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
1. 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
2. Pi is a mathematical constant which is the ratio of a circle's circumference to its diameter. It is denoted by π
π≈3.14≈227
3. 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 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 = πr^{2} =1/4πd^{2} Circumference = 2πr = πd
 
Sector of Circle r = radius θ = central angle  Area = (θ/360) *π*r*r
 
Ellipse Major axis length = 2a Minor axis length = 2b  Area = πab Perimeter ≈  
Rectangular Solid l = length w = width h = height  Total Surface Area 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 Total Surface Area Volume = (π 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=πrs Total Surface Area  
Sphere r = radius d = diameter  d = 2r Surface Area =4πr*r=πd*d Volume =4/3πr*r*r=16πd*d*d 
4. Important properties of Geometric Shapes
a. Properties of Triangle
Sum of the angles of a triangle = 180°
b. Properties of Quadrilaterals
B. Square
C. Parallelogram
D. Rhombus
Other properties of quadrilaterals
c. 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
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?
A. 4.04 %
B. 2.02 %
C. 4 %
D. 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%
2. A towel, when bleached, lost 20% of its length and 10% of its breadth. What is the percentage of decrease in area?
A. 30 %
B. 28 %
C. 32 %
D. 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%
3. If the length of a rectangle is halved and its breadth is tripled, what is the percentage change in its area?
A. 25 % Increase
B. 25 % Decrease
C. 50 % Decrease
D. 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%
4. 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?
A. 14 metres
B. 20 metres
C. 18 metres
D. 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
5. 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?
A. equal to ½
B. equal to ¾
C. greater than 1
D. 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 θ = h/a
=> h = a sin θ
Area of the rhombus = ah = a × a sin θ = a^{2} sin θ ...(2)
From (1) and (2)
Area of the square/Area of the rhombus= a^{2} /a^{2}sinθ=1/sinθ
Since 0° < θ < 90°, 0 < sin θ < 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.
6. 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.
A. 37500 m^{2}
B. 30500 m^{2}
C. 32500 m^{2}
D. 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
7. What is the percentage increase in the area of a rectangle, if each of its sides is increased by 20%?
A. 45%
B. 44%
C. 40%
D. 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 /Original Area)×100=(4400/10000)×100=44%
8. 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?
A. 814
B. 802
C. 836
D. 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
9. 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?
A. 126 sq. ft.
B. 64 sq. ft.
C. 100 sq. ft.
D. 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.
10. 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?
A. 400
B. 365
C. 385
D. 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.
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