Quantitative Aptitude - SPLessons

Mensuration Problems

Chapter 39

SPLessons 5 Steps, 3 Clicks
5 Steps - 3 Clicks

Mensuration Problems

shape Introduction

Mensuration Problems deals with the area, perimeter, diagonal, height, circumference etc. of figures like square, rectangle, parallelogram, rhombus, trapezium, triangle, equilateral triangle, cube, cuboid, cylinder, cone, sphere, hemi-sphere, prism, circle, semi circle, sector.


    Mensuration: It is a branch of mathematics which deals with the lengths of lines, areas of surfaces and volumes of solids.


    Plane Mensuration: It deals with the sides, perimeters and areas of plane figures of different shapes.


    Solid Mensuration: It deals with the areas and volumes of solid objects


shape Methods

Square: It is a regular quadrilateral which consists four equal sides and four equal angles. Every angle is 90 degrees.

    Here AB = BC = CD = AD = 5 m = a (Let)

    1. Perimeter of square = 4 x sides = 4a
    = 4 x 5 = 20m

    2. Area of a square = \((sides)^{2}\)
    = \(a^{2}\) = \((5)^{2}\) = 25 sq. m

    3. Side of a square = \(\sqrt{area}\) = \(\sqrt{25}\) = 5m or,
    \(\frac{Perimeter}{4}\) = \(\frac{20}{4}\) = 5m

    4. Diagonal of a square = \(\sqrt{2}\) x side = \(\sqrt{2}\) a
    = \(\sqrt{2}\) x 5 = 5\(\sqrt{2}\) m

    5. Side of a square = \( \frac{diagonal}{\sqrt{2}}\) = \(\frac{5\sqrt{2}}{\sqrt{2}}\) = 5m


Rectangle: A rectangle is a plane, whose opposite sides are equal and diagonals are equal. Each angle is equal to 90˚.


    Here AB = CD; length l = 4m

    AD = BC; breadth b = 3m

    1. Perimeter of a rectangle = 2(length + breadth)

    = 2(l + b)

    = 2(4 + 3) = 14 m


    2. Area of rectangle = length x breadth = l x b = 4 x 3

    = 12 \(m^{2}\)


    3. Length of a rectangle : \(\frac{area}{breadth}\) = \(\frac{A}{b}\) = \(\frac{12}{3}\) = 4 m

    Or, [\(\frac{perimeter}{2}\) – breadth] = (\(\frac{14}{2}\) – 3) = 4m

    Breath of a rectangle : \(\frac{area}{length}\) = \(\frac{A}{l}\) = \(\frac{12}{4}\) = 3 m

    Or, [\(\frac{perimeter}{2}\) – length] = (\(\frac{14}{2}\) – 4) = 3 m


Parallelogram: It is a rectilinear figure with opposite sides parallel.


    (i) Area of parallelogram = base x height

    = b x h

    = 8 x 5 = 40 sq.cm.

    (ii) Perimeter of a parallelogram = 2(AB + BC)

    = 2( 8 + 5 ) = 26 cm


Rhombus: It is a simple quadrilateral in which all sides have same length. A rhombus is often called as ‘Diamond’.


    i) Area of rhombus = \(\frac{1}{2}\) x (product of diagonals)

    = \(\frac{1}{2} (d_{1} . d_{2}) \) = \(\frac{1}{2}\) x 8 x 6 = 24\(cm^{2}\)

    ii) Perimeter of rhombus = 4 x side = 4a

    here AB = BC = CD = AD = 4a

    AC = \(d_{1}\), BD = \(d_{2}\)


Trapezium: It is a quadrilateral with one of pair of sides parallel.


    Area if a trapezium = \(\frac{1}{2}\) x ( sum of parallel sides ) x height

    = \(\frac{1}{2}\) x ( a + b ) x h

    = \(\frac{1}{2}\) x (15 + 17) x 10

    = \(\frac{1}{2}\) x 32 x 10 = 160 \(cm^{2}\)


Triangle: It is a polygon with three edges and three vertices.


    1. Area of triangle = \(\frac{1}{2}\) x base x height

    = \(\frac{1}{2}\) x 15 x 12 = 90 sq. cm

    here AD = 12 cm = height, BC = 15 cm = base

    2. Semi perimeter of a triangle

    S = \(\frac{a + b + c}{2}\) = \(\frac{10 + 8 + 6}{2}\) = 12 cm

    here BC = a, AC = b, AB = c

    3. Area of triangle = \(\sqrt{s(s – a)(s – b)(s – c)}\)

    Where a = 10cm, b = 8cm, c = 6cm, s = 12cm

    = \(\sqrt{12(12 – 10)(12 – 8)(12 – 6)}\)

    = \(\sqrt{12 \times 2 \times 4 \times 6}\)

    = 24 \(cm^{2}\)

    4. Perimeter of a triangle = 2s = (a + b + c)

    = 10 + 8 + 6 = 24 cm


Equilateral triangle: It is defined as a polygon with three equal sides. Each and every angle in it is 60 degrees.


    1. Area of an equilateral triangle = \(\frac{\sqrt{3}}{4}\) x \((side)^{2}\)

    = \(\frac{\sqrt{3}}{4}\) x \((4\sqrt{3})^{2}\)

    = \(\frac{\sqrt{3}}{4}\) x 48 = \(12\sqrt{3}\) \(cm^{2}\)

    2. Height of an equilateral triangle = \(\frac{\sqrt{3}}{2}\) x \((side)^{2}\)

    = \(\frac{\sqrt{3}}{2}\) x \(4\sqrt{3}\)

    = 6cm

    3. Perimeter of an equilateral triangle = 3 x (side)

    = 3 x \(4\sqrt{3}\) = \(12\sqrt{3}\) cm


Cube: A cube is a three dimensional figure and has six square faces which meet each other at right angles. It has eight vertices and twelve edges.


    1. Volume of cube = \((side)^{3}\)

    = \(12^{3}\)

    = 1728 cubic cm

    Cube: All sides are equal = 12 cm


    2. Side of a cube = \(\sqrt[3]{Volume}\)

    = \(\sqrt[3]{1728}\)

    = 12 cm


    3. Diagonal of cube = \(\sqrt{3}\) x (Side) = \(\sqrt{3}\) x 12 = 12 \(\sqrt{3}\) cm


    4. Total surface area of a cube = 6 x \((Side)^{2}\) = 6 x \((12)^{2}\) = 864 sq.cm


Cuboid (Rectangular parallelopiped): A solid body having six rectangular faces, is called cuboid. (or) A parallelopiped whose faces are rectangles is called rectangular parallelopiped or cuboid.


    1. Total surface area of cuboid = 2 (lb + bh + hl) sq. unit

    Here l = length, b = breadth, h = height

    = 2(12 x 8 + 8 x 6 + 6 x 12)

    = 2(96 + 48 + 72) = 2 x 216 = 432 sq. cm.


    2. Volume of a cuboid = (length × breadth × height) = lbh

    = 12 × 8 × 6 = 576 cuboic cm


    3. Diagonal of a cuboid = \(\sqrt{l^{2} + b^{2} + h^{2}}\) = \(\sqrt{12^{2} + 8^{2} + 6^{2}}\)

    = \(\sqrt{144 + 64 + 36}\) = \(\sqrt{244}\) = \(2\sqrt{61}\) cm.


    4. Length of cuboid = \(\frac{Volume}{Breadth \times Height}\) = \(\frac{v}{b \times h}\)


    5. Breadth of cuboid = \(\frac{Volume}{Length \times Height}\) = \(\frac{v}{l \times h}\)


    6. Height of cuboid = \(\frac{Volume}{length \times breadth}\) = \(\frac{v}{l \times b}\)


Cylinder: A solid geometrical figure with straight parallel sides and a circular or oval cross section is called cylinder.


    1. Area of curved surface = (perimeter of base) x height

    = \(2 \pi r h \) sq. unit

    = 2 x \(\frac{22}{7}\) x 7 x 15 = 660 sq. cm


    2. Total surface area = area of circular ends + curved surface area

    = \(2 \pi r^{2}\) + \(2 \pi r(r + h) \) sq. unit

    = 2 x \(\frac{22}{7}\) x 7(15 + 7)

    = 2 x 22 x 22

    = 968 sq. cm.


    3. Volume = (area of base) x height

    = \((\pi r^{2})\) x h = \(\pi r^{2}h\)

    = \(\frac{22}{7}\) x 7 x 7 x 15 = 2310 cubic cm.


    4. Volume of a hollow cylinder = \(\pi R^{2}h – \pi r^{2}h\)

    = (\(\pi h (R^{2} – r^{2})\)) = \( \pi h(R + r)(R – r)\)

    = \( \pi \times height \times (sum of radii)(difference of radii)\)

    Here R, r are outer and inner radii respectively and h is the height.


Cone: A solid (3-dimensional) object with a circular flat base joined to a curved side that ends in an apex point is called cone.


    1. In right angled \(\bigtriangleup\) OAC, we have

    \(l^{2}\) = \(h^{2}\) + \(r^{2}\)

    (Here r = 35 cm, l = 37 cm, h = 12 cm)

    Or, l = \(\sqrt{h^{2} + r^{2}}\)

    h = \(\sqrt{l^{2} – r^{2}}\), r = \(\sqrt{l^{2} – h^{2}}\)

    Where l = slant height, h = height, r = radius of base


    2. Curved surface area = \(\frac{1}{2}\) x (perimeter of base) x slant height

    = \(\frac{1}{2}\) x \(2 \pi r\) x l = \(\pi r l\) sq. unit

    = \(\frac{22}{7}\) x 35 x 37 = 4070 sq. cm


    3. Total surface area S = area of circular base + curved surface area = \((\pi r^{2} + \pi r l)\) = \(\pi r (r + l)\) sq. unit

    = \(\frac{22}{7}\) x 35 (37 + 35) = 7920 sq. cm


    4. Volume of cone = \(\frac{1}{3}\) (area of base) x height

    = \(\frac{1}{3} (\pi r^{2})\) x h = \(\frac{1}{3} \pi r^{2} h\) cubic unit

    = \(\frac{1}{3}\) x \(\frac{22}{7}\) x 35 x 35 x 12

    = 15400 cubic cm


    Frustum of Cone:


    5. Volume of frustum = \(\frac{1}{3} \pi h (R^{2} + r^{2} + Rr)\) cubic unit


    6. Lateral surface = \(\pi l (R + r)\)

    Where \(l^{2}\) = \(h^{2}\) + \((R – r)^{2}\)


    7. Total surface area = \(\pi [R^{2} + r^{2} + l(R + r)]\)

    R, r be the radius of base and top the frustum

    ABB ‘A’ h and l be the vertical height and slant

    Height respectively.


Sphere: A 3-dimensional object shaped like a ball is called sphere and every point on the surface is the same distance from the centre.


    1. Surface area = \(4 \pi r^{2}\)

    = 4 x \(\frac{22}{7}\) x \((10.5)^{2}\) = 1386 sq. cm

    Here, d = 21 cm

    Therefore, r = 10.5 cm


    2. Radius of sphere = \(\sqrt{ \frac{surface area}{4 \pi}}\) = \(\sqrt{ \frac{1386 \times 7}{4 \times 22}}\) = 10.5 cm


    3. Diameter of sphere = \(\sqrt{ \frac{surface}{4 \pi}}\) = \(\sqrt{ \frac{surface}{4 \pi}}\) = \(\sqrt{ \frac{1386 /times 7}{22}}\) = 21 cm


    4. Volume of sphere V = \(\frac{4}{3} \pi r^{3}\) = \(\frac{4}{3} \pi (\frac{d}{2})^{3}\) = \(\frac{1}{6} \pi d^{3}\)


    = \(\frac{1}{6}\) x \(\frac{22}{7}\) x 21 x 21 x 21 = 4831 cubic cm



    5. Radius of sphere = \(\sqrt{ \frac{3}{4 \pi} \times volume of shpere}\)


    6. Diameter = \(\sqrt[3]{ \frac{6 \times v}{\pi}}\)


    7. Volume of spherical ring = \(\frac{4}{3} \pi (R^{3} – r^{3}\)


    Hemisphere: In geometry, hemisphere is an exact half of sphere.



    8. Curved surface of hemisphere = 2 \(\pi r^{2}\)


    9. Volume of hemisphere = \(\frac{2}{3} \pi r^{3}\)


    10. Total surface area of hemisphere = \(3 \pi r^{2}\)


Circle: It is defined as, if a straight line is bent until its ends join. It is a simple closed curve and the distance between any of the points and centre of circle is called radius.


    1. Circumference of a circle = \(\pi\) x diameter

    = \(\pi\) x \(2r\) = \(2 \pi r\)

    = 2 x \(\frac{22}{7}\) x 42 = 264 cm


    2. Radius of a circle = \(\frac{circumference}{2 \pi}\) = \(\frac{264 \times 7}{2 \times 22}\) = 42 cm


    3. Area of a circle = \(\pi \times r^{2}\) = \(\frac{22}{7}\) x \(42^{2}\) = \(\frac{22}{7}\) x 42 x 42 = 5544 \(cm^{2}\)



    4. Radius of a circle = \(\sqrt{\frac{area}{\pi}}\)

    = \(\sqrt{\frac{5544}{22} \times 7}\) = \(\sqrt{1764}\) = 42cm


    Semi-circle: It is a two-dimensional geometric shape that also includes the diameter segment from one end of the arc to the other as well as all the interior points.


    5. Area of a semi circle = \(\frac{1}{2} \pi r^{2}\) = \(\frac{1}{8} \pi d^{2}\)

    = \(\frac{1}{2}\) x \(\frac{22}{7}\) x \(42^{2}\) = \(2772 cm^{2}\)


    6. Circumference of semi circle = \(\frac{22}{7}\) x 42 = 132 cm


    7. Perimeter of semi circle = \((\pi r + 2r)\) = \((\pi + 2)r\) = \((\pi + 2)\frac{d}{2}\)



    8. Area of sector OAB = \(\frac{x}{360}\) x \(\pi r^{2}\)

    (x being the central angle)

    = \(\frac{30^{0}}{360^{0}}\) x \(\frac{22}{7}\) x 3.5 x 3.5 = 3.21 sq. m.


    9. Central angle by arc AB = \(360^{0}\) x \(\frac{area of OAB}{are of circle}\)

    = \(360^{0}\) x \(\frac{3.21}{\frac{22}{7} \times 3.5 x 3.5}\) = \(\frac{360 \times 321}{22 \times 35 \times 5}\) = \(30^{0}\)(approx)


    10. Radius of circle = \(\sqrt{\frac{360^{0}}{central \ angle \ by \ arc} \times \frac{area \ of \ OAB}{\pi}}\)

    = \(\sqrt{\frac{360^{0}}{30^{0}} \times \frac{3.21}{\frac{22}{7}}}\) = \(\sqrt{\frac{134.82}{11}}\) = \(\sqrt{12.23}\) = 3.5 m.



    11. Area of ring = difference of the area of two circle

    = \(\pi R^{2}\) – \(\pi r^{2}\) = (\(R^{2} – r^{2}\))

    = \(\pi\)(R + r)(R – r)

    = (sum of radius)(diff. of radius)

    = \(\frac{22}{7}\) x (4 + 3)(4 – 3) = \(\frac{22}{7}\) x 7 x 1

    = 22 sq. cm.

shape Formulae

1. Square:
Area = \((side)^2\)
Perimeter = 4 x side
Diagonal = \(\sqrt{2(side)^2}\)


2. Rectangle:
Area = Length x Breadth
Perimeter = 2(Length x Breadth)
Diagonal = \(\sqrt{l^2 + b^2}\)


3. Parallelogram:
Area = Base x Height
Perimeter = 2(Length x Breadth)
Base = \(\frac{Area}{Height}\)


4. Rhombus:
Area = \(\frac{1}{2}* d_{1} d_{2}\)
Where, \(d_{1} \ and \ d_{2}\) are diagonals.
Perimeter = 4 x side
Diagonal = \(\frac{2 * area}{other diagonal}\)


5. Trapezium:
Area = \(\frac{1}{2}(a + b) * h\)
Perimeter = Sum of the sides


6. Triangle:
Area = \(\frac{1}{2} * Base * Height\) (or) \(\sqrt{s(s – a)(s – b)(s – c)}\)
Where a, b, c are sides of triangle and s = \(\frac{a + b + c}{2}\)
Perimeter = Sum of the sides


7. Equilateral Triangle:
Area = \(\frac{\sqrt{3}}{2}(side)^2\)
Perimeter = 3(Side)


8. Cube:
Let each edge of a cube be of length \(a\). Then,
Volume = \(a^3\) cubic units
Surface area = 6\(a^2\) sq. units
Diagonal = \(\sqrt{3}a\)units


9. Cuboid:
Let l – length, b – breadth, h – height. Then,
Volume = (l x b x h) cubic units
Surface area = 2(lb + bh + lh) sq. units
Diagonal = \(\sqrt{l^2 + b^2 + h^2}\) units


10. Cylinder:
Let radius of base = r and height( or length) = h. Then,
Volume = \(\pi r^2 h \) cubic units
Curved Surface area = 2\(\pi r h\) sq. units
Total surface area = 2(\(\pi rh + 2\pi r^2\)) = 2\(\pi r(h + r)\)sq. units


11. Cone:
Let radius of base = r and height = h. Then,
Slant height, l = \(\sqrt{h^2 + r^2}\) units
Volume = \(\frac{1}{3}\pi r^2 h\) cubic units
Curved surface area = \(\pi r l\) sq. units
Total surface area = \(\pi r l + \pi r^2\) sq. units


12. Sphere:
Let radius of the sphere be r. Then,
Volume = (\(\frac{4}{3} \pi r^3\)) cubic units
Surface area = \(\pi r^2\) sq. units


13. Hemisphere:
Let the radius of a hemisphere be \(r\).
Volume = \(\frac{2}{3} \pi r^3\) cubic units.
Curved surface area = 2\(\pi r^2\) sq. units.
Total surface area = 3\(\pi r^2 \) sq. units.


14. Prism:
Volume of right prism = (Area of the base * height)cu. units
Lateral surface area of a right prism = (perimeter of the base * height)sq. units
Total surface area of a right prism = lateral area + 2(area of one base)sq. units


15. Circle:
Area = \(\pi r^2\)
where, r = radius
Circumference = \(2 \pi r \)


16. Semi-circle:
Area = \(\frac{1}{2} \pi r^2\)
Circumference = \(\pi r + 2r\)


17. Sector:
Area = \(\frac{\theta}{360} * \pi r^2\)
Circumference = l + 2r[l = \(\frac{\theta}{360} * 2\pi r\)]
where l = length of arc.

shape Samples

1. A wheel makes 1000 revolutions in covering a distance of 88 km. Find the radius of the wheel?

Solution:

    Given that,

    Number of revolutions made by wheel = 1000

    Distance = 88 km

    So, distance covered in 1 revolution = \(\frac{88 * 1000}{1000}\) m = 88 m

    Therefore, \(2 \pi r \) = 88

    ⇒ 2 x \(\frac{22}{7}\) x r = 88

    ⇒ r = 88 x \(\frac{7}{44}\)

    ⇒ r = 14 m

    Therefore, radius of the wheel = 14m.


2. Expenditure incurred in cultivating a square field at the rate of Rs. 170 per hectare is Rs. 680. What would be the cost of fencing the field at the rate of Rs. 3 per metre.

Solution:

    Given that,

    Expenditure per hectare = Rs. 680

    Area = \(\frac{680}{170}\) = 4 hectares = 4 x 10000 sq.m.

    So, Sides of the field = \(\sqrt{40000}\) = 200 m.

    Perimeter of the field = 4 x side = 4 x 200 = 800 m.

    Therefore, Cost of putting the fence around it is

    = 800 x 3 = Rs. 2400.


3. A man walking at the rate of 6 km per hour crosses a square field diagonally in 9 seconds. Find the area of the field ?

Solution:

    Given that,

    Time = 9 seconds

    Rate = 6 km

    Now, Distance covered in \(\frac{6 * 1000}{3600} * 9\) = 15 m

    Diagonal of the square field = 15 m.

    Hence, Area of the square field = \(\frac{(15)^2}{2}\) = \(\frac{225}{2}\) = 112.5 sq.m.


4. Find the diagonal of a cuboid whose dimensions are 22 cm, 12 cm, and 7.5 cm?

Solution:

    Given that,

    Length = 22 cm

    Breadth = 12 cm

    Height = 7.5 cm

    Now,

    Diagonal of the cuboid = \(\sqrt{l^2 + b^2 + h^2}\) = \(\sqrt{(22)^2 + (12)^2 + (7.5)^2}\) = \(\sqrt{684.25}\) = 26.15 cm.

    Therefore, Diagonal of the cuboid = 26.15 cm.


5. The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2. If the total surface area is 616\({cm}^2\), then find the volume of the cylinder?

Solution:

    GGiven that,

    Ratio between the curved surface area and total surface area of a right circular cylinder is 1 : 2

    Total surface area = 616\({cm}^2\)

    Now, \(\frac{2 \pi r h}{2 \pi r (h + r)}\) = \(\frac{1}{2}\)

    ⇒ \(\frac{h}{h + r}\) = \(\frac{1}{2}\)

    ⇒ h = r

    Hence, total surface area = \(2 \pi r (h + r)\) = 4 \(\pi r^2\) (since, h = r)

    Therefore, 4 \(\pi r^2\) = 616

    ⇒ \( r^2\) = 616 x \(\frac{1}{4}\) x \(\frac{7}{22}\) = 49

    ⇒ \( r\) = 7

    Thus, h = 7 cm and r = 7 cm

    Therefore, Volume = \(\pi r^2 h\)

    = \(\frac{22}{7}\) x 7 x 7 x 7 = 1078 \({cm}^3\).