MENSURATION
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Mensuration is the branch of mathematics which deals with the study of different geometrical shapes, their areas and Volume. In the broadest sense, it is all about the process of measurement. It is based on the use of algebraic equations and geometric calculations to provide measurement data regarding the width, depth and volume of a given object or group of objects
 Pythagorean Theorem (Pythagoras’ theorem)
In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides
c^{2} = a^{2} + b^{2} where c is the length of the hypotenuse and a and b are the lengths of the other two sides
 Pi is a mathematical constant which is the ratio of a circle’s circumference to its diameter. It is denoted by ?
??3.14?227
 Geometric Shapes and solids and Important Formulas
Geometric Shapes  Description  Formulas 
Rectangle
l = Length b = Breadth d= Length of diagonal 
Area = lb
Perimeter = 2(l + b) d = ?l2+b2 

Square
a = Length of a side d= Length of diagonal 
Area= a*a=1/2*d*d
Perimeter = 4a d = 2?a 

Parallelogram
b and c are sides b = base h = height 
Area = bh
Perimeter = 2(b + c) 

Rhombus
a = length of each side b = base h = height d1, d2 are the diagonal 
Area = bh(Formula 1)
Area = ½*d1*d2 (Formula 2 ) Perimeter = 4a 

Triangle
a , b and c are sides b = base h = height 
Area = ½*b*h (Formula 1) Area(Formula 2) = ?S(S?a)(S?b)(S?c where S is the semiperimeter S =(a+b+c)/2 (Formula 2 for area Heron’s formula) Perimeter = a + b + c Radius of incircle of a triangle of area A =AS 

Equilateral Triangle
a = side 
Area = (?3/4)*a*a Perimeter = 3a
Radius of incircle of an equilateral triangle of side a = a/2*?3 Radius of circumcircle of an equilateral triangle


Base a is parallel to base b  Trapezium(Trapezoid in American English)
h = height 
Area = 12(a+b)h

Circle
r = radius d = diameter 
d = 2r
Area = ?r2 = 14?d2 Circumference = 2?r = ?d


Sector of Circle
r = radius ? = central angle 
Area = (?/360) *?*r*r Arc Length, s = (?/180)* ?*r In the radian system for angular measurement,


Ellipse
Major axis length = 2a Minor axis length = 2b 
Area = ?ab
Perimeter ? 

Rectangular Solid
l = length w = width h = height 
Total Surface Area = 2lw + 2wh + 2hl = 2(lw + wh + hl) Volume = lwh 

Cube
s = edge 
Total Surface Area = 6s^{2}
Volume = s^{3} 

Right Circular Cylinder
h = height r = radius of base 
Lateral Surface Area = (2 ? r)h Total Surface Area Volume = (? r^{2})h 

Pyramid
h = height B = area of the base 
Total Surface Area = B + Sum of the areas of the triangular sides
Volume = 1/3*B*h 

Right Circular Cone
h = height r = radius of base 
Lateral Surface Area=?rs where s is the slant height =?r*r+h*h Total Surface Area =?rs+?r2 

Sphere
r = radius d = diameter 
d = 2r
Surface Area =4?r*r=?d*d Volume =4/3?r*r*r=16?d*d*d 




 Important properties of Geometric Shapes
 Properties of Triangle
 Sum of the angles of a triangle = 180°
 Sum of any two sides of a triangle is greater than the third side.
 Properties of Triangle
 The line joining the midpoint of a side of a triangle to the positive vertex is called the median
 The median of a triangle divides the triangle into two triangles with equal areas
 Centroid is the point where the three medians of a triangle meet.
 Centroid divides each median into segments with a 2:1 ratio
 Area of a triangle formed by joining the midpoints of the sides of a given triangle is onefourth of the area of the given triangle.
 An equilateral triangle is a triangle in which all three sides are equal
 In an equilateral triangle, all three internal angles are congruent to each other
 In an equilateral triangle, all three internal angles are each 60°
 An isosceles triangle is a triangle with (at least) two equal sides
 In isosceles triangle, altitude from vertex bisects the base.
 Properties of Quadrilaterals
 Rectangle
 The diagonals of a rectangle are equal and bisect each other
 opposite sides of a rectangle are parallel
 opposite sides of a rectangle are congruent
 opposite angles of a rectangle are congruent
 All four angles of a rectangle are right angles
 The diagonals of a rectangle are congruent
 Square
 All four sides of a square are congruent
 Opposite sides of a square are parallel
 The diagonals of a square are equal
 The diagonals of a square bisect each other at right angles
 All angles of a square are 90 degrees.
 A square is a special kind of rectangle where all the sides have equal length
 Parallelogram
 The opposite sides of a parallelogram are equal in length.
 The opposite angles of a parallelogram are congruent (equal measure).
 The diagonals of a parallelogram bisect each other.
 Each diagonal of a parallelogram divides it into two triangles of the same area
 Rhombus
 All the sides of a rhombus are congruent
 Opposite sides of a rhombus are parallel.
 The diagonals of a rhombus bisect each other at right angles
 Opposite internal angles of a rhombus are congruent (equal in size)
 Any two consecutive internal angles of a rhombus are supplementary; i.e. the sum of their angles = 180° (equal in size)
 If each angle of a rhombus is 90°, it is a square
Other properties of quadrilaterals
 The sum of the interior angles of a quadrilateral is 360 degrees
 If a square and a rhombus lie on the same base, area of the square will be greater than area of the rhombus (In the special case when each angle of the rhombus is 90°, rhombus is also a square and therefore areas will be equal)
 A parallelogram and a rectangle on the same base and between the same parallels are equal in area.
 Of all the parallelogram of given sides, the parallelogram which is a rectangle has the greatest area.
 Each diagonal of a parallelogram divides it into two triangles of the same area
 A square is a rhombus and a rectangle.
 Sum of Interior Angles of a polygon
 The sum of the interior angles of a polygon = 180(n – 2) degrees where n = number of sides Example 1 : Number of sides of a triangle = 3. Hence, sum of the interior angles of a triangle = 180(3 – 2) = 180 × 1 = 180 ° Example 2 : Number of sides of a quadrilateral = 4. Hence, sum of the interior angles of any quadrilateral = 180(4 – 2) = 180 × 2 = 360.
Solved Examples
Level 1
 An error 2% in excess is made while measuring the side of a square. What is the percentage of error in the calculated area of the square?
 4.04 %
 2.02 %
 4 %
 2 %
Answer : Option A
Explanation :
Error = 2% while measuring the side of a square.
Let the correct value of the side of the square = 100
Then the measured value = (100×(100+2))/100=102 (? error 2% in excess)
Correct Value of the area of the square = 100 × 100 = 10000
Calculated Value of the area of the square = 102 × 102 = 10404
Error = 10404 – 10000 = 404
Percentage Error = (Error/Actual Value)×100=(404/10000)×100=4.04%
 A towel, when bleached, lost 20% of its length and 10% of its breadth. What is the percentage of decrease in area?
 30 %
 28 %
 32 %
 26 %
Answer : Option B
Explanation :
Let original length = 100 and original breadth = 100
Then original area = 100 × 100 = 10000
Lost 20% of length
=> New length =( Original length × (100?20))/100
=(100×80)/100=80
Lost 10% of breadth
=> New breadth= (Original breadth × (100?10))/100
=(100×90)/100=90
New area = 80 × 90 = 7200
Decrease in area
= Original Area – New Area
= 10000 – 7200 = 2800
Percentage of decrease in area
=(Decrease in Area/Original Area)×100=(2800/10000)×100=28%
 If the length of a rectangle is halved and its breadth is tripled, what is the percentage change in its area?
 25 % Increase
 25 % Decrease
 50 % Decrease
 50 % Increase
Answer : Option D
Explanation :
Let original length = 100 and original breadth = 100
Then original area = 100 × 100 = 10000
Length of the rectangle is halved
=> New length = (Original length)/2=100/2=50
breadth is tripled
=> New breadth= Original breadth × 3 = 100 × 3 = 300
New area = 50 × 300 = 15000
Increase in area = New Area – Original Area = 15000 – 10000= 5000
Percentage of Increase in area =( Increase in Area/OriginalArea)×100=(5000/10000)×100=50%
 The area of a rectangle plot is 460 square metres. If the length is 15% more than the breadth, what is the breadth of the plot?
 14 metres
 20 metres
 18 metres
 12 metres
Answer : Option B
Explanation:
lb = 460 m^{2} ——(Equation 1)
Let the breadth = b
Then length, l =( b×(100+15))/100=115b/100——(Equation 2)
From Equation 1 and Equation 2,
115b/100×b=460b2=46000/115=400?b=?400=20 m
 If a square and a rhombus stand on the same base, then what is the ratio of the areas of the square and the rhombus?
 equal to ½
 equal to ¾
 greater than 1
 equal to 1
Answer : Option C
Explanation :
If a square and a rhombus lie on the same base, area of the square will be greater than area of the rhombus (In the special case when each angle of the rhombus is 90°, rhombus is also a square and therefore areas will be equal)
Hence greater than 1 is the more suitable choice from the given list
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Note : Proof
Consider a square and rhombus standing on the same base ‘a’. All the sides of a square are of equal length. Similarly all the sides of a rhombus are also of equal length. Since both the square and rhombus stands on the same base ‘a’,
Length of each side of the square = a
Length of each side of the rhombus = a
Area of the sqaure = a^{2} …(1)
From the diagram, sin ? = h/a
=> h = a sin ?
Area of the rhombus = ah = a × a sin ? = a^{2} sin ? …(2)
From (1) and (2)
Area of the square/Area of the rhombus= a^{2} /a^{2}sin?=1/sin?
Since 0° < ? < 90°, 0 < sin ? < 1. Therefore, area of the square is greater than that of rhombus, provided both stands on same base.
(Note that, when each angle of the rhombus is 90°, rhombus is also a square (can be considered as special case) and in that case, areas will be equal.
 The breadth of a rectangular field is 60% of its length. If the perimeter of the field is 800 m, find out the area of the field.
 37500 m^{2}
 30500 m^{2}
 32500 m^{2}
 40000 m^{2}
Answer : Option A
Explanation :
Given that breadth of a rectangular field is 60% of its length
?b=(60/100)* l =(3/5)* l
perimeter of the field = 800 m
=> 2 (l + b) = 800
?2(l+(3/5)* l)=800?l+(3/5)* l =400?(8/5)* l =400?l/5=50?l=5×50=250 m
b = (3/5)* l =(3×250)/5=3×50=150 m
Area = lb = 250×150=37500 m2
 What is the percentage increase in the area of a rectangle, if each of its sides is increased by 20%?
 45%
 44%
 40%
 42%
Answer : Option B
Explanation :
Let original length = 100 and original breadth = 100
Then original area = 100 × 100 = 10000
Increase in 20% of length.
=> New length = (Original length ×(100+20))/100=(100×120)/100=120
Increase in 20% of breadth
=> New breadth= (Original breadth × (100+20))/100=(100×120)/100=120
New area = 120 × 120 = 14400
Increase in area = New Area – Original Area = 14400 – 10000 = 4400
Percentage increase in area =( Increase in Area /OriginalArea)×100=(4400/10000)×100=44%
 What is the least number of squares tiles required to pave the floor of a room 15 m 17 cm long and 9 m 2 cm broad?
 814
 802
 836
 900
Answer : Option A
Explanation :
l = 15 m 17 cm = 1517 cm
b = 9 m 2 cm = 902 cm
Area = 1517 × 902 cm^{2}
Now we need to find out HCF(Highest Common Factor) of 1517 and 902.
Let’s find out the HCF using long division method for quicker results
902) 1517 (1
902
—————–
615) 902 (1
 615
————–
287) 615 (2
574
—————–
41) 287 (7
287
————
0
————
Hence, HCF of 1517 and 902 = 41
Hence, side length of largest square tile we can take = 41 cm
Area of each square tile = 41 × 41 cm^{2}
Number of tiles required = (1517×902)/(41×41)=37×22=407×2=814
Level 2
 A rectangular parking space is marked out by painting three of its sides. If the length of the unpainted side is 9 feet, and the sum of the lengths of the painted sides is 37 feet, find out the area of the parking space in square feet?
 126 sq. ft.
 64 sq. ft.
 100 sq. ft.
 102 sq. ft.
Answer : Option A
Explanation :
Let l = 9 ft.
Then l + 2b = 37
=> 2b = 37 – l = 37 – 9 = 28
=> b = 282 = 14 ft.
Area = lb = 9 × 14 = 126 sq. ft.
 A large field of 700 hectares is divided into two parts. The difference of the areas of the two parts is onefifth of the average of the two areas. What is the area of the smaller part in hectares?
 400
 365
 385
 315
Answer : Option D
Explanation :
Let the areas of the parts be x hectares and (700 – x) hectares.
Difference of the areas of the two parts = x – (700 – x) = 2x – 700
onefifth of the average of the two areas = 15[x+(700?x)]2
=15×7002=3505=70
Given that difference of the areas of the two parts = onefifth of the average of the two areas
=> 2x – 700 = 70
=> 2x = 770
?x=7702=385
Hence, area of smaller part = (700 – x) = (700 – 385) = 315 hectares.
 The length of a rectangle is twice its breadth. If its length is decreased by 5 cm and breadth is increased by 5 cm, the area of the rectangle is increased by 75 sq.cm. What is the length of the rectangle?
 18 cm
 16 cm
 40 cm
 20 cm
Answer : Option C
Explanation :
Let breadth = x cm
Then length = 2x cm
Area = lb = x × 2x = 2x^{2}
New length = (2x – 5)
New breadth = (x + 5)
New Area = lb = (2x – 5)(x + 5)
But given that new area = initial area + 75 sq.cm.
=> (2x – 5)(x + 5) = 2x^{2} + 75
=> 2x^{2} + 10x – 5x – 25 = 2x^{2} + 75
=> 5x – 25 = 75
=> 5x = 75 + 25 = 100
=> x = 1005 = 20 cm
Length = 2x = 2 × 20 = 40cm
 The ratio between the length and the breadth of a rectangular park is 3 : 2. If a man cycling along the boundary of the park at the speed of 12 km/hr completes one round in 8 minutes, then what is the area of the park (in sq. m)?
 142000
 112800
 142500
 153600
Answer : Option D
Explanation :
l : b = 3 : 2 —(Equation 1)
Perimeter of the rectangular park
= Distance travelled by the man at the speed of 12 km/hr in 8 minutes
= speed × time = 12×860 (? 8 minute = 860 hour)
= 85 km = 85 × 1000 m = 1600 m
Perimeter = 2(l + b)
=> 2(l + b) = 1600
=> l + b = 16002 = 800 m —(Equation 2)
From (Equation 1) and (Equation 2)
l = 800 × 35 = 480 m
b = 800 × 25 = 320 m (Or b = 800 – 480 = 320m)
Area = lb = 480 × 320 = 153600 m^{2}
 It is decided to construct a 2 metre broad pathway around a rectangular plot on the inside. If the area of the plots is 96 sq.m. and the rate of construction is Rs. 50 per square metre., what will be the total cost of the construction?
 Rs.3500
 Rs. 4200
 Insufficient Data
 Rs. 4400
Answer : Option C
Explanation :
Let length and width of the rectangular plot be l and b respectively
Total area of the rectangular plot = 96 sq.m.
=> lb = 96
Width of the pathway = 2 m
Length of the remaining area in the plot = (l – 4)
breadth of the remaining area in the plot = (b – 4)
Area of the remaining area in the plot = (l – 4)(b – 4)
Area of the pathway
= Total area of the rectangular plot – remaining area in the plot
= 96 – [(l – 4)(b – 4)]
= 96 – [lb – 4l – 4b + 16]
= 96 – [96 – 4l – 4b + 16]
= 96 – 96 + 4l + 4b – 16
= 4l + 4b – 16
= 4(l + b) – 16
We do not know the values of l and b and hence area of the pathway cannot be found out. So we cannot determine total cost of the construction.
 A circle is inscribed in an equilateral triangle of side 24 cm, touching its sides. What is the area of the remaining portion of the triangle?
 144?3?48? cm^{2}
 121?3?36? cm^{2}
 144?3?36? cm^{2}
 121?3?48? cm^{2}
Answer : Option A
Explanation :
Area of an equilateral triangle = (3/?4)*a *a where a is length of one side of the equilateral triangle
Area of the equilateral ? ABC = (3/?4)*a *a = (3/?4)*24*24=144?3 cm2? (1)
Area of a triangle = 12bhwhere b is the base and h is the height of the triangle
Let r = radius of the inscribed circle. Then
Area of ? ABC
= Area of ? OBC + Area of ? OCA + area of ? OAB
= (½ × r × BC) + (½ × r × CA) + (½ × r × AB)
= ½ × r × (BC + CA + AB)
= ½ x r x (24 + 24 + 24)
= ½ x r x 72 = 36r cm^{2} —(2)
From (1) and (2),
144?3=36r?r=144?3/36=4?3????(3)
Area of a circle = ?r2 where = radius of the circle
From (3), the area of the inscribed circle = ?r2=?(4?3)* (4?3)=48??(4)
Hence, area of the remaining portion of the triangle
= Area of ? ABC – Area of inscribed circle
144?3?48? cm2
 What will be the length of the longest rod which can be placed in a box of 80 cm length, 40 cm breadth and 60 cm height?
 ?11600 cm
 ?14400 cm
 ?10000 cm
 ?12040 cm
Answer : Option A
Explanation :
The longest road which can fit into the box will have one end at A and other end at G (or any other similar diagonal).
Hence the length of the longest rod = AG
Initially let’s find out AC. Consider the right angled triangle ABC
AC^{2} = AB^{2} + BC^{2} = 40^{2} + 80^{2} = 1600 + 6400 = 8000
?AC = ?8000 cm
Consider the right angled triangle ACG
AG^{2} = AC^{2} + CG^{2}
(?8000)^{ 2}+60^{2}=8000+3600=11600
=> AG = ?11600 cm
=> Length of the longest rod = ?11600cm
 A rectangular plot measuring 90 metres by 50 metres needs to be enclosed by wire fencing such that poles of the fence will be kept 5 metres apart. How many poles will be needed?
 30
 44
 56
 60
Answer : Option C
Explanation :
Perimeter of a rectangle = 2(l + b)
where l is the length and b is the breadth of the rectangle
Length of the wire fencing = perimeter = 2(90 + 50) = 280 metres
Two poles will be kept 5 metres apart. Also remember that the poles will be placed along the perimeter of the rectangular plot, not in a single straight line which is very important.
Hence number of poles required = 280/5 = 56
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