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Mensuration - Volume of Cube, Cuboid, and Cylinder

Grade 8ICSE

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

🔑Concepts

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The volume of a 3D solid is the measure of the space occupied by it. For a Cube, all edges are of equal length aa. The volume is calculated as V=a×a×a=a3V = a \times a \times a = a^3.

A cube diagram showing all sides equal to length a.
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A Cuboid is a 3D shape with length ll, breadth bb, and height hh. Its volume is the product of these three dimensions: V=l×b×hV = l \times b \times h.

A cuboid diagram labeled with length l, breadth b, and height h.
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A Cylinder consists of two circular bases and a curved surface. The volume is the product of the area of the circular base πr2\pi r^2 and the height hh. Thus, V=πr2hV = \pi r^2 h.

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Conversion of units is vital in volume calculations. Common conversions include: 1 cm3=1 mL1 \text{ cm}^3 = 1 \text{ mL}, 1000 cm3=1 L1000 \text{ cm}^3 = 1 \text{ L}, and 1 m3=1,000,000 cm3=1000 L1 \text{ m}^3 = 1,000,000 \text{ cm}^3 = 1000 \text{ L}.

📐Formulae

Volume of a Cube=a3 (where a is the side length)\text{Volume of a Cube} = a^3 \text{ (where } a \text{ is the side length)}

Volume of a Cuboid=l×b×h\text{Volume of a Cuboid} = l \times b \times h

Volume of a Cylinder=πr2h\text{Volume of a Cylinder} = \pi r^2 h

Base Area of a Cylinder=πr2\text{Base Area of a Cylinder} = \pi r^2

Radius (r)=Diameter2\text{Radius } (r) = \frac{\text{Diameter}}{2}

1 Litre=1000 cm31 \text{ Litre} = 1000 \text{ cm}^3

1 m3=1000 Litres1 \text{ m}^3 = 1000 \text{ Litres}

💡Examples

Problem 1:

Find the volume of a cylinder with a base radius of 7 cm7 \text{ cm} and a height of 10 cm10 \text{ cm}. (Use π=227\pi = \frac{22}{7})

Solution:

  1. Identify given values: r=7 cmr = 7 \text{ cm}, h=10 cmh = 10 \text{ cm}.
  2. Use the formula: V=πr2hV = \pi r^2 h.
  3. Substitute the values: V=227×7×7×10V = \frac{22}{7} \times 7 \times 7 \times 10.
  4. Cancel out 77 from numerator and denominator: V=22×7×10V = 22 \times 7 \times 10.
  5. Calculate the final product: V=154×10=1540 cm3V = 154 \times 10 = 1540 \text{ cm}^3.

Explanation:

To find the cylinder's volume, we calculate the area of the circular base first using πr2\pi r^2 and then multiply it by the height. Substituting the radius and height into the formula gives the total space occupied in cubic centimeters.

Problem 2:

A cuboidal water tank is 5 m5 \text{ m} long, 4 m4 \text{ m} wide, and 3 m3 \text{ m} deep. Find its capacity in litres.

Solution:

  1. Calculate volume in cubic meters: V=l×b×h=5×4×3=60 m3V = l \times b \times h = 5 \times 4 \times 3 = 60 \text{ m}^3.
  2. Convert m3m^3 to litres: Since 1 m3=1000 litres1 \text{ m}^3 = 1000 \text{ litres}.
  3. Capacity = 60×1000=60,000 litres60 \times 1000 = 60,000 \text{ litres}.

Explanation:

We first find the volume of the tank by multiplying length, breadth, and depth. Since the question asks for capacity in litres, we use the conversion factor where 11 cubic meter equals 10001000 litres.

Problem 3:

Calculate the volume of a cube whose total surface area is 216 cm2216 \text{ cm}^2.

Cube with side 6 cm.

Solution:

  1. Let the side of the cube be aa.
  2. Total Surface Area of a cube =6a2= 6a^2.
  3. Given: 6a2=2166a^2 = 216
  4. a2=2166=36a^2 = \frac{216}{6} = 36
  5. a=36=6 cma = \sqrt{36} = 6 \text{ cm}
  6. Volume =a3=63=216 cm3= a^3 = 6^3 = 216 \text{ cm}^3.

Explanation:

We first use the surface area formula to find the length of one side of the cube, then raise that side length to the power of 3 to find the volume.

Problem 4:

A cylindrical pillar has a diameter of 14 cm14 \text{ cm} and a height of 2 m2 \text{ m}. Find the volume of the pillar in cm3\text{cm}^3. (Take π=227\pi = \frac{22}{7})

Diagram of a cylinder with diameter 14 cm and height 2 m.

Solution:

  1. Identify the given values: Diameter (d)=14 cm(d) = 14 \text{ cm} Height (h)=2 m=200 cm(h) = 2 \text{ m} = 200 \text{ cm}

  2. Calculate the radius: r=d2=142=7 cmr = \frac{d}{2} = \frac{14}{2} = 7 \text{ cm}

  3. Use the formula for volume of a cylinder: V=πr2hV = \pi r^2 h V=227×7×7×200V = \frac{22}{7} \times 7 \times 7 \times 200 V=22×7×200V = 22 \times 7 \times 200 V=154×200V = 154 \times 200 V=30,800 cm3V = 30,800 \text{ cm}^3

Final Answer: The volume of the pillar is 30,800 cm330,800 \text{ cm}^3.

Explanation:

To find the volume, ensure all units are consistent. Since the diameter was in cm and height in m, we converted the height to cm (1 m=100 cm1 \text{ m} = 100 \text{ cm}). Then, we used the radius (half the diameter) in the volume formula for a cylinder.