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Class: Subject: Topic: VIII Mathematics Mensuration
I. Areas of square, Rectangle, triangle and circle.
You should know:
What is a Triangle?
What is a Square?
What is a Rectangle?
What is a Circle?
You will learn:
how to calculate area and perimeter of all above figures.
The list of Subtopics Why these? Why should I care? Common Mistakes
Areas of square, rectangle,triangle and circle.
Area of a Trapezium.
Area of a Quadrilateral andspecial quadrilateralsRhombus & parallelogram.
Area of Cube, Cuboidscylinder
Volume of cube cuboidcylinder.
This topic involves a lot ofcalculation - so many errorsare possible, in calculating.Inafility to covelate the givendata to arrive at an answer -for instance “What is given”“What can I assume” “Whatdo I need find” etc.Confusion between variousdiffeent types ofQuadrilaterls - theirproperties and formulas.
This topic is very useful incalculation of areas andvolumes in mathematics aswell as in physics.Important for board examsas well as competetiveexams in future.Study of mensurationexpands our understandingof other irregular shapestoo.Useful in making modelsets.
All around us there areobjects - some havingrgular shape some irregularshape, is it not wonderful tobe able to measure thesurface area or volume andcompare.For instance you can planout wonderful interiors,furnishings etc. bymeasuring the size of yourroom, walls etc.Look at the clothes thatyou wear or the bags thatyou carry, they have beenmade by using the conceptsof mensuration.
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Fill up the blanks:
Figure Perimeter Area
1)
2)
3)
4)
4 x side
2 x ( ) x
Sum of 12 x x
x
The area of a square whose side is 4 cm is
The Length of a recangle is twice the breadth. If area is 200 sqcm.
Find its perimeter
If base and height of a triangle are 13cm and 6cm find area
The area and perimeter of a circle happen to be equal in magnitude. Then the diameter
of this circle =
The sides of a triangle are 6cm, 8cm and 10cm. The area is
Now consider this question:
A B
D C
P Q
S R
45
35
80
60
In this given figure PQRS is a Rectangle of sides80cm & 60cm.
ABCD is a rectangle inside PQRSof sides 45cmand 35cm.
Find area of the shaded portion. Also findperimeter of the shaded portion.
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Let us solve:
Area of the shaded portion is the difference in areas between outer rectangle and innerrectangle.
is area of shaded = (Area of PQRS) - (Area of ABCD)
Portion = (60 x 80) - (35 x 45)
= 4800 - 1575
= 3225 sqcm.
The perimeter of the shaded portion is the sum of the perimeters of the 2 rectangles.
= 2 (60 + 80) + 2 (35 + 45)
= 440cm.
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WORKSHEET
Now you solve these questions:
1) Find areas of
b) 7cm7cm
find area of shadedportion
D
A
C
B8cm
8cm
a)
c)
find area of ABC
A
BC
12cm
20cm
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e) 20
10
4
4
Find area of shaded portion.
d)
25
25cm3cm
5cm5cm
5cm
Find area of shaded portion.
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2) Find the areas of the following figures:
P
Q
R
S
T
UX
Y
UQRT is rectangle
UQ = 25 cm
UT = 10 cm
PS = 32 cm
PX = YS
a)
b) A B
D C
ABCD is a rectangle of sides
AB = 49 cm
AD = 14 cm
The semi circles are drawn using the length andbreadth as diameter.
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3) A floor is in the shape of a rectangle of sides 15m and 12m. Square tiles of side 1.5m areto be used for flooring. If the cost of a tile is Rs.50/- find total cost of flooring.
c)
35cm
Find area of shaded portion.
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4)
D 25cm
A
CB
12cm
30cm
In this le ABC,
AD is attitude on BC.
Find the magnitude of the attitude on AC.
5) There is a Triangle park which has sides AB = 12m, BC = 20m and AC = 16m. Findthe least distance a person has to travel from A to reach the road connecting B & C.
6) The area of a square is equal to five times the area of a rectangle of dimentions 20cmby 64cm. Find the perimeter of this square.
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II. Area of a Trapezium
You should know:
What is a trapezium?
You will learn:
How to calculate the area of a trapezium?
A
D C
Ba cm
b cm
hcm
ABCD is a trapezium with AB // DC
AB = a cm
DC = b cm
and the distance between AB & DC is h cm
Area = 12 x ( a + b ) x h
write in words: Area of trapezium equals
____________________
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1) Find the Area of the following Trapeziums.
A
D C
B
6 cm
6 cm12
cm
A B
CD
7cm
20cm
( a )
Solution: 12 x ( AD + BC ) x DC
= 12 x (12 + 6) x 6
= 54 sqcm.
( b )
2) The area of a trapezium whose parallel sides measure 25cm & 35cm is 300 sq.cm.Find the distance between the parallel sides.
3) There is a trapezium in which the longer of the two parallel sides is twice the shorterside and the distance between the two parallel sides is half the shorter side. If area ofthis trapezium.
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4) The area of a trapezium is 126 sq.cm. The height of the trapezium is 7cm. If one of thebases is longer than the other by 6cm, find the lengths of the bases.
5) In the adjoining figure AB || DC and DA is perpendicular to AB. Further DC = 7cm,CB = 10 cm and AB = 13 cm. Find the area of the quadrilateral ABCD.
6) The area of a field in the shape of a trapezium measures 1440 m2. The perpendiculardistance between its parallel sides is 24 m. If the ratio of the parallel sides is 5 : 3,find the length of the longer parallel side.
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7) The lengths of the parallel sides of a trapezium are in the ratio 5 : 3 and the distancebetween them is 12.5cm. If the area of the trapezium is 450 cm2, find the lengths ofits parallel sides.
8) The lengths of parallel sides of a trapezium are x & y cm and area of the trapezium is
12 (x2 - y2) cm2. Find the distance between the parallel sides (in cm) in terms of x & y.
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III. Areas of Quadrilaterals and special Quadrilaterals - Rhombus & Parallelogram.
You should know:
What is a Quadrilaterals?
What is a Parallelogram?
What is a Rhombus?
Identify the similarities & differences between a parallelogram of a Rhombus.
You wil learn:
Areas of a parallelogram = base x height
Notice that Area of le = 12 base x height that is because a parallelogram readly comprises
of two congruent triangles of base & height as the parallelogram.
height
base
height
base
1
2height
base
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Area of a Quadrilaterals is 12 x diagonal x (sum of the two offsets)
12 x DB x (AX + CY)
A
B
C
DX
Y
Area of a Rhombus:
The formula for a parallelogram is also applicable to the Rhombus
to Area = base x height
base
height
Additinally Area = 12 x d1 x d2
d1 & d2 are length of the two diagonals.
d1
d2
Note: A square is also a Rhombus
area of sq = side x side and also
12 (diagonal)
For example:
Find area of a square whose diagonal is 8 cm.
Ans: 12 8 8 = 32 sqcm.
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WORKSHEET
AB
C
D
L
K
Find area of ABCD is
BD = 15cm
AK = 10cm
CL = 12cm
1)
Solution:
We know area = 12 BD (AK + CL)
= 12 15 ( 10 + 12 ) = 165 sqcm.
2)C
B
A D
E
F
PQ
RS
Find area of this polygon ABCDEF if
AD = 18 cm
AS = 14 cm
AR = 12 cm
AQ = 8 cm
AP = 4 cm
also BP = 5 cm
CR = 6 cm
FQ = 5 cm
ES = 5 cm
Hint: Find areas of le ABP, CRD, SED, AQF also find areas of trapeziums BCRP & QFESadd them up.
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3)A
B
C
D
E
F X
Y
Find area of the given figure ABCDEF
if AD = 36 cm
BC = 12 YD
FE // AD // BC
AB = 20 cm
YB = 12 cm
FX = 8 cm
FE = AY
4) A
B
CD
h1
h2
In this Quadrilateral
area = 351 sq.cm
DB = 27 cm
h1 = 12 cm
find h2
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5) Find areas of
7cm
10 cm(a)
(b)
8cm
7cm
6
3
6
10
(c)
6) Find the area of a Rhombus whose side measures 13cm and one of its diagonals is10cm.
(Hint: Use the pythogoras theorem)
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7) Find the perimeter of a Rhombus whose diagonals measure 12cm and 16cm.
8) The diagonals of two square are in the ratio of 2 : 5. Find the ratio of their areas.
9) What is the perimeter of a square, if the length of its diagonals is 12 2cm?
10) One of the diagonals of a rhombus is double the other diagonal. Its area is 25 sq.cm.Find the sum of the diagonals.
11) The perimeter of a rhombus is 56 m and its height is 5 m. Its area is:
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12) Find the perimeter, diagonal and area of a square having each side 8m
long. Take 2 1.41
13) A field in the form of a parallelogram has one of its diagonals 42m long and theperpendicular distance of this diagonal form either of the outlying vertices is 10.8m.Find the area of the field.
14) A parallelogram has sides of 15 cm and 12 cm. If the distance between its shortersides is 7.5 cm, find the distance between its longer sides.
15) One diagonal of a parallelogram is 70 cm and the perpendicular distance of thisdiagonal from either of the outlying vertices is 27 cm. Find the area of the parallelogram.
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16) Matrix Matching:
Column - A Column - B
1) Area of Square a) Sum of length of three sides
2) Perimeter of square b)12 d2
3) Area of Rectangle c) r2
4) Perimeter of Rectangle d) length breadth
5) Area of le e)12 d1 d2
6) Perimeter of le f)12 d (h1 + h2)
7) Area of Circle g)12 h (a + b)
8) Perimeter of Circle h) twice the sum of length of breadth
9) Area of Quadrilater i) 2 r
10) Area of parallelogram j)12 base height
11) Area of Trapezium k) base height
12) Area of Rhombus l) fourtimes the side
13) Area of kite
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Some HOTS questions:
1) There is a Rhombus whose area is 21sqcm and the perimeter is 40cm. Find the sumof lengths of diagonals.
2) If the length and breadth of a rectangle are increased by 3cm, its area increases by72sqcm. If the length alone is increased by 1cm the area increases by 9sqcm.Find length & breadth.
3) A square shed of side 7m is in the middle of a huge grass field. A cow is tied to one ofits corners outside the shed, with a rope of length 14m. What is the area that the cowcan graze assuming that if cannot enter inside the shed.
4) A wire is shaped into a square, it enclosed an area of 100sqcm. If this wire is remodelledto form a semicircle find area of the semicircle.
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IV. Areas of cube, cuboid, cylinder.
You should know:
Cube, cuboid, cylinders belong to the family of prisms. Prisms are these dimensional objectswhose bases are parallel to the tops and the base & top are congruent shapes.
All prisms have two types of areas.
a) Lateral surface area - the area of the walls (L.S.A)
b) Total surface area is the sum of Lateral surface area with twice the area of the base (T.S.A)
Explain the following with appropriate figures:
a) cube b) cuboid c) cylinder
You will learn:
L.S.A of cube = 4 (side)2
T.S.A of cube = 6 (side)2
L.S.A of cuboid = 2 ( l + b ) h
T.S.A of cuboid = 2 (lh + bh + lh)
L.S.A of cylinder = 2 r h
T.S.A of cylinder = 2 r (r + h)
- S - l
h
b
Cube Cuboid
h
- r -
Cylinder
You may also understand that the
L.S.A = Perimeter of the base height.
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For example:
a) Find L.S.A and T.S.A of a cube of side 8cm.
Ans: L.S.A = 4 S2 = 4 8 8 = 256 sqcm.
T.S.A = 6 S2 = 6 8 8 = 384 sqcm.
b) Find L.S.A & T.S.A of a cuboid whose height is twice the breadth and length is twicethe height. And the breadth = 10 cm.
Solution:
b = 10cm
h = 2 10 = 20cm.
l = 2 h = 2 20= 40cm.
L.S.A = 2 ( l + b ) h
= 2 (40 + 10 ) 20 = 2000 sqcm.
T.S.A = 2 ( l b + bh + l h )
= 2 ( 40 10 + 10 20 + 40 20 )
= 2 ( 400 + 200 + 800 )
= 2800 sqcm.
c) A rectangle sheet of dimensions 88cm by 20cm is rolled along its length so that thetwo breadths are joined to form a cylinder.
Find L.S.A & T.S.A of this cylinder.
Solution:
If one can imagine in this cylinder.
88
2020
Perimeter = 88cm
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Perimeter of base = 2 r = 88cm.
2 227 r = 88
r = 7 x 882 x 22 = 14cm.
L.S.A = 2 r h
= 2 227 14 20 = 88 20
= 1760 sqcm.
(Note: The area of the rect.sheet is the same as L.S.A.of this cylinder)
T.S.A = 2 r (r + h)
= 2 227 14 (14 + 20)
= 2992 sqcm.
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Now solve the following questions yourself:
1) A road roller is in the shape of a cylinder. The radius of the cross section is 14cm andlength is 1m. What is the area covered by it in making 200 revlutions.
2) The breadth and height of a cuboid 4cm and 2cm respectively. If the total surfacearea of the cuboid is 88 sqcm find length.
3) A metallic trunk is to be made from a rectangle sheet of metal bearing
dimensions l = 80 cm
b = 60 cm
h = 50 cm.
If the metal costs Rs. 250/- per 100 sqcm and making charges for the trunk is Rs.500/-. Find total cost incurred.
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4) Two cubes of side 7cm are joined along a face to form a cuboid. Find L.S.A & T.S.A ofthis cuboid.
5) The T.S.A of a cuboid is 214 sqcm. The areas of 2 of its faces are 42 sqcm and 35sqcm. Find the length of the cuboid if it is the greatest of the dimensions.
6) There is a square sheet of side 30cm. From each of its four corners 4 squares of side5cm are cut away. The rest of the sheet is folded to form an open cuboidal box. FindL.S.A of this box.
5cm
5cm
5cm5cm
Fold along the alloted line to get a open box.
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V. Volume of Cube, Cuboid, Cylinder.
You should know:
What is Cube,
What is Cuboid,
What is Cylinder.
You wil learn:
Formula for volume of
Cube, Cuboid, Cylinder
Volume of a Cube = (Side)3
Volume of a Cuboid = l b h
Volume of Cylinder = r2h
Note: a) Volume has its units as cubic units
b) Volume of a prism area of base height.
1) Find volume of
a) Cube of side 5cm
b) Cuboid of l = 10cm, b = 6cm, h = 4cm.
c) Cylinder of base area 154 sqcm and height 10cm.
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2) There is a reservoir of the shape of a cylinder whose base radius is 21m and height is15m. How many hours 2il it take to fill it with water flows at the rate of 45m3 perhour.
3) The ratio of volumes of 2 cubes is 729 : 1331. Find ratio of their T.S.A’s.
4) There is a cubical wooden block of side 7cm. A cylinder which has a height of 7cmand base rasting on one of the faces of the cube is carved out. Find the volume ofwood that is left.
5) A square sheet has a side of 60cm. Four small squares of side 6cm are cut from thefour corners. The rest is then folded to form an open cuboid box. Find the volume ofthis box.
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6) A wall has dimensions 15m x 10m x 8m. 10% of this wall is occupied by martar. Therest 90% is occupied by bricks. The cost of 1000 bricks of dimensions 10cm x 8cm x4cm is 400/-. Find total cost of making this wall.
7) A cylindrical vessel of diameter 48cm has water to a heght of 10cm. A metal cube of14cm edge is immersed in it. Calculate the height to which the water in the vesselrises.
8) A swimming pool 150m long and 50m wide in deep at the shallow end and 6m deep atthe dep end. Find volume of the pool.
9)
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Name these figures:
( a ) ( b ) ( c )
Is this a cylinder
If not then find out its name?
Fill up the blanks:
1) A cube has edges.
2) Amount of region occuped by a solid is called
3) 1 liter = cm3
1 m3 = litres..
4) Sum of areas of faces of a solids is called its
5) Cube, cublid & cylinder belong to the familes of
