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Mensuration: Surface Area and Volume - Compute total surface area and volume of cuboids and cubes

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A cuboid is a three-dimensional solid with six rectangular faces. Its dimensions are represented by length (ll), breadth (bb), and height (hh).

3D diagram of a cuboid showing length, breadth, and height.
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A cube is a special case of a cuboid where all three dimensions are equal (l=b=h=al = b = h = a). All six faces of a cube are congruent squares with area a2a^2.

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Surface Area is the sum of the areas of all faces. For a cuboid, Total Surface Area (TSA) includes all six faces, while Lateral Surface Area (LSA) excludes the top and bottom faces (2h(l+b)2h(l+b)).

A flattened net of a cube showing 6 square faces.
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The Volume of a cuboid or cube measures the amount of space occupied by the solid. It is calculated by the product of its base area and its height.

📐Formulae

Total Surface Area (TSA) of a Cuboid = 2(lb+bh+hl)2(lb + bh + hl)

Lateral Surface Area (LSA) of a Cuboid = 2h(l+b)2h(l + b)

Total Surface Area (TSA) of a Cube = 6a26a^2

Lateral Surface Area (LSA) of a Cube = 4a24a^2

Length of Diagonal of a Cuboid = sqrtl2+b2+h2\\sqrt{l^2 + b^2 + h^2}

Length of Diagonal of a Cube = asqrt3a\\sqrt{3}

💡Examples

Problem 1:

Find the total surface area and the lateral surface area of a cuboid whose length is 15cm15 cm, breadth is 10cm10 cm, and height is 20cm20 cm.

Solution:

Given: l=15cml = 15 cm, b=10cmb = 10 cm, h=20cmh = 20 cm\

  1. Total Surface Area (TSA) = 2(lb+bh+hl)2(lb + bh + hl)
    TSA=2(15times10+10times20+20times15)TSA = 2(15 \\times 10 + 10 \\times 20 + 20 \\times 15)
    TSA=2(150+200+300)=2(650)=1300cm2TSA = 2(150 + 200 + 300) = 2(650) = 1300 cm^2\
  2. Lateral Surface Area (LSA) = 2h(l+b)2h(l + b)
    LSA=2times20(15+10)=40(25)=1000cm2LSA = 2 \\times 20(15 + 10) = 40(25) = 1000 cm^2

Explanation:

To find the TSA, we calculate the area of all six faces using the dimensions provided. For LSA, we only calculate the area of the four vertical faces using the perimeter of the base multiplied by the height.

Problem 2:

The total surface area of a cube is 1350cm21350 cm^2. Find its side length and its lateral surface area.

Solution:

Given: TSA=1350cm2TSA = 1350 cm^2\

  1. Use the TSA formula for a cube: 6a2=13506a^2 = 1350
    a2=frac13506=225a^2 = \\frac{1350}{6} = 225
    a=sqrt225=15cma = \\sqrt{225} = 15 cm\
  2. Now find the Lateral Surface Area (LSA):
    LSA=4a2LSA = 4a^2
    LSA=4times(15)2=4times225=900cm2LSA = 4 \\times (15)^2 = 4 \\times 225 = 900 cm^2

Explanation:

We first use the relationship between TSA and the side length (aa) to isolate aa. Once the side length is found by taking the square root, we plug it into the LSA formula to find the area of the four side faces.

Problem 3:

A plastic box 1.5m1.5 m long, 1.25m1.25 m wide and 65cm65 cm deep is to be made. It is open at the top. Find the area of the sheet required for making the box and the cost of the sheet if a sheet measuring 1m21 m^2 costs Rs 2020.

An open-top rectangular box with dimensions 1.5m by 1.25m by 0.65m.

Solution:

  1. Convert all units to meters: Length l=1.5ml = 1.5 m Breadth b=1.25mb = 1.25 m Height h=65cm=0.65mh = 65 cm = 0.65 m

  2. Area of sheet required (Box is open at the top): Area = Lateral Surface Area + Area of Base Area = 2h(l+b)+lb2h(l + b) + lb Area = 2(0.65)(1.5+1.25)+(1.5×1.25)2(0.65)(1.5 + 1.25) + (1.5 \times 1.25) Area = 1.3(2.75)+1.8751.3(2.75) + 1.875 Area = 3.575+1.875=5.45m23.575 + 1.875 = 5.45 m^2

  3. Cost calculation: Total Cost = Area ×\times Rate Total Cost = 5.45×20=1095.45 \times 20 = 109

Final Answer: The area of the sheet is 5.45m25.45 m^2 and the cost is Rs 109.

Explanation:

Since the box is open at the top, we only calculate the area of the four side walls (LSA) and the bottom rectangular base. We ensure all units are consistent (meters) before calculation.

Problem 4:

Calculate the volume of a wooden crate in the shape of a cube if the length of its longest diagonal is 63cm6\sqrt{3} cm.

A cube with a diagonal line drawn from one bottom corner to the opposite top corner.

Solution:

  1. Find the side length aa: Diagonal of Cube = a3a\sqrt{3} 63=a36\sqrt{3} = a\sqrt{3} a=6cma = 6 cm

  2. Calculate the volume: Volume = a3a^3 Volume = 636^3 Volume = 216cm3216 cm^3

Final Answer: The volume of the crate is 216cm3216 cm^3.

Explanation:

The diagonal of a cube is given by the formula a3a\sqrt{3}. Once the side 'a' is found, the volume is a3a^3.