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Mensuration
Areas
Areas of some well-known figures are given below:
Perimeter in Do You Know?
S. Area in
Name Figure units of
No square units
length Square:
Diagonal = a 2
b Rectangle:
1. Rectangle 2(a + b) ab
a Diagonal = a 2 + b2
a = length
Parallelogram:
b = breadth
The diagonals bisec
a a2 teach other. Sum of
2. Square 4a o
1 adjacent angles = 180
a (diagonal)2
a = side 2 Rhombus:
a The diagonals cut at
b
right angles
h b
1 2 1
a2 = ( d1) + ( d2)2
Parallelogra a 2 2
3. 2(a + b) ah
m a = side
b = side adjacent to a
h = distance between
the opp. parallel sides
a
d1 Quadrilateral
a 1 T
d2 a d1d2 Parallelogram r
2 a
Rhombus a p
4a Rectangle e
4. a = side of rhombus; z
d1,d2 are the two i
Square u
diagonals m
D C
h1
h2
A B
1
5 Quadrilateral Sum of its (AC) (h1 +
AC is one of its 2
four sides
diagonals and h1, h2 h2)
are the altitudes on
AC from D, B
respectively.
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b
1
Trapezium Sum of its four h(a + b)
a 2
sides
6. a, b, are parallel sides
and h is the distance
between parallel sides
a c
h
1
b
bh
a + b + c = 2s 2
7. Triangle b is the base and h is where s is the or
the altitude. semi perimeter. s(s a)(s b)(s c)
a, b, c are three sides
of .
d
h
Right
8. b+h+d 1
triangle b bh
d(hypotenuse) 2
= b 2 + h2
a a
h
1
(i) ah
Equilateral a 2
9. 3a
triangle 3 2
a = side (ii) a
4
3
h = altitude = a
2
a a
Isosceles c 4a 2 c 2
10. 2a + c
triangle c 4
c = unequal side
a = equal side
d
a
a
d(hypotenuse) 1 2
11. 2a + d a
Isosceles =a 2 2
right triangle a = Each of equal
sides.
The angles are 90o,
45o, 45o.
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r
12. Circle 2r r2
r = radius of the circle
22
= or 3.1416
7
13. Semicircle
r r 1 2
r + 2r r
r = radius of the circle 2
R
Ring r
14. (shaded . (R2 r2)
region)
R = outer radius
r = inner radius
r o r
l + 2r
Sector of a l where l =
15. r2
circle o = central angle of 360
2r
the sector 360
r = radius of the
sector l = length of
the arc
Ex.1 A horizontal pipe for carrying floodwater
has diameter 1 m. When water in it is 10 cm
deep then the width of water surface AB is equal
A B 10 cm
to ________ cm.
(1) 10 cm (2) 25 cm
(3) 60 cm (4) 70 cm
(5) None
100
Sol. Let O be the centre of circle.
O
In OAD
50 cm 40 cm 50 cm
502 = 402 + x2
2
2500 1600 = x A x D x B
x2 = 900
10 cm
x = 30
AB = 2x = 60 cm. Answer: (3)
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Ex.2 Area of the shaded region is
r
4 3 3 4 2 4 8 3 2
(1) r (2) r
4 4
4 3 3 + 4 2 4 3 3 4 2 60o 60
o
(3) r (4) r
4 4
(5) None of these
1 3 3 2 3 3 2
Sol. Area of the semi-hexagonal patch = r = r
2 2 4
r
Area of the semi-square patch = 2r = r2
2
3 3 2
Area of shaded region = r2 r2 r
4
( 4 4 3 3 ) 2
= r. Answer: (1)
4
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Volume MENION
Nature Lateral/ Total surface
S. No of the Shape of the solid curved area Volume Abbreviations
solid surface area Used
l = length
h b = breath
1. Cuboid 2h (l + b) 2(lb + bh + lh) lbh h = height
b
l
a a = length of
2 2 3
4a 6a a edge
2. Cube a
a
2 (area of one
(perimeter of
Right end) + lateral Area of
3. base)
prism surface area base
Height
height
r
r = radius of
Right h base
4. circular 2rh 2r(r + h) r2h h = height of
cylinder r the cylinder
1 1
(Perimeter Area of the (Area of
2 3
5. of the base) base + lateral
Right base)
(slant height) surface area
pyramid height
h l h = height
1 2
6. Right r(l + r) r h r = radius
r
r l 3 l = slant height
circular
cone
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Lateral/
S. Nature of Shape of the Total surface
curved Volume Abbreviations
No the solid solid area
surface area Used
r = radius
r
4 3
7. 4r2 r
3
Sphere
r = radius
r
2 3
8. 2r2 3r2 r
Hemi- 3
sphere
R
R = outer radius
r r = inner radius
4
9. 2
4 (R r )
2
(R3r3)
Spherical 3
shell
R = larger radius
R r = smaller
Volume of h 2 2
10. h (R + r + Rr) radius
bucket 3
h = height
r
Length of a diagonal:
The length of diagonal of a cuboid = l2 + b 2
+ h 2
The length of a diagonal of a cube = a 3
Ex.1 An insect is crawling to climb a 20 metre long pipe of diameter 3.5 metres. If the insect take 4
circular rounds to reach the top of the building, then what will be total distance (in nearer
metres) traveled by insect to reach the top?
(1) 47 m (2) 48 m (3) 49 m (4) 50 m
Sol. Since the insect is moving in circular way, so it will cover the distance which will be 4 times the distance
of the longest side AB of a right angled triangle. In the figure AO is equal to the circumference of the
pipe.
5
22
AO = d = x3.5 = 11 m
7
5
BO = 5 m
AB = 112 + 5 2 = 146 = 12.08 m 5
Total distance traveled = 12.08 4 = 48.32 m B
5
Hence distance in nearer meters = 48 m
A 2r O
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Ex.2 The surface area of a cube is 726 sq metres. Find the volume of the cube.
(1) 1234 m2 (2) 1431 m2 (3) 1341 m (4) 1331 m (5) None of these
2
Sol. Surface area S = 726 m Volume V = ?
In a cube,
S = 6a2 = 726 a2 = 121 a = 11 m
So, V = a3 = 113 = 1331 m3. Answer: (4)
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