Mathematics solution NCERT
Class 9 - Chapter 6: Measuring Space: Perimeter and Area
1. The perimeter of a circle is 44 cm. What is its radius?
Solution:
We know that the perimeter (circumference) of a circle is:
C = 2πr
Given: C = 44 cm
Therefore,
2πr = 44
r = 44 / 2π
Taking π = 22/7,
r = 44 / (2 × 22/7)
r = 44 × 7/44
r = 7 cm
Answer: The radius of the circle is 7 cm.
2. Calculate, correct to 3 significant figures, the circumference of a circle with:
(i) Radius 7 cm
Solution:
We know that circumference C = 2πr.
Given: r = 7 cm
C = 2 × π × 7
C = 14π
C ≈ 43.9823 cm
Correct to 3 significant figures, C = 44.0 cm.
Answer: 44.0 cm
(ii) Radius 10 cm
Solution:
Given: r = 10 cm
C = 2πr
C = 2 × π × 10
C = 20π
C ≈ 62.8319 cm
Correct to 3 significant figures, C = 62.8 cm.
Answer: 62.8 cm
(iii) Radius 12 cm
Solution:
Given: r = 12 cm
C = 2πr
C = 2 × π × 12
C = 24π
C ≈ 75.3982 cm
Correct to 3 significant figures, C = 75.4 cm.
Answer: 75.4 cm
3. Calculate the length of the arc of a circle if:
We use the formula:
Arc length = (θ/360°) × 2πr
where θ is the angle at the centre and r is the radius.
(i) Radius = 3.5 cm and angle = 60°
Solution:
Given: r = 3.5 cm and θ = 60°
Arc length = (60°/360°) × 2π × 3.5
= (1/6) × 7π
= 7π/6 cm
Using π = 22/7,
Arc length = 7 × (22/7) / 6
= 22/6 cm
≈ 3.67 cm
Answer: The arc length is approximately 3.67 cm.
(ii) Radius = 6.3 m and angle = 120°
Solution:
Given: r = 6.3 m and θ = 120°
Arc length = (120°/360°) × 2π × 6.3
= (1/3) × 12.6π
= 4.2π m
Using π = 22/7,
Arc length = 4.2 × 22/7
= 13.2 m
Answer: The arc length is 13.2 m.
4. Find the perimeter of a sector of a circle of radius 14 cm and sector angle 75°.
Solution:
The perimeter of a sector consists of its arc length and two radii.
Therefore,
Perimeter of a sector = Arc length + 2r
Given: r = 14 cm and θ = 75°
First, calculate the arc length.
Arc length = (θ/360°) × 2πr
= (75°/360°) × 2 × π × 14
= (75/360) × 28π
= (5/24) × 28π
= 35π/6 cm
Taking π = 22/7,
Arc length = (35/6) × (22/7)
= 55/3 cm
≈ 18.33 cm
Now, the perimeter of the sector is:
Perimeter = Arc length + 2r
= 18.33 + 2 × 14
= 18.33 + 28
= 46.33 cm (approximately)
Answer: The perimeter of the sector is approximately 46.33 cm.
5. Find the perimeters of the following shapes (taking the arcs to be quarter or half or three-quarters of a circle, as appropriate) (Fig. 6.14i to 6.14ix):
(i) Find the perimeter of the given shape.
Solution:
The shape consists of two straight lines of length 80 m each and two semicircles of diameter 60 m.
Two semicircles together form one complete circle.
Perimeter = Sum of straight lengths + Circumference of the circle
= 80 + 80 + πd
= 160 + π × 60
= 160 + 60π
Taking π = 22/7,
= 160 + 60 × 22/7
= 160 + 188.57
= 348.57 m (approximately)
Answer: The perimeter is 160 + 60π m, or approximately 348.57 m.
(ii) Find the perimeter of the given shape.
Solution:
The shape consists of an outer semicircle of diameter 12 cm, an inner semicircle of diameter 8 cm, and two straight line segments.
Outer radius = 12/2 = 6 cm
Inner radius = 8/2 = 4 cm
Each straight segment = (12 - 8)/2 = 2 cm
Total length of straight segments = 2 + 2 = 4 cm
Length of the outer semicircular arc = πR
= 6π cm
Length of the inner semicircular arc = πr
= 4π cm
Perimeter = 6π + 4π + 4
= 10π + 4
Taking π = 22/7,
= 10 × 22/7 + 4
= 220/7 + 28/7
= 248/7 cm
≈ 35.43 cm
Answer: The perimeter is 10π + 4 cm, or approximately 35.43 cm.
(iii) Find the perimeter of the given shape.
Solution:
The shape consists of four semicircles, each having a diameter of 10 cm.
Radius of each semicircle = 10/2 = 5 cm
Length of one semicircular arc = πr
= 5π cm
Total perimeter = 4 × 5π
= 20π cm
Taking π = 22/7,
= 20 × 22/7
= 440/7 cm
≈ 62.86 cm
Answer: The perimeter is 20π cm, or approximately 62.86 cm.
(iv) Find the perimeter of the given shape.
Solution:
The shape consists of three semicircles constructed outward on the three equal sides of an equilateral triangle.
Diameter of each semicircle = 12 cm
Radius of each semicircle = 12/2 = 6 cm
Length of one semicircular arc = πr
= 6π cm
Total perimeter = 3 × 6π
= 18π cm
Taking π = 22/7,
= 18 × 22/7
= 396/7 cm
≈ 56.57 cm
Answer: The perimeter is 18π cm, or approximately 56.57 cm.
(v) Find the perimeter of the given shape.
Solution:
Solution:
The boundary of the shape consists of four quadrants of radius 14 cm and four semicircles of radius 7 cm.
Radius of each quadrant = 14 cm
Radius of each semicircle = 7 cm
Four quadrants together form one complete circle.
Length of four quadrants = 2πr
= 2 × π × 14
= 28π cm
Four semicircles together form two complete circles.
Length of four semicircles = 4πr
= 4 × π × 7
= 28π cm
Total perimeter = 28π + 28π
= 56π cm
Taking π = 22/7,
= 56 × 22/7
= 8 × 22
= 176 cm
Answer: The perimeter of the shape is 176 cm.
(vi) Find the perimeter of the given shape.
Solution:
The shape consists of one large semicircle and four small semicircles along the same baseline.
Diameter of the large semicircle = 28 cm
Radius of the large semicircle = 28/2 = 14 cm
Length of the large semicircular arc = π × 14
= 14π cm
The diameter of the large semicircle is divided into four equal parts.
Diameter of each small semicircle = 28/4 = 7 cm
Radius of each small semicircle = 7/2 = 3.5 cm
Length of one small semicircular arc = π × 3.5
= 3.5π cm
Total length of four small semicircular arcs = 4 × 3.5π
= 14π cm
Total perimeter = 14π + 14π
= 28π cm
Taking π = 22/7,
= 28 × 22/7
= 88 cm
Answer: The perimeter is 88 cm.
(vii) Find the perimeter of the given shape.
Solution:
The shape consists of three semicircles constructed outward on the three sides of a right-angled triangle.
The two perpendicular sides are 6 cm and 8 cm.
Let the hypotenuse be c.
By the Pythagorean theorem,
c² = 6² + 8²
c² = 36 + 64
c² = 100
c = 10 cm
Thus, the diameters of the three semicircles are 6 cm, 8 cm and 10 cm.
Perimeter = (π/2) × (6 + 8 + 10)
= (π/2) × 24
= 12π cm
Taking π = 22/7,
= 12 × 22/7
= 264/7 cm
≈ 37.71 cm
Answer: The perimeter is 12π cm, or approximately 37.71 cm.
(viii) Find the perimeter of the given shape.
Solution:
The shape consists of one large semicircle and three smaller semicircles.
Diameter of the large semicircle = 4 + 4 + 4 = 12 cm
Radius of the large semicircle = 12/2 = 6 cm
Length of the large semicircular arc = π × 6
= 6π cm
Diameter of each small semicircle = 4 cm
Radius of each small semicircle = 4/2 = 2 cm
Length of one small semicircular arc = π × 2
= 2π cm
Total length of three small semicircular arcs = 3 × 2π
= 6π cm
Total perimeter = 6π + 6π
= 12π cm
Taking π = 22/7,
= 12 × 22/7
= 264/7 cm
≈ 37.71 cm
Answer: The perimeter is 12π cm, or approximately 37.71 cm.
(ix) Find the perimeter of the given shape.
Solution:
The shape consists of one large semicircle and two smaller semicircles.
Diameter of the large semicircle = 10 + 10 = 20 cm
Radius of the large semicircle = 20/2 = 10 cm
Length of the large semicircular arc = π × 10
= 10π cm
Diameter of each small semicircle = 10 cm
Radius of each small semicircle = 10/2 = 5 cm
Length of one small semicircular arc = π × 5
= 5π cm
Total length of two small semicircular arcs = 2 × 5π
= 10π cm
Total perimeter = 10π + 10π
= 20π cm
Taking π = 22/7,
= 20 × 22/7
= 440/7 cm
≈ 62.86 cm
Answer: The perimeter is 20π cm, or approximately 62.86 cm.
6. If the diameter of a car tyre is 56 cm, then: (i) How far does the car need to travel for the tyre to complete one revolution? (ii) How many revolutions does the tyre make if the car travels 10 km?
(i) How far does the car need to travel for the tyre to complete one revolution?
Solution:
Given: Diameter of the tyre = 56 cm
Distance travelled in one revolution = Circumference of the tyre
We know that,
Circumference = πd
Taking π = 22/7,
Circumference = (22/7) × 56
= 22 × 8
= 176 cm
Answer: The car travels 176 cm (or 1.76 m) in one revolution.
(ii) How many revolutions does the tyre make if the car travels 10 km?
Solution:
Distance travelled by the car = 10 km
Converting kilometres into centimetres,
10 km = 10 × 1000 × 100 cm
= 10,00,000 cm
Distance travelled in one revolution = 176 cm
Number of revolutions = Total distance travelled / Distance travelled in one revolution
= 10,00,000 / 176
= 56,81,81/100?
= 10,00,000 / 176
= 5681.8181...
Answer: The tyre makes approximately 5,681.82 revolutions.
Since a revolution count is often expressed as a whole number, the tyre completes 5,681 full revolutions and is partway through the next revolution after travelling 10 km.
7. Find the total perimeter of all the petals in each of the given flowers.
(i) Flower with a square of side 14 cm
Solution:
Side of the square = 14 cm
The centres of the arcs are the midpoints of the sides of the square.
Radius of each arc = 14/2 = 7 cm
Each petal consists of two quarter-circle arcs.
Therefore, the perimeter of one petal is equal to the length of two quarter-circle arcs, which together form a semicircle.
Length of one semicircular arc = πr
= (22/7) × 7
= 22 cm
Number of petals = 4
Total perimeter of all the petals = 4 × 22
= 88 cm
Answer: The total perimeter of all the petals is 88 cm.
(ii) Flower with a regular hexagon of side 42 cm
Solution:
Side of the regular hexagon = 42 cm
Radius of each arc = 42 cm
The flower consists of 6 identical petals. Each petal is formed by two arcs, each subtending an angle of 60° at its centre.
Length of one arc = (θ/360°) × 2πr
Substituting θ = 60° and r = 42 cm,
Length of one arc = (60°/360°) × 2 × (22/7) × 42
= (1/6) × 2 × 22 × 6
= 44 cm
Each petal consists of two such arcs.
Perimeter of one petal = 2 × 44
= 88 cm
Number of petals = 6
Total perimeter of all the petals = 6 × 88
= 528 cm
Answer: The total perimeter of all the petals is 528 cm.
8. The ratio of the perimeters of two circles is 5 : 4. What is the ratio of their radii?
Solution:
Let the radii of the two circles be r1 and r2, respectively.
We know that the perimeter (circumference) of a circle is:
C = 2πr
According to the question,
C1 : C2 = 5 : 4
Therefore,
2πr1 : 2πr2 = 5 : 4
Cancelling the common factor 2π, we get:
r1 : r2 = 5 : 4
Answer: The ratio of their radii is 5 : 4.