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Mensuration - Surface Area and Volume of 3D Solids (Cube, Cuboid, Cylinder)

Grade 9ICSE

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

🔑Concepts

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The cuboid is a three-dimensional solid with six rectangular faces. Its dimensions are length (ll), breadth (bb), and height (hh). The diagonal of a cuboid is the longest line segment connecting two opposite vertices, calculated as l2+b2+h2\sqrt{l^2 + b^2 + h^2}.

A cuboid showing length, breadth, and height.
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A cube is a special type of cuboid where all edges are equal in length (side =a= a). The volume is the cube of the side, and the total surface area is exactly 66 times the area of one square face.

A cube with side length a.
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A right circular cylinder has two parallel circular bases and a curved surface. The distance between the centers of the bases is the height (hh), and the radius of the circular base is rr. The Curved Surface Area (CSA) corresponds to the area of the rectangle formed if the cylinder is unrolled.

Cylinder showing radius r and height h.
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Cross-section concepts: For solids like prisms and cylinders, the Volume can be generalized as Area of Cross-section×Length\text{Area of Cross-section} \times \text{Length}. For a cylinder, the cross-section is a circle with area πr2\pi r^2.

📐Formulae

Volume of a Cuboid: V=l×b×hV = l \times b \times h

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

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

Diagonal of a Cuboid: d=l2+b2+h2d = \sqrt{l^2 + b^2 + h^2}

Volume of a Cube: V=a3V = a^3

Total Surface Area (TSA) of a Cube: TSA=6a2TSA = 6a^2

Lateral Surface Area (LSA) of a Cube: LSA=4a2LSA = 4a^2

Diagonal of a Cube: d=a3d = a\sqrt{3}

Volume of a Cylinder: V=πr2hV = \pi r^2 h

Curved Surface Area (CSA) of a Cylinder: CSA=2πrhCSA = 2\pi rh

Total Surface Area (TSA) of a Cylinder: TSA=2πr(r+h)TSA = 2\pi r(r + h)

💡Examples

Problem 1:

A rectangular water tank is 5m5 m long, 3m3 m wide, and 2m2 m high. Find its capacity in liters and the total surface area of its outer walls and base (excluding the top).

Solution:

  1. Volume (VV) = l×b×h=5×3×2=30m3l \times b \times h = 5 \times 3 \times 2 = 30 m^3.
  2. Since 1m3=10001 m^3 = 1000 liters, Capacity = 30×1000=30,00030 \times 1000 = 30,000 liters.
  3. Area of 4 walls (LSA) = 2h(l+b)=2×2(5+3)=4×8=32m22h(l + b) = 2 \times 2(5 + 3) = 4 \times 8 = 32 m^2.
  4. Area of the base = l×b=5×3=15m2l \times b = 5 \times 3 = 15 m^2.
  5. Total area required = Area of walls + Area of base = 32+15=47m232 + 15 = 47 m^2.

Explanation:

To find the capacity, we first calculate the volume in cubic meters and then convert it to liters. For the surface area, we calculate the lateral surface area for the four walls and add only the area of the bottom rectangular face, as the top is excluded.

Problem 2:

A solid metallic cylinder has a radius of 7cm7 cm and a height of 20cm20 cm. Calculate its Curved Surface Area (CSA) and its Volume. (Take π=227\pi = \frac{22}{7})

Solution:

  1. Given: r=7cmr = 7 cm, h=20cmh = 20 cm.
  2. Curved Surface Area (CSA) = 2πrh2\pi rh
  3. CSA=2×227×7×20=2×22×20=880cm2CSA = 2 \times \frac{22}{7} \times 7 \times 20 = 2 \times 22 \times 20 = 880 cm^2.
  4. Volume (VV) = πr2h\pi r^2 h
  5. V=227×7×7×20=22×7×20=154×20=3080cm3V = \frac{22}{7} \times 7 \times 7 \times 20 = 22 \times 7 \times 20 = 154 \times 20 = 3080 cm^3.

Explanation:

We apply the direct formulas for a cylinder. The radius 77 cancels out with the denominator of π\pi, simplifying the multiplication for both the curved surface area and the total volume.

Problem 3:

A cube has a total surface area of 486cm2486 cm^2. Find the length of its edge and its volume.

A cube with unknown side a.

Solution:

Given, Total Surface Area (TSA) =6a2=486cm2= 6a^2 = 486 cm^2 a2=4866a^2 = \frac{486}{6} a2=81a^2 = 81 a=81=9cma = \sqrt{81} = 9 cm Now, Volume of the cube V=a3V = a^3 V=93=9×9×9V = 9^3 = 9 \times 9 \times 9 V=729cm3V = 729 cm^3

Explanation:

Since the cube has 6 identical square faces, we find the area of one face first, then the side length. The volume is the side length raised to the power of three.

Problem 4:

A hollow cylindrical pipe is made of metal and has an external radius of 12cm12 cm and an internal radius of 10cm10 cm. If the height of the pipe is 14cm14 cm, calculate the volume of metal used in making the pipe. (Take π=227\pi = \frac{22}{7})

Cross section of a hollow cylinder showing inner radius r and outer radius R.

Solution:

R=12cmR = 12 cm r=10cmr = 10 cm h=14cmh = 14 cm

Volume of metal = External Volume - Internal Volume V=πR2h−πr2h=πh(R2−r2)V = \pi R^2 h - \pi r^2 h = \pi h(R^2 - r^2) V=227×14×(122−102)V = \frac{22}{7} \times 14 \times (12^2 - 10^2) V=22×2×(144−100)V = 22 \times 2 \times (144 - 100) V=44×44V = 44 \times 44 V=1936cm3V = 1936 cm^3

Explanation:

To find the volume of the material in a hollow cylinder, we subtract the volume of the inner cavity from the total external volume. The formula is V=π(R2−r2)hV = \pi(R^2 - r^2)h, where RR is the outer radius and rr is the inner radius.