Exercise
2.3
1. Find the area of a circular pathway
whose outer radius is 32 cm and inner radius is 18 cm.
Outer radius R = 32 cm
Inner radius r = 18 cm
Area of the circular pathway = π (R2 – r2)
= 22/7 (322–182) cm2
= 22/7 (32 +18)(32 –18) cm2
= 22/7 × 50 × 14 cm2
= 2200 cm2
Area of the Circular pathway = 2200 cm2.
2. There is a circular lawn of radius
28 m. A path of 7 m width is laid around the lawn. What will be the
area of the path?
Inner radius of the circle r = 28 m
Width of the path w = 7m
Outer radius of the circle R = r + w
= 28 + 7 = 35 m
Area of the pathway = π (R2 – r2)
= π (R+ r)(R+ r)
= 22/7 (35 + 28)(35 – 28) m2
= 22/7 × 63 ⊂ 7m2
= 1386 m2
Area of the pathway = 1386 m2.
3. A circular carpet whose radius is 106 cm is laid on a circular hall of radius 120 cm. Find the area of the hall uncovered by the carpet.
Radius of the circular Hall R = 120 cm
Radius of the circular carpet r = 106 cm
The area of the hall uncovered by the carpet = πR2 – πr2
= π (R2 – r2)
= π (R+ r)(R – r)
= 22/7 (120 +106)(120 –106) cm2
= 22/7 × 226 × 14 cm2
= 9944 cm2
The area of the hall uncovered by the Carpet = 9944 cm2.
4.
A school ground is in the shape of a circle with radius 103 m. Four tracks
each of 3 m wide has to be constructed inside the ground for the purpose
of track events. Find the cost of constructing the track at the rate of ₹50 per
sq.m.
Outer radius of the circular ground R = 103 m
Width of the track w = 3m
Inner radius of the ground r = R – w
= (103 –3) m = 100 m
Area of the track = π (R2 – r2)
= 22/7 (1032 – 1002) m2
= 22/7 (10609 – 10000) m2
= 22/7 × 609 = 1914 m2
Area of the track = 1914 m2
The cost of constructing the track for 1 sq.m = ₹ 50
The cost of constructiog 1914 squ.m = ₹ 1914 × 50
= ₹ 95700
The cost of constructing the track = ₹ 95700.
5. The figure shown is the aerial view
of the pathway.Find the area of the pathway.
Area of the outer rectangle = 80 × 50 m2
= 4000 m2
Area of the inner rectangle = 70 × 40 = 2800 m2
The area of the pathway = Area of the outer rectangle – Area of
the inner rectangle.
= 4000m2 – 2800m2 = 1200 m2
6.
A rectangular garden has dimensions 11 m × 8 m. A path of 2 m wide
has to be constructed along its sides. Find the area of the path.
Area of the outer rectangular garden = 11 × 8 = 88 m2
Width of the path = 2 m
Area of the inner rectangular garden = (11 – 4) (8 – 4) m2
= 7 × 4 = 28 m2
Area of the path = Outer Area – Inner Area
= (88 – 28)m2 =
60 m2
Area of the path = 60 m2.
7. A picture is painted on a ceiling
of a marriage hall whose length and breadth are 18 m and 7 m respectively.
There is a border of 10 cm along each of its sides. Find the area of the
border.
Area of the ceiling = 18 × 7 = 126 m2
Width ofthe border w = 10
cm = 0.1m
Inner area of the picture = (18 – 0.2) (7 – 0.2)m2
= 17.8 × 6.8 = 121.04 m2
Area ofthe border = Area ofthe Ceiling – Area ofthe Picture
= (126 – 121.04)m2 = 4.96 m2
The area of the border = 4.96 m2.
8.
A canal of width 1 m is constructed all along inside the field which is 24
m long and 15 m wide. Find (i) the area of the canal (ii) the cost
of constructing the canal at the rate of ₹12 per sq.m.
Outer area of the field = 24 × 15 m2 = 360 m2
Width of the Canal = 1 m
Inner area of the field = (24 – 2) (15 – 2) m2
= 22 × 13m2 =
286m2
i) Area of the canal = Outer area of the field – Inner area of
the field.
= (360 – 286) m2 = 74m2
ii) The cost of constructing the canal for 1 sq.m = ₹ 12
The cost of constructing the canal for 74 m2 = ₹ 12 ×
74
= ₹ 888
The cost of constructing the canal = ₹ 888
Objective type questions
9.
The formula to find the area of the circular path is
(i) π(R 2 −
r2 ) sq. units
(ii) πr2 sq. units
(iii) 2πr2 sq. units
(iv) πr2 +
2r sq. units
Answer: (i)
π (R2 – r2) sq. units
10.
The formula used to find the area of the rectangular path is
(i) π(R2 −
r2 ) sq. units
(ii) (L ×
B) − (l × b) sq. units
(iii) LB sq. units
(iv) lb sq. units
Answer: (ii)
(L × B) – (l × b) sq. units
11.
The formula to find the width of the circular path is
(i) ( L −
l ) units
(ii) ( B −
b ) units
(iii) ( R −
r ) units
(iv) ( r −
R ) units
Answer: (iii)
(R – r) units
ANSWERS:
Exercise 2.3
1. 2200 cm2
2. 1386 m2
3. 9944 cm2
4. ₹95700
5. 1200 m2
6. 60 m2
7. 4.96 m2
8. (i) 74 m2 (ii) ₹888
Objective type questions
9. (i) π( R2
− r2 ) sq.units
10. (ii) (L × B) − (l
× b) sq.units
11. (iii) R − r
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