krit.club logo

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

•

A cuboid is a three-dimensional solid with six rectangular faces. Visually, it has three pairs of identical opposite faces that meet at right angles. The dimensions are defined by length (ll), breadth (bb), and height (hh).

•

A cube is a special case of a cuboid where all dimensions are equal (l=b=h=al = b = h = a). Visually, it appears as a solid where all six faces are identical squares, such as a standard playing die.

•

A right circular cylinder is a solid with two congruent circular bases connected by a curved surface. Visually, it looks like a pipe or a soup can, characterized by the radius (rr) of the base and the perpendicular height (hh) between the two circular faces.

•

Surface Area is categorized into Lateral/Curved Surface Area (LSA/CSA) and Total Surface Area (TSA). LSA/CSA refers only to the area of the vertical or surrounding faces (excluding top and bottom), while TSA includes the area of all surfaces including the top and bottom bases.

•

Volume represents the total amount of three-dimensional space occupied by the object. For solids with a uniform cross-section (like the cube, cuboid, and cylinder), the volume is calculated as Area of Base×Height\text{Area of Base} \times \text{Height}.

•

The diagonal of a cuboid or cube is the longest straight line segment that can be placed inside the solid. Visually, this line connects two opposite vertices, passing through the interior center of the solid.

•

Cross-section refers to the shape exposed by making a straight cut through a solid. For these solids, if you cut parallel to the base, the cross-section is identical to the base (a rectangle for a cuboid, a square for a cube, and a circle for a cylinder).

•

Units of measurement must be consistent across all dimensions before calculation. Area is measured in square units (cm2,m2cm^2, m^2), and volume is measured in cubic units (cm3,m3cm^3, m^3 or liters, where 1 liter=1000cm31 \text{ liter} = 1000 cm^3).

📐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. \n2. Since 1m3=10001 m^3 = 1000 liters, Capacity = 30×1000=30,00030 \times 1000 = 30,000 liters. \n3. 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. \n4. Area of the base = l×b=5×3=15m2l \times b = 5 \times 3 = 15 m^2. \n5. 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. \n2. Curved Surface Area (CSA) = 2πrh2\pi rh \n3. 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. \n4. Volume (VV) = πr2h\pi r^2 h \n5. 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.