{"id":2852,"date":"2023-04-26T10:00:02","date_gmt":"2023-04-26T04:30:02","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=2852"},"modified":"2023-04-26T09:26:20","modified_gmt":"2023-04-26T03:56:20","slug":"surface-area-volume-cuboid","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/surface-area-volume-cuboid\/","title":{"rendered":"How do you find the Surface Area and Volume of a Cuboid"},"content":{"rendered":"

How do you find the Surface Area and Volume of a Cuboid<\/strong><\/h2>\n

\"\"<\/p>\n

If \u2113, b and h denote respectively the length, breadth and height of a cuboid, then<\/p>\n

    \n
  1. Total surface area of the cuboid = 2 (\u2113b + bh + \u2113h) square units.<\/li>\n
  2. Volume of the cuboid\u00a0= Area of the base \u00d7 height = \u2113\u00a0\u00d7 b \u00d7 h cubic units.
    \nwhere base area = Breadth \u00d7 length<\/li>\n
  3. Diagonal of the cuboid or longest rod\u00a0= \\(\\sqrt {{\\ell ^2} + {b^2} + {h^2}} \\) units. and Total length of its edges = 4 (\u2113 +\u00a0b +\u00a0h)<\/li>\n
  4. Area of four walls of a room\u00a0= 2 (\u2113 + b) h sq. units.<\/li>\n<\/ol>\n

    Surface Area and Volume of a Cuboid Example Problems with Solutions<\/strong><\/h2>\n

    Example 1:<\/strong>\u00a0 \u00a0 \u00a0The length, breadth and height of a cuboid are in the ratio 6 : 4 : 5. If the total surface area of \u00a0the cuboid is 2368 cm2<\/sup> ; find its dimension.
    \nSolution: \u00a0 \u00a0\u00a0<\/strong>Let length (\u2113) = 6x cm, breadth (b) = 4x cm and height (h) = 5x cm,
    \n\u2234\u00a0\u00a0\u00a0 Total surface area
    \n= 2(\u2113 \u00d7 b + b \u00d7 h \u00d7 h \u00d7 \u2113)
    \n= 2(6x \u00d7 4x + 4x \u00d7 5x + 5x \u00d7 6x)cm2<\/sup>
    \n= 2(24x2<\/sup>\u00a0+ 20x2<\/sup>\u00a0+ 30x2<\/sup>) cm2<\/sup>
    \n=148x2<\/sup>\u00a0cm2<\/sup>
    \nGiven : Total surface = 2368 cm2<\/sup>
    \n\u21d2\u00a0\u00a0 148x2<\/sup>\u00a0= 2368
    \n\u21d2\u00a0\u00a0 x2<\/sup>\u00a0= \\(\\frac{{2368}}{{148}}\\)\u00a0= 16
    \nand x =\u00a0\\(\\sqrt {16} \u00a0= 4\\)
    \n\u2234\u00a0\u00a0\u00a0 length = 6x cm = 6 \u00d7 4 cm = 24 cm,
    \nbreadth = 4x cm = 4 \u00d7 4 cm = 16 cm and
    \nheight = 5x cm = 5 \u00d7 4 cm = 20 cm<\/p>\n

    Example 2:<\/strong>\u00a0 \u00a0 \u00a0A plastic box 1.5 m long, 1.25 m wide and 65 cm deep is to be made. It is to be open at the top. Ignoring the thickness of the plastic sheet, determine :
    \n(i)\u00a0\u00a0 The area of the sheet required for making the box.
    \n(ii)\u00a0 The cost of sheet for it, if a sheet measuring 1 m2<\/sup>\u00a0costs Rs. 20.
    \nSolution: \u00a0 \u00a0\u00a0<\/strong>Given. length (\u2113) = 1.5 m, breadth (b) = 1.25 m and depth i.e., height (h) = 65 cm = 0.65 m.
    \n(i)\u00a0\u00a0 Since, the box is open it has five faces in which four faces are the walls forming lateral surface area and one face is the base.
    \n\u2234\u00a0\u00a0\u00a0 The area of the sheet required for making the box.
    \n= \u00a0\u00a0 Lateral surface area of the box + area of its base
    \n= 2(\u2113 + b) \u00d7 h + \u2113 \u00d7 b
    \n= 2(1.5m + 1.25m) \u00d7 0.65m + 1.5m \u00d7 1.25 m
    \n= 2 \u00d7 2.75 m \u00d7 0.65 m + 1.875 m2<\/sup>
    \n= 3.575 m2<\/sup>\u00a0+ 1.875 m2 = 5.45 m2<\/sup>
    \n(ii)\u00a0 Since, a sheet measuring 1 m2<\/sup>\u00a0costs Rs.20
    \n\u2234 \u00a0\u00a0 The cost of sheet for the box
    \n= 5.45 \u00d7 Rs 20 =\u00a0 Rs. 109<\/p>\n

    Example 3:<\/strong>\u00a0 \u00a0 \u00a0The length, breadth and height of a room are 5m, 4m and 3m respectively. Find the cost of white washing the walls of the room and the ceiling at the rate of Rs. 7.50 per m2<\/sup>.
    \nSolution: \u00a0 \u00a0\u00a0<\/strong>Given. \u2113 = 5m, b = 4m and h = 3m
    \nSince, the area of the walls of the room
    \n= its lateral surface area = 2(\u2113 + b) \u00d7 h
    \nAnd, the area of the ceiling of the room
    \n= \u2113 \u00d7 b
    \n\u2234\u00a0\u00a0\u00a0 Total area to be white washed
    \n= Area of the walls + area of the ceiling
    \n= 2(\u2113 + b) \u00d7 h + \u2113 \u00d7 b.
    \n= 2(5m + 4m) \u00d7 3m + 5m \u00d7 4m
    \n= 2 \u00d7 9m \u00d7 3m + 20m2<\/sup>
    \n= 54m2<\/sup>\u00a0+ 20m2<\/sup>\u00a0= 74m2<\/sup>
    \nSince, the rate of white washing
    \n= Rs. 7.50 per m2<\/sup>
    \n\u2234 \u00a0Cost of white washing
    \n= 74 \u00d7 Rs.7.50 = Rs. 555<\/p>\n

    Example 4:<\/strong>\u00a0 \u00a0 \u00a0The floor of a rectangular hall has a perimeter 250 m. If the cost of painting the four walls at the rate of Rs. 10 per m2<\/sup>\u00a0is Rs : 15,000, find the height of the hall.
    \nSolution: \u00a0 \u00a0\u00a0<\/strong>We know, the perimeter of a rectangle
    \n= 2(length + breadth) = 2 (\u2113 + b)
    \nAnd, given the floor of a rectangular hall has perimeter 250 m
    \n\u21d2 \u00a0 2(\u2113 + b) = 250 m
    \nArea of the four walls of the hall
    \n= Lateral surface area of the hall
    \n= 2(\u2113 + b) \u00d7 h = 250 m \u00d7 hm = 250 hm2<\/sup>
    \nSince, the rate of painting four walls is
    \nRs. 10 per m2<\/sup>.
    \n\u2234\u00a0\u00a0\u00a0 Cost of painting 250 hm2<\/sup>\u00a0= 250 h \u00d7 Rs.10
    \nAccording to the given statements :
    \n250 h \u00d7 Rs. 10 = Rs. 15,000
    \n\u21d2\u00a0\u00a0 h = \\(\\frac{{15,000}}{{250 \\times 10}}m\\)\u00a0= 6 m
    \n\u2234\u00a0\u00a0\u00a0 The height of the hall = 6 m<\/p>\n