Mensuration

May 30 • Bank, Banking Awareness Notes • 404 Views • No Comments on Mensuration

Mensuration Notes

Area: Area of a bounded figure is the space covered by it.
Perimeter: The length of the boundary of a closed figure is called perimeter

Triangle:
 Figure: Triangle

Area of a triangle = Area of Triangle
Area of Triangle
Where a, b, c are the lengths of the sides of triangle

 

  1. Area of an Equilateral Triangle =  Area of an equilateral triangle
    Perimeter of an Equilateral Triangle with side ‘a’ = 3a
    equilateral triangle
  1. Area of Isosceles triangle = Area of isosceles triangle
    And height of Isosceles triangle,height of isosceles triangle
    Perimeter of Isosceles triangle = 2b + a
     isosceles triangle

Square:
Square

Area of a Square = Area of Square Or, 

area where d is the diagonal of a square
Perimeter of a square = 4a
Diagonal of a square = Diagonal of a square
Area of a path which is outside of a square of uniform width,
uniform width x
Area of a path which is inside of a square of uniform width,
Capture1
Perimeter of a path which is outside of a square of uniform width,
Capture2
Perimeter of a path which is inside of a square of uniform width,
Capture3 

Rectangle:
Rectangle

Area of a rectangle = l * b, where l is the length of a rectangle and b is the breadth of a rectangle
Perimeter of a rectangle = 2(l + b)
Diagonal of a rectangle,
Diadonal
Area of a path which is outside of a rectangle of uniform width,
Capture
Area of a path which is inside of a rectangle of uniform width,
Capture1
Perimeter of a path which is outside of a square of uniform width,
Capture2 

Perimeter of a path which is outside of a square of uniform width,
Capture3

 

Parallelogram:
Parallelogram:

Area of Parallelogram =Area of parallelogram

Rhombus:

rhombus

  • It is a parallelogram whose all sides are equal
  • Its diagonal bisect each other at right angle.

Area of rhombus = rhombus

Area = rhombus

Where b is the side of rhombus, h is the height and d1,d2are the diagonals of the rhombus

Perimeter of rhombus = 4b

Side of rhombus = rhombus

 

Trapezium:

 

Area of trapezium  = 12height (sum of parallel sides)

Median of trapezium =  12(sum of parallel sides)

Median is the line segment joining the midpoints of non parallel side

 

Circle:

 

Area of a circle= r2

Diameter (d) = 2r

Circumference = 2r

Area of a sector =360 r2

Length of arc = 360 2r

Area of the ring or circular path = (R+r) (R-r)

 

Cube:

Volume of a cube = a3

Diagonal of a cube = 3a

Surface area of a cube = 6a2

Area of four walls = 4a2

 

Cuboid:

 

Volume of a cuboid = lbh

Diagonal of a cuboid = l2+ b2+h2

Surface area of a cuboid = 2(lb + bh + hl)

Area  of four walls = 2(lh + bh)

 

Cylinder:

 

Volume of a cylinder = r2h

Curved surface area  = 2rh

Total surface area = 2r(r + h)

 

Cone:

 

Volume of a cone = 13r2h

Curved surface area = rl

Total surface area = r (l + r)

Slant height = h2+ r2

 

Sphere:

 

Volume of a sphere = 43r3

Surface area = 4r2

 

Hemisphere:

 

Volume of a hemisphere = 23r3

Curved surface area = 2r2

Total surface area = 3r2

 

Frustum of a cone

 

Volume = 13h(r2+R2+ r.R)

Surface area = (r + R)(R -r)2+h2

Total surface area =  (r + R)(R -r)2+h2 + r2 + R2

Slant height, l = (R -r)2+h2

 

Right pyramid

Volume of a pyramid = 13(area of the base)height

Lateral surface  area of a pyramid = 12(perimeter of base) slant height

Total surface area of a pyramid = lateral surface area + area of the base

 

                  

 

                         Work out problems

 

  1. The area of a rectangular field is 460 square metres. If the length is 15 percent more than the breadth, what is breadth of the rectangular field ?
  1. 15
  2. 26
  3. 34.5
  4. Cannot be determined
  5. None of these

Answer: e) none of these

 

Solution: Let breadth of a rectangle be x

Then length = x + 15x/100

                   = x + 3x/20

                   = 23x/20

Area of a rectangle = 460

x 23x20 = 460

xx = 400

x = 20

 

  1. If the length of the diagonal AC of a square ABCD is 5.2 cm, then the area of the square is :
  1. 15.12 sq. cm
  2. 13..52 sq.cm
  3. 12.62 sq.cm
  4. 10.00 sq. cm

Answer: b) 13.52 sq.cm

 

Solution: since the square is a rhombus and the diagonals of a square are equal

Area of a rhombus =  12 product of diagonals

                              = 125.2 5.2

                              = 2.65.2

                              = 13.52

 

3.if the diagonal of two squares are in the ratio of 2:5, their area will be in the ratio of

  1. 2:5
  2. 2:5
  3. 4:25
  4. 4:5

Answer: c) 4:25

 

Solution: let the diagonals of the square be 2x and 5x respectively

Area of one square =  122x 2x

                          = 4x2/2

Area of other square =  125x 5x

                                 = 25x2/2

Ratio of areas of both the square = 4x22/ 25x22

                                                     = 4/25

                                                    = 4:25

 

  1. From four corners of a square sheet of side 4 cm, four pieces, each in the shape of arc of a circle with radius 2 cm, are cut out. The area of the remaining portion is
  1. 8 –
  2. 16 – 4
  3. 16 – 8
  4. 4 – 2

Answer: b) 16 – 4

 

Solution: area of a square sheet = 44 = 16 sq. cm

Since the angle between two sides of a square is 90 degree and the radius of arc is 2 cm

S, area of 4 arc will be = 49036022

                                    = 4

Area of remaining portion = 16 – 4

 

  1. The cost of building a fence around a circular field is rupee 7700 at the rate of  rupee 14 per foot . what is the area of the circular field ?
  1. 24062.5
  2. 23864.4
  3. 24644.5
  4. Cannot be determined
  5. None of these

Answer: a) 24062.5

 

Solution: length of a fence = 7700/14

                                         = 550

Circumference of a circle field = 2r

              2r = 550

              2 227r = 550

             r = 175/2

Area of circular field = r2

                               = 22717521752

                               = 24062.5

 

  1. The sides of a triangle are 6 cm, 8 cm and 10 cm. The area (in cm2) of the triangle formed by joining the mid_points of this triangle is:
  1. 12
  2. 24
  3. 6
  4. 18

Answer: c) 6

 

Solution: a = 10, b = 8, c = 6

    s= (6 + 8 + 10)/2

    S = 12

Area of a triangle ABC = 12(12-10) (12-6) (12-8)

                                     = 12(2) (6)) (4)

                                     = 24

All the four triangles ADE, BDF,CEF and DEF are congruent, so their area are equal

Hence, area of triangle DEF = 24/4

                                             = 6 sq cm.

 

  1. If D and E are the mid_points of the side AB and AC respectively of the ABC in the figure given here, the shaded region of the triangle is what percent of the whole triangular region?

 

  1. 50%
  2. 25%
  3. 80%
  4. 75%

Answer: d) 75%

 

Solution: Since D and E are the mid_ points of AB and AC respectively

So, DE is parallel to BC and 2DE = BC

ADE ABC

area ADEarea ABC = (DE)2 (BC)2   

               = (DE)2 (2DE)2   

               = 1/4

area of shaded partarea ABC = ¾

% = 3/4100

   = 75%

 

  1. If the length of a rectangle is increased by 20% and breadth is decreased by 20% . then its area
  1. Increases by 4%
  2. Decreases by 4%
  3. Decreases by 1%
  4. Remains unchanged

Answer: b) decreased by 4%

 

Solution: let the length of rectangle be 10

And its breadth be 10

Then, area = 100

When the length is increased by 20%, new length = 12

When its breadth is decreased by 20%, new breadth = 8

then , new area = 96

So, area decreased by 4%

  1. If the surface area of a cube is 294 sq. cm, then its volume(in sq. cm) is
  1. 216
  2. 125
  3. 343
  4. 512

Answer  c) 343

 

Solution surface area of a cube = 294

                     6a2= 294

                      a2= 49

                      a = 7

Volume of a cube = a3

                            = 777

                            = 343

 

  1. A right triangle with sides 3 cm, 4 cm and 5 cm is rotated about the side 3 cm to form a cone, the volume of the cone so formed is
  1. 16
  2. 12
  3. 15
  4. 20

Answer: a) 16

 

Solution:

 

when a triangle of side 3cm, 4cm and 5cm is rotated about the side 3 cm, then a cone of radius 4 cm and height 3 cm is formed

Volume of a cone = 13r2h

                            =  13443

                            = 16

 

  1. A wooden box measures 20 cm by 12 cm by 10 cm. Thickness of the wood is 1 cm. Volume of wood to make the box (in cubic cm) is
  1. 960
  2. 519
  3. 2400
  4. 1120

Answer a) 960

 

Solution outer dimension of wooden box = 20 cm , 12 cm and 10 cm

Outer volume of box = 201210

                                = 2400 cc ( cubic centimeter)

Thickness of wood = 1 cm

Inner dimension of wooden box = 20 – 2 = 18 cm, 12 – 2 = 10 cm and 10 – 2 = 8 cm

Inner volume of box = 18108

                               = 1440 cc

Volume of wood in the box = 2400 – 1440  = 960 cc

 

  1. Water flows into a tank which is 200m long and 150m wide, through a pipe of cross-section 0.3m0.2m at 20 km/hour. Then the time (in hours) for the water level in the tank to reach 8m is
  1. 50
  2. 120
  3. 150
  4. 200

Answer  d) 200

 

Solution: speed of water = 20 km/hour

                                      = 20000 m/hour

Area of base = 0.3m0.2m

                    = 0.06 sq. meter

Volume of water flows in 1 hour = area of base speed of water in 1 hour

                                                  = 0.06 20000

                                                  = 1200 cubic meter

Volume of tank upto 8 m height = 2001508

                                                 = 240000 cubic meter

Time to reach the water 8 m high = 240000/1200

                                                     = 200 hours  

 

  1. The base of a right prism is a quadrilateral ABCD. given that AB = 9 cm, BC = 14 cm, CD = 13 cm, DA = 12 cm and angle DAB = 90 degree. If the volume of the prism be 2070 cm3, then the area of the lateral surface is
  1. 720
  2. 810
  3. 1260
  4. 2070

Answer a) 720

 

Solution Area of quadrilateral ABCD = area of ABD + area of BCD

Since   ABD is a right angled triangle

BD = AB2 + AD2

      = 92 + 122

     = 225

     = 15 cm

Area of ABD = 12912

                           = 54 sq.cm

 

For BCD, s = (13 + 15 + 14)/2

                     = 21

area of BCD =  21(21 – 13) (21 – 14) (21 – 15)                                                               

                        =  21876

                        = 84 sq. cm

Area of quadrilateral ABCD = 54 + 84

                                            = 138 sq. cm  

Volume of prism = area of baseheight

                 2070 = 138h

                    Height = 2070/138                                                                

                               = 15 cm

Perimeter of quadrilateral ABCD = 9 + 14 + 13 + 12

                                                    = 48 cm

Area of lateral surface = perimeter of baseheight

                                    = 48 15

                                   = 720 sq. cm

                              

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