Mensuration Basic Fundas - Vikas Saini



  • Sphere

    Volume (V) = (4/3) x π x (radius)^3
    Surface area (SA) = 4 x π x (radius)^2

    1. The radius of a sphere is 14.7 cm, what is its volume?
      Solution :-
      Volume = (4/3) x (22/7) x (14.7)^3 = 13311.144 cu. Cm

    2. The volume of a sphere is 457.34π cu. cm. What is its radius?
      Solution :-
      (4/3) x π x (radius)^3 = 457.34 π
      Radius^3 = 457.34 x 3 / 4
      Radius = 7.

    3. If the surface area of a sphere is 616 sq. cm, find its radius.
      Solution :- 4 π radius^2 = 616
      Radius = 7.

    4. The volume of a sphere is 7241.15 cu. cm, find its surface area.
      Solution :-
      (4/3) x π x (radius)^3 = 7241.15
      Radius = 12.
      SA = 4 π (radius)^2 = 1810.29 sq cm.

    Hemi sphere

    Volume (V) = (2/3) x π x (radius)^3
    Surface area (SA) = 2 x π x (radius)^2

    1. The diameter of a hemisphere is 30 cm, find its volume. (π = 3.14)
      Solution :-
      Radius = 30/2 = 15 cm.
      V = (2/3) x π x (radius)^3
      V = 7065 cu cm.

    2. Find the total surface area of a hemisphere whose volume is 452.16 cu. cm. (π = 3.14).
      Solution :-
      452.16 = (2/3) x π x (radius)^3
      Radius = 6.
      CSA =2 x π x (radius)^2 = 339.12 sq cm.
      Cube
      Volume = side^3
      Surface area = 6 x side^2

    3. The volume of a cube is 46656 cu. cm. What is its total surface area?
      Solution :-
      V = Side^3
      Side = (46656)^(1/3)
      Side = 36.
      Surface area = 6 x 36^2 = 7776 sq. Cm.

    Cuboid

    Volume = length x breadth x height
    Surface area = 2 (length x breadth + breadth x height + length x height)

    1. The length, breadth and height of a parallelepiped are 5 cm, 8 cm and 12 cm respectively. Find its total surface area and volume.
      Solution :-
      Surface area = 2 (5 x 8 + 8 x 12 + 5 x 12)
      =392 sq cm
      Volume = 5 x 8 x 12 = 480 cu. cm.

    Prism

    Area of polygon = (n/4) x (side)^2 Cot (180/n)
    n = no of sides in polygon

    1. Find the volume of a prism with a regular pentagonal base. Side of its base is 3.81 cm and its height is 8 cm. cot (36°) = 1.3763
      Area of pentagonal base = (5/4) (3.81)^2 Cot (180 / 5)
      = (5/4) (3.81)^2 x 1.3763
      = 25 sq cm (approximately)
      Volume = area x height
      = 25 x 8
      = 200 cu. cm

    2. Find the volume of a prism with a regular hexagonal base of side 12 cm and height 15 cm.
      Area of hexagonal base = (6/4) x (12)^2 x Cot (180/6)
      =216( 3)^1/2 .
      Volume = 216(3)^1/2 x 15 = 3240 (3)^1/2.

    3. Find the surface area of the vertical faces of a prism with a regular pentagonal base of side 50 cm and height 14 cm.
      Perimeter = 50 x 5 = 250.
      Surface area = perimeter x height = 250 x 14 = 3500.

    Pyramid

    1. The side of a regular hexagonal base of a pyramid is 18 cm. The slant height and height of the pyramid are 30 cm and 28 cm respectively. Find its volume and total surface area.
      Solution :-
      Area of hexagonal base = (6/4) x (18)^2 x Cot (180/6)
      =841.75 sq cm.
      Volume of pyramid = (1/3) x base area x height
      = (1/3) x 841.75 x 28
      =7856.33 cu cm.
      Surface area = 1/2 x perimeter x slant height + base area
      =2461.75 sq cm.

    2. The volume and surface area of the lateral faces of a pyramid with a regular pentagonal base of side 12 cm is 1984 cu. cm and 900 sq. cm. What are its height and slant height?
      cot (36°) = 1.3763
      Solution :-
      Base area = (5/4) x (12)^2 x Cot (180/5) = 248.4 sq cm.
      Volume = 1/3 x base area x height
      Height = 3 x 1984 / 248.4 = 24 cm (approximately)
      Surface area = 1/2 x perimeter x slant height + base area
      900 – 248.4 = 1/2 x (5 x 12) x slant height
      Slant height = 30 cm.


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