Class 9 Maths Ganita Manjari Chapter 6 Exercise 6.1 Solutions – Measuring Space: Perimeter and Area
Perimeter is the total distance around the boundary of a shape, and the perimeter of a circle is called circumference.
Class 9 Maths Ganita Manjari Chapter 6 Exercise 6.1 connects this idea with arc length, sector perimeter, curved shapes, tyre revolutions and circle ratios.
Chapter 6, Measuring Space: Perimeter and Area Class 9, begins with the 4 × 100 m relay track, where athletes in outer lanes start ahead because curved lanes have larger radii. This idea leads to circumference, arc length and curved boundaries. Class 9 Maths Ganita Manjari Chapter 6 Exercise 6.1 Solutions cover radius from circumference, circumference for given radii, arc length, sector perimeter, mixed curved-shape perimeters, car tyre revolutions, flower-petal perimeters and ratio of radii. The textbook asks students to use 22/7 for π unless stated otherwise.
Class 9 Maths Ganita Manjari Chapter 6 Exercise 6.1 Solutions
Q1. The perimeter of a circle is 44 cm. What is its radius?
The radius of the circle is 7 cm.
Given:
Perimeter of circle = 44 cm
Use:
Circumference = 2Ï€r
Ï€ = 22/7
So:
44 = 2 × 22/7 × r
44 = 44r/7
r = 44 × 7 / 44
r = 7 cm
Answer:
The radius of the circle is 7 cm.
Q2. Calculate, correct to 3 significant figures, the circumference of a circle with: (i) radius 7 cm (ii) radius 10 cm (iii) radius 12 cm.
The circumferences are 44.0 cm, 62.9 cm and 75.4 cm to 3 significant figures.
Use:
Circumference = 2Ï€r
Ï€ = 22/7
Q2(i). Radius = 7 cm
Circumference = 2 × 22/7 × 7
Circumference = 44 cm
To 3 significant figures:
Circumference = 44.0 cm
Q2(ii). Radius = 10 cm
Circumference = 2 × 22/7 × 10
Circumference = 440/7
Circumference = 62.857...
To 3 significant figures:
Circumference = 62.9 cm
Q2(iii). Radius = 12 cm
Circumference = 2 × 22/7 × 12
Circumference = 528/7
Circumference = 75.428...
To 3 significant figures:
Circumference = 75.4 cm
Answer:
(i) 44.0 cm
(ii) 62.9 cm
(iii) 75.4 cm
Q3. Calculate the length of the arc of a circle if: (i) the radius is 3.5 cm and the angle at the centre is 60°, and (ii) the radius is 6.3 m and the angle at the centre is 120°.
The arc lengths are 11/3 cm and 13.2 m.
Use:
Arc length = 2πr × θ/360
Q3(i). Radius = 3.5 cm, angle = 60°
Arc length = 2 × 22/7 × 3.5 × 60/360
Arc length = 22 × 1/6
Arc length = 22/6
Arc length = 11/3 cm
Answer:
Arc length = 11/3 cm
Q3(ii). Radius = 6.3 m, angle = 120°
Arc length = 2 × 22/7 × 6.3 × 120/360
Arc length = 2 × 22/7 × 6.3 × 1/3
Arc length = 13.2 m
Answer:
Arc length = 13.2 m
Q4. Find the perimeter of a sector of a circle of radius 14 cm and sector angle 75°.
The perimeter of the sector is 139/3 cm.
Given:
Radius = 14 cm
Sector angle = 75°
Use:
Sector perimeter = arc length + 2r
First find arc length:
Arc length = 2πr × θ/360
Arc length = 2 × 22/7 × 14 × 75/360
Arc length = 88 × 75/360
Arc length = 55/3 cm
Now add the two radii:
Sector perimeter = 55/3 + 14 + 14
Sector perimeter = 55/3 + 28
Sector perimeter = 55/3 + 84/3
Sector perimeter = 139/3 cm
Answer:
The perimeter of the sector is 139/3 cm.
Q5. Find the perimeters of the following shapes in Fig. 6.14.
Use π = 22/7.
Q5(i)
The perimeter is 2440/7 m, or 348.6 m approximately.
The shape has two straight portions of 80 m each and two semicircles of diameter 60 m.
Two semicircles make one full circle.
Perimeter = 80 + 80 + π × 60
Perimeter = 160 + 22/7 × 60
Perimeter = 160 + 1320/7
Perimeter = 1120/7 + 1320/7
Perimeter = 2440/7 m
Perimeter = 348.57 m
Answer:
Perimeter = 348.6 m
Q5(ii)
The perimeter is 248/7 cm.
The shape has an outer semicircle of diameter 12 cm, an inner semicircle of diameter 8 cm and two straight base parts.
Outer semicircle length = 1/2 × π × 12
Outer semicircle length = 6Ï€
Inner semicircle length = 1/2 × π × 8
Inner semicircle length = 4Ï€
Total straight parts = 12 - 8
Total straight parts = 4 cm
Perimeter = 6Ï€ + 4Ï€ + 4
Perimeter = 10Ï€ + 4
Perimeter = 10 × 22/7 + 4
Perimeter = 220/7 + 28/7
Perimeter = 248/7 cm
Answer:
Perimeter = 248/7 cm
Q5(iii)
The perimeter is 440/7 cm.
The boundary is made of four semicircles, each with diameter 10 cm.
One semicircle length = 1/2 × π × 10
One semicircle length = 5Ï€
Perimeter = 4 × 5π
Perimeter = 20Ï€
Perimeter = 20 × 22/7
Perimeter = 440/7 cm
Answer:
Perimeter = 440/7 cm
Q5(iv)
The perimeter is 396/7 cm.
The boundary is made of three semicircular arcs, each with diameter 12 cm.
One semicircle length = 1/2 × π × 12
One semicircle length = 6Ï€
Perimeter = 3 × 6π
Perimeter = 18Ï€
Perimeter = 18 × 22/7
Perimeter = 396/7 cm
Answer:
Perimeter = 396/7 cm
Q5(v)
The perimeter is 88 cm.
The boundary is made of four semicircles, each with diameter 14 cm.
One semicircle length = 1/2 × π × 14
One semicircle length = 7Ï€
Perimeter = 4 × 7π
Perimeter = 28Ï€
Perimeter = 28 × 22/7
Perimeter = 88 cm
Answer:
Perimeter = 88 cm
Q5(vi)
The perimeter is 80 cm.
The boundary has one large semicircle of diameter 28 cm and two smaller semicircular arcs. The remaining boundary includes two straight parts.
Large semicircle length = 1/2 × π × 28
Large semicircle length = 14Ï€
Two small semicircles together have arc length:
7Ï€
Straight parts together:
14 cm
Perimeter = 14Ï€ + 7Ï€ + 14
Perimeter = 21Ï€ + 14
Perimeter = 21 × 22/7 + 14
Perimeter = 66 + 14
Perimeter = 80 cm
Answer:
Perimeter = 80 cm
Q5(vii)
The perimeter is 264/7 cm.
The figure uses semicircles on the three sides of a right triangle with sides 6 cm, 8 cm and 10 cm.
Hypotenuse:
10 cm
Perimeter = 1/2 × π × 6 + 1/2 × π × 8 + 1/2 × π × 10
Perimeter = 3Ï€ + 4Ï€ + 5Ï€
Perimeter = 12Ï€
Perimeter = 12 × 22/7
Perimeter = 264/7 cm
Answer:
Perimeter = 264/7 cm
Q5(viii)
The perimeter is 264/7 cm.
The large semicircle has diameter 12 cm.
The three smaller semicircles each have diameter 4 cm.
Large semicircle length = 1/2 × π × 12
Large semicircle length = 6Ï€
Three small semicircles:
3 × 1/2 × π × 4 = 6π
Perimeter = 6Ï€ + 6Ï€
Perimeter = 12Ï€
Perimeter = 264/7 cm
Answer:
Perimeter = 264/7 cm
Q5(ix)
The perimeter is 440/7 cm.
The figure has one large semicircle of diameter 20 cm and two smaller semicircular arcs of diameter 10 cm each.
Large semicircle length = 1/2 × π × 20
Large semicircle length = 10Ï€
Two smaller semicircles:
2 × 1/2 × π × 10 = 10π
Perimeter = 10Ï€ + 10Ï€
Perimeter = 20Ï€
Perimeter = 440/7 cm
Answer:
Perimeter = 440/7 cm
Q6. 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?
The car travels 176 cm in one revolution, and the tyre makes 62500/11 revolutions in 10 km.
Given:
Diameter = 56 cm
Use:
Distance in one revolution = circumference
Circumference = πd
Circumference = 22/7 × 56
Circumference = 176 cm
Q6(i)
Answer:
The car travels 176 cm in one revolution.
Q6(ii)
Convert 10 km into cm:
10 km = 10 × 1000 m
10 km = 10000 m
10000 m = 10000 × 100 cm
10000 m = 1000000 cm
Number of revolutions:
Revolutions = total distance ÷ distance in one revolution
Revolutions = 1000000 ÷ 176
Revolutions = 1000000/176
Revolutions = 62500/11
Revolutions = 5681.8 approximately
Answer:
The tyre makes 62500/11 revolutions, or about 5682 revolutions.
Q7. Find the total perimeter of all the petals in each of the given flowers.
Q7(i)
The total perimeter of all petals is 88 cm.
The square has side 14 cm.
Each petal boundary is made of two quarter-circle arcs of radius 7 cm.
There are 4 petals.
Total arcs = 4 × 2
Total arcs = 8 quarter-circle arcs
One quarter-circle arc length:
Arc length = 1/4 × 2πr
Arc length = πr/2
Arc length = 7Ï€/2
Total perimeter of all petals:
8 × 7π/2 = 28π
28π = 28 × 22/7
28Ï€ = 88 cm
Answer:
The total perimeter of all petals is 88 cm.
Q7(ii)
The total perimeter of all petals is 528 cm.
The flower has 6 petals inside a regular hexagon of side 42 cm.
Each petal has two arcs.
Total arcs:
6 × 2 = 12 arcs
Each arc has radius 42 cm and centre angle 60°.
Arc length = 2πr × 60/360
Arc length = 2π × 42 × 1/6
Arc length = 14Ï€
Total perimeter:
12 × 14π = 168π
168π = 168 × 22/7
168Ï€ = 528 cm
Answer:
The total perimeter of all petals is 528 cm.
Q8. The ratio of the perimeters of two circles is 5:4. What is the ratio of their radii?
Answer:
Let the radii of the two circles be r1​ and r2​.
The ratio of the perimeters of the two circles = 5 : 4
Thus,
=
=
Hence, the ratio of their radii is 5 : 4.