Geometry and Mensuration
Mensuration: Mensuration is the branch of mathematics which deals with the study of
Geometric shapes , Their area , Volume and different parameters in geometric objects.
Some important mensuration formulas are:
1.Area of rectangle (A) = length(l) * Breath(b);
2.Perimeter of a rectangle (P) = 2 * (Length(l) + Breath (b))
3.Area of a square (A) = Length (l) * Length (l)
4.Perimeter of a square (P) = 4 * Length (l)
5.Area of a parallelogram(A) = Length(l) * Height(h)
6.Perimeter of a parallelogram (P) = 2 * (length(l) + Breadth(b))
7.Area of a triangle (A) = (Base(b) * Height(b)) / 2
And for a triangle with sides measuring “a” , “b” and “c” , Perimeter = a+b+c
and s = semi perimeter = perimeter / 2 = (a+b+c)/2
And also . Area of triangle,
This formulas is also knows as “Hero’s formula”.
8.Area of triangle(A)
=
9. Area of isosceles triangle =
Where , a = length of two equal side , b= length of base of isosceles triangle.
10.Area of trapezium (A) =(a+b)/2
Where , “a” and “b” are the length of parallel sides.
11.Perimeter of a trapezium (P) = sum of all sides
12.Area f rhombus (A) = Product of diagonals / 2
13.Perimeter of a rhombus (P) = 4 * l
where l = length of a side
14.Area of quadrilateral (A) = 1/2 * Diagonal * (Sum of offsets)
15. Area of a Kite (A) = 1/2 * product of it’s diagonals
16. Perimeter of a Kite (A) = 2 * Sum on non-adjacent sides
17. Area of a Circle (A) =
, where , r= radius of the circle
18. Circumference of a Circle =
r= radius of circle
d= diameter of circle
19. Total surface area of
cuboid = 2(lb+bh+lh)
where , l= length , b=breadth , h=height
20. Total surface area of
cuboid = 6l2
where , l= length
21. length of diagonal of cuboid =
22. length of diagonal of cube =
23. Volume of cuboid = l * b * h
24. Volume of cube = l * l* l
25. Area of base of a cone =
26. Curved surface area of a cone =C
Where , r = radius of base , l = slanting height of cone
27. Total surface area of a cone =
28. Volume of right circular cone =
Where , r = radius of base of cone , h= height of the cone (perpendicular to base)
29. Surface area of triangular prism = (P * height) + (2 * area of triangle)
Where , p = perimeter of base
30. Surface area of polygonal prism = (Perimeter of base * height ) + (Area of polygonal
base * 2)
31. Lateral surface area of prism = Perimeter of base * height
32. Volume of Triangular prism = Area of the triangular base * height
33. Curved surface area of a cylinder =
Where , r = radius of base , h = height of cylinder
34. Total surface area of a cylinder =
35. Volume of a cylinder =
36. Surface area of sphere =
where , r= radius of sphere , d= diameter of sphere
37. Volume of a sphere =
38. Volume of hollow cylinder =
where , R = radius of cylinder , r= radius of hollow , h = height of cylinder
39. Surface area of a right square pyramid =
Where , a = length of base , b= length of equal side ;
of the isosceles triangle forming the slanting face.
40. Volume of a right square pyramid =
41. Area of a regular hexagon =
42. area of equilateral triangle =
43. Curved surface area of a Frustums =
44. Total surface area of a Frustums =
45. Curved surface area of a Hemisphere =
46. Total surface area of a Hemisphere =
47. Volume of a Hemisphere =
48. Area of sector of a circle =
The four walls and ceiling of a room of length 25 ft., breadth 12 ft. and height 10 ft. are to be
painted. Painter A can Paint 200 sqr.ft in 5 days, painter B can paint 250 sqr.ft in 2 days. If A &
B work together, in how many days do they finish the job?
A5 8/13
B5 11/12
C6 10/33
D7 6/11
Answer: Option C
Explanation:
Total area to be painted = 25*12 +2(10*12 + 10*25) = 1040 sqr.ft
A paints = 200/5 = 40 sqr.ft per day
B paints = 250/2 = 125 sqr.ft per day
A + B = 40 + 125 = 165 sqr.ft
Number of days = 1040/165 = 6 10/33
1.There
is a rectangular Garden whose length and width is 60m X 20m.There is a walkway of
uniform width around garden. Area of walkway is 516m2. Find width of walkway?
A1
B2
C3
D4
Answer: Option C
Explanation:
let the width of rectangle be x so the length & breath is increased by 2x
so new total area along with walkway is (60+2x)*(20+2x)
so (60+2x)*(20+2x)-60*20=516
=>(60+2x)*(20+2x)=1716
=>(30+x)*(10+x)=429=33*13
=>x=3
2.In a trapezoid ABCD, the diagonals intersect at E. If area of triangle ∆ABE = 72 and area of
triangle ∆CDE = 50, then area of trapezium ABCD is
A342
B242
C184
D424
Answer: Option B
Explanation:
Let h1 and h2 be the heights of the triangles CDE and ABE respectively .
The area of CDE ∆CDE = ½*CD* h1 = 50 . . . (i) ,
And area of triangle ABE, ∆ABE = ½*AB* h2 =72. . . (ii)
Also triangles ABE and CDE are similar hence CD/AB = h1/h2 = k .
Hence we get on dividing (i) by (ii)
=> CD*h1/AB*h2 = 50 / 72
=> k 2 = 25/36 (putting AB/CD = h1/h2 = k)
we get k = 5/6.
Hence area of trapezium ABCD = ½*(AB+CD)*(h1+h2) = ½*AB*h 1 (1+k)^ 2 = 72*(1+5/6)^
2= 242
3.The
dimensions of a certain machine are 48" X 30" X 52". If the size of the machine is
increased proportionately until the sum of its dimensions equals 156". What will be the increase
in the shortest side?
A6
B26
C32
DData insufficient
Answer: Option A
Explanation:
Sum of present dimension 48+30+52=130.
New dimension =156.
Increase in dimension = 26.
Ratio of dimensions = 48:30:52 =>24:15:26
Therefore increase in the shortest side15*(26)/(24+15+26)=6
4.The
length of the side of a square is represented by x+2. The length of the side of an equilateral
triangle is 2x. If the square and the equilateral triangle have equal perimeter, then the value of x
is _______
A5
B14
C17
D4
5.A cone of height 9 cm with diameter of its base 18 cm is carved out from a wooden solid
sphere of radius 9 cm. The percentage of the wood wasted is :
A25%
B30%
C50%
D75%
Answer: Option D
Explanation:
Volume of sphere = [((4/3π)*9*9*9)]cm3.
Volume of cone = [(1/3π)*9*9*9]cm3.
Volume of wood wasted = [((4/3π)*9*9*9)-((1/3π)*9*9*9)]cm3
= (π*9*9*9)cm3.
Therefore required percentage = [((π*9*9*9)/((4/3π)*9*9*9))*100]% = ((3/4)*100)% = 75%.
6.A
parallelogram has sides 30 m and 14 m and one of its diagonals is 40 m long. Then, its area
is :
A336 m2
B168 m2
C480 m2
D372 m2
Answer: Option A
Explanation:
Let ABCD be a ||gm in which AB = 30 m, BC = 14 m & AC = 40 m.
Clearly, area of || gm ABCD = 2(area of ∆ ABC).
Let a = 30, b = 14 & c = 40.
Then, s=(1/2)(a+b+c) = 42
Therefore area of ∆ ABC = √s(s-a)(s-b)(s-c)
=√42*12*28*2 = 168 m2.
Therefore area of ||gm = (2*168) m2 = 336 m2
edge of a cuboid are in the ratio 1 : 2 : 3 and its surface area is 88 cm2. The volume of the
cuboid is :
A24 cm3
B48 cm3
C64 cm3
D120 cm3
Answer: Option B
Explanation:
7.The
Let the dimensions of the cuboid be x, 2x and 3x.
Then, 2(x*2x + 2x*3x + x*3x) = 88
= 2x2 + 6x2 + 3x2 = 44 = 11x2 = 44 = x2 = 4 = x = 2.
Therefore volume of the cuboid = (2*4*6) cm3 = 48 cm3.
8.If
the radius of a circle is diminished by 10%, then its area is diminished by :
A10%
B19%
C20%
D36%
Answer: Option B
Explanation:
Let the original radius be R cm.
New radius = (90% of R) cm =[(90/100)*R] cm = 9R/10 cm.
Original area = πR2.
Diminished area = [πR2-π(9R/10)2] cm2 = [(1-(81/100) πR2] cm2 = ((19/100 πR2) cm2.
Decrease% = [(19πR2/100)*(1/πR2)*100]% = 19%.
9.A
man standing at a point P is watching the top of a tower, which makes an angle of elevation
of 30º with the man's eye. The man walks some distance towards the tower to watch its top and
the angle of the elevation becomes 60º. What is the distance between the base of the tower and
the point P?
A43 units
B8 units
C12 units
DData inadequate
Answer: Option D
Explanation:
One of AB, AD and CD must have given.
So, the data is inadequate