 cm, and each of the other bands is 10.5 cm wide. Find the area of each of the five scoring regions.

Solution: The width of each scoring band is 10.5 cm 

Starting with the gold region 

Question 4: 

Question 4: The wheels of a car are of a diameter of 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is traveling at a speed of 66 km per hour?

Question 5:  Tick the correct answer in the following and justify your choice: If the perimeter and the area of a circle are numerically equal, then the radius of the circle is (A) 2 units, (B) π units (C) 4 units, (D) 7 units

Solution

If the perimeter = area of the circle

2pi*r = pi*r*r

=> r = 2 units

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                                                                              Exercise 12.2

Question 1: Find the area of a sector of a circle with radius 6 cm if the angle of the sector is 60°.

 

Question 2: Find the area of a quadrant of a circle whose circumference is 22 cm. 

Solution:  Circumference of the circle = 2pi *r = 22cm

 2*22/7 * r = 22

r = 22*7/22*2 = 7/2 cm

Area of the quadrant of the circle = (1/4) pi * r * r

                                                     = 1/4 * (22/7) *(7/2)*(7/2) 

                                                     = 77/8 cm

To be continued...