Time, Speed and Distance 75
F Shortcut Approach
Æ If speed of stream is a and a boat (swimmer) takes n times as long
to row up as to row down the river, then
a(n + 1)
Speed of boat (swimmer) in still water =
(n - 1)
Note: This formula is applicable for equal distances.
Æ If a man capable of rowing at the speed (u) m/sec in still water, rows
the same distance up and down a stream flowing at a rate of (v) m/
sec, then his average speed through the journey is
Upstream ´ Downstream (u - v) (u + v)
= =
Man 's rate in still water u
Æ If boat's (swimmer's) speed in still water is a km/h and river is
flowing with a speed of b km/h, then average speed in going to a
certain place and coming back to starting point is given by
(a + b)(a - b)
km/h.
a
ebooks Reference Page No.
Solved Examples – S-37-42
Exercises with Hints & Solutions – E-71-77
Chapter Test – 17-18
Past Solved Papers
76 Mensuration
Chapter
10 Mensuration
MENSURATION
Mensuration is the science of measurement of the lenghts of lines, areas
of surfaces and volumes of solids.
Perimeter
Perimeter is sum of all the sides. It is measured in cm, m, etc.
Area
The area of any figure is the amount of surface enclosed within its boundary
lines. This is measured in square unit like cm2, m2, etc.
Volume
If an object is solid, then the space occupied by such an object is called its
volume. This is measured in cubic unit like cm3, m3, etc.
Basic Conversions :
I. 1 m = 10 dm
1 dm = 10 cm
1 cm = 10 mm
1 m = 100 cm = 1000 mm
1 km = 1000 m
5
II. 1 km = miles
8
1 mile = 1.6 km
1 inch = 2.54 cm
III. 100 kg = 1 quintal
10 quintal = 1 tonne
1 kg = 2.2 pounds (approx.)
IV. l litre = 1000 cc
1 acre = 100 m2
1 hectare = 10000 m2 (100 acre)
Mensuration 77
PART I : PLANE FIGURES
TRIANGLE
B
a c
h
C A
b
Perimeter (P) = a + b + c
Area (A) = s ( s – a )( s – b )( s – c )
a +b+c
where s = and a, b and c are three sides of the triangle.
2
1
Also, A = ´ bh ; where b ® base
2
h ® altitude
Equilateral triangle
a a
a
Perimeter = 3a
3 2
A= a ; where a ® side
4
78 Mensuration
Right triangle
p h
b
1
A = pb and h 2 = p2 + b2 (Pythagoras triplet)
2
where p ® perpendicular
b ® base
h ® hypotenuse
RECTANGLE
Perimeter = 2 ( l + b)
Area = l × b; where l ® length
b ® breadth
F Shortcut Approach
If the length and breadth of a rectangle are increased by a% and b%,
æ ab ö
respectively, then are will be increased by ç a + b + ÷ %.
è 100 ø
Mensuration 79
SQUARE
Perimeter = 4 × side = 4a
Area = (side)2 = a2; where a ® side
PARALLELOGRAM
a h a
Perimeter = 2 (a + b)
Area = b × h; where a ® breadth
b ® base (or length)
h ® altitude
RHOMBUS
a
d1 d2
Perimeter = 4 a
1
Area = d1 ´ d 2 where a ® side and
2
d1 and d2 are diagonals.
80 Mensuration
IRREGULAR QUADRILATERAL
p h1 q
d
s h2 r
Perimeter = p + q + r + s
1
Area = ´ d ´ ( h1 + h 2 )
2
TRAPEZIUM
a
m
h n
b
Perimeter = a + b + m + n
Area = 1 ( a + b ) h ; where (a) and (b) are two parallel sides;
2
(m) and (n) are two non-parallel sides;
h ® perpendicular distance
between two parallel sides.
AREA OF PATHWAYS RUNNING ACROSS THE
MIDDLE OF A RECTANGLE
a b
A = a ( l + b) – a2; where l ® length
b ® breadth,
a ® width of the pathway.
Mensuration 81
Pathways outside
b + 2a b
+ 2a
A = (l + 2a) (b + 2a) – lb; where l ® length
b ® breadth
a ® width of the pathway
Pathways inside
l – 2a
b b – 2a
a
l
A = lb – (l – 2a) (b – 2a); where l ® length
b ® breadth
a ® width of the pathway
F Shortcut Approach
Æ If a pathway of width x is made inside or outside a rectangular plot
of length l and breadth b, then are of pathway is
(i) 2x (l + b + 2x), if path is made outside the plot.
(ii) 2x (l + b – 2x), if path is made inside the plot.
Æ If two paths, each of width x are made parallel to length (l) and
breadth (b) of the rectangular plot in the middle of the plot, then
area of the paths is x(l + b – x)
82 Mensuration
CIRCLE
Perimeter (Circumference) = 2pr = pd
Area = pr2; where r ® radius
d ® diameter
r
22
and p = or 3.14
7
F Shortcut Approach
The length and breadth of a rectangle are l and b, then are of circle of
pb 2
maximum radius inscribed in that rectangle is .
4
SEMICIRCLE
Perimeter = pr + 2r
1
Area = ´ pr 2
2 r
F Shortcut Approach
The are a of the largest triangle incribed in a semi-circle of radius r is
equal to r 2.
SECTOR OF A CIRCLE
q O
Area of sector OAB = ´ pr 2
360
q r
q
Length of an arc (l) = ´ 2 pr A B
360 l Segment
Area of segment = Area of sector – Area of triangle OAB
q 1
= ´ pr 2 – r 2 sin q
360° 2
Mensuration 83
Perimeter of segment = length of the arc + length of segment
prq q
AB = + 2r sin
180 2
RING
R2
R1
(
Area of ring = p R 22 – R12 )
PART-II SOLID FIGURE
CUBOID
A cuboid is a three dimensional box.
Total surface area of a cuboid = 2 (lb + bh + lh)
Volume of the cuboid = lbh
l b
Area of four walls = 2(l + b) × h
F Shortcut Approach
If length, breadth and height of a cuboid are changed by x%, y% and
z% respectively, then its volume is increased by
é xy + yz + zx xyz ù
= êx + y + z + + ú%
ë 100 (100)2 û
84 Mensuration
Note: Increment in the value is taken as positive and decrement in
value is taken as negative. Positive result shows total increment and
negative result shows total decrement.
CUBE
A cube is a cuboid which has all its edges equal.
Total surface area of a cube = 6a2
Volume of the cube = a3
a
a
a
RIGHT PRISM
A prism is a solid which can have any polygon at both its ends.
Lateral or curved surface area = Perimeter of base × height
Total surface area = Lateral surface area + 2 (area of the end)
Volume = Area of base × height
RIGHT CIRCULAR CYLINDER
It is a solid which has both its ends in the form of a circle.
Lateral surface area = 2prh
Total surface area = 2pr (r + h)
Volume = pr2h; where r is radius of the base and h is the height
Mensuration 85
PYRAMID
A pyramid is a solid which can have any polygon at its base and its edges
converge to single apex.
Lateral or curved surface area
1
= (perimeter of base) × slant height
2
Total surface area = lateral surface area + area of the base
1
Volume = (area of the base) × height
3
RIGHT CIRCULAR CONE
It is a solid which has a circle as its base and a slanting lateral surface that
converges at the apex.
Lateral surface area = prl
Total surface area = pr (l + r)
1
Volume = pr 2 h ; where r : radius of the base
3
h : height
l : slant height
86 Mensuration
l
h
SPHERE
It is a solid in the form of a ball with radius r.
Lateral surface area = Total surface area = 4pr2
34
Volume = pr ; where r is radius.
3
HEMISPHERE
It is a solid half of the sphere.
Lateral surface area = 2pr2
Total surface area = 3pr2
Volume = 2 pr 3 ; where r is radius
3
r
r
Mensuration 87
F Shortcut Approach
If side of a cube or radius (or diameter) of sphere is increased by x%,
éæ x ö
3
ù
then its volume increases by ç 100 ÷ - 1ú ´100%
ê 1 +
ëêè ø ûú
F Shortcut Approach
If in a cylinder or cone, height and radius both change by x%, then
éæ x ö
3
ù
volume changes by êç 100 ÷ - 1ú ´100%
1 +
ëêè ø ûú
FRUSTUM OF A CONE
When a cone cut the left over part is called the frustum of the cone.
Curved surface area = pl (r 1 + r2)
Total surface area = pl ( r1 + r2 ) + pr12 + pr22 r1
where l = h 2 + ( r1 – r2 ) 2
h
( )
1 2 2 r2
Volume = ph r1 + r1r2 + r2
3
ebooks Reference Page No.
Solved Examples – S-43-47
Exercises with Hints & Solutions – E-78-87
Chapter Test – 19-20
Past Solved Papers
88 Clock and Calendar
Chapter
11 Clock and Calendar
CLOCK
Introduction
• A clock has two hands : Hour hand and Minute hand.
• The minute hand (M.H.) is also called the long hand and the hour
hand (H.H.) is also called the short hand.
• The clock has 12 hours numbered from 1 to 12.
Also, the clock is divided into 60 equal minute divisions. Therefore, each
hour number is separated by five minute divisions. Therefore,
F Shortcut Approach
360
Æ One minute division =
60
= 6° apart. ie. In one minute, the minute
hand moves 6°.
Æ One hour division = 6° × 5 = 30° apart. ie. In one hour, the hour hand
moves 30° apart.
30° 1°
Also, in one minute, the hour hand moves = = apart.
60° 2
Æ Since, in one minute, minute hand moves 6° and hour hand moves
1° 1°
, therefore, in one minute, the minute hand gains 5 more
2 2
than hour hand.
1°
Æ In one hour, the minute hand gains 5 ´ 60 = 330° over the hour
2
hand. i.e. the minute hand gains 55 minutes divisions over the hour
hand.
