NCERT Solutions for Class 10 Maths Chapter 12 - Facts that Matter and Areas Related to Circles (Excercise 12.1)

Facts that Matter

• The total distance (perimeter) around a circle is called its circumference.
• The plane surface enclosed in a circle is called its area.
• If ‘r’ be the radius of a circle, then
  (i) Circumference of the circle = 2πr
  (ii) Area of the circle = πr²

Note:
(i) The interior of a circle along with its boundary is called the circular region of the circle.
(ii) By the area of a circle, we mean the area of the circular region.

➢ Area of Sector and Segment of a Circle 

• The portion (or part) of the circular region enclosed by two radii and the corresponding arc is called a sector of the circle.
  
• The portion (or part) of the circular region enclosed between a chord and the corresponding arc is called a segment of the circle.
  

Note:
(i) ∠AOB is called the ‘angle of sector’.
(ii) OAPB is the ‘minor sector’ and OAQB is the ‘major sector’.
(iii) APB is the ‘minor segment’ and AQB is the ‘major segment’.
(iv) When we write ‘sector’ and ‘segment’ we will mean the ‘minor-sector’ and the ‘minor-segment’ respectively.

➢ Let us remember that

• Area of the sector of ‘angle
• Length of the arc of a sector of angle  
• Area of a segment = [Area of the corresponding sector] − [Area of the corresponding triangle]

Page No. 225

Exercise 12.1

[Unless stated otherwise, use π = 22/7]
Q.1. The radii of the two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has circumference equal to the sum of the circumferences of the two circles.
Sol. We have, r₁ = 19 cm
► r₂ = 9 cm
► Circumference of circle-I = 2πr₁ = 2π (19) cm
► Circumference of circle-II = 2πr₂ = 2π (9) cm
► Sum of the circumferences of circle-I and circle-II = 2π (19) + 2π (9)
= 2π (19 + 9) cm = 2π (28) cm
Let R be the radius of the circle-III.
∴ Circumference of circle-III = 2πR
According to the condition:
► 2πR = 2π (28)

Thus, the radius of the new circle = 28 cm

Q.2. The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having area equal to the sum of the areas of the two circles.
Sol. We have, Radius of circle-I, r₁ = 8 cm
► Radius of circle-II, r₂ = 6 cm
► Area of circle-I = πr₁² = π (8)² cm²
► Area of circle-II = πr₂² = π (6)² cm²
Let the area of the circle-III be R
∴ Area of circle-III = πR²
Now, according to the condition:

π r₁² + π r₂² = πR²
i.e. π (8)² + π (6)² = πR²
⇒ π (8² + 6²)= πR²
⇒ 8² + 6² = R²
⇒ 64 + 36 = R²
⇒ 100 = R²
⇒ 102 = R² ⇒ R = 10
Thus, the radius of the new circle = 10 cm.

Q.3. Figure depicts an archery target marked with its five scoring regions from the centre outwards as Gold, Red, Blue, Black and White. The diameter of the region representing 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.
Sol. Diameter of the innermost region = 21 cm
Radius of the innermost (Gold Scoring) region = 21/2 = 10.5 cm
► Area of Gold region = π (10.5)² cm²

► Area of the Red region


► Area of Blue region

► Area of Black region

► Area of White region
= π [(42 + 10.5)2] = (42)² cm

Q.4. The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is travelling at a speed of 66 km per hour?
Sol. The diameter of a wheel = 80 cm
∴ Radius of the wheel = 80/2 = 40 cm
∴ Circumference of the wheel

⇒ Distance covered by a wheel in one revolution  
Distance travelled by the car in 1hr = 66 km = 66 × 1000 × 100 cm
∴ Distance travelled in 10 minutes

Now,
Number of revolutions

Thus, the required number of revolutions = 4375

Q.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
Ans. (a)
Sol. We have
[Numerical area of the circle] = [Numerical circumference of the circle]
⇒ π r² = 2πr
⇒π r² − 2πr = 0
⇒ r² − 2r = 0
⇒ r (r − 2) = 0r = 0 or r = 2
But r cannot be zero
∴ r = 2 units.
Thus, option (a) is correct.