In-Depth: 3D Geometry (Volume/Surface Area)

CLAT Application & Relevance

Importance: Medium. Similar to 2D geometry, 3D geometry problems in CLAT QT are typically integrated into Data Interpretation caselets. You'll need to calculate volumes or surface areas of common 3D shapes (cubes, cuboids, cylinders, spheres, cones) when dimensions or capacities are described in a passage. These problems often relate to storage, construction, or resource allocation.

How it's tested: Calculating the capacity of a water tank; finding the cost of painting the external surface of a building; determining how many smaller items can fit into a larger container; problems involving melting and recasting shapes.

Section 1: Core Concepts & Formulas

3D (three-dimensional) geometry deals with solid objects that have length, width, and height. Key concepts include Volume (the space occupied by the object) and Surface Area (the total area of all surfaces of the object).

Key Formulas for Common 3D Shapes

Cube:
- Side = 'a'
- Volume = a³
- Total Surface Area (TSA) = 6a²
- Lateral Surface Area (LSA) / Area of 4 walls = 4a²
Cuboid:
- Length = 'l', Width = 'w', Height = 'h'
- Volume = l × w × h
- Total Surface Area (TSA) = 2(lw + wh + hl)
- Lateral Surface Area (LSA) / Area of 4 walls = 2h(l + w)
Cylinder:
- Radius of base = 'r', Height = 'h'
- Volume = πr²h
- Curved Surface Area (CSA) = 2πrh
- Total Surface Area (TSA) = 2πrh + 2πr² = 2πr(h + r)
Cone:
- Radius of base = 'r', Height = 'h', Slant height = 'l'
- l² = r² + h²
- Volume = (1/3)πr²h
- Curved Surface Area (CSA) = πrl
- Total Surface Area (TSA) = πrl + πr² = πr(l + r)
Sphere:
- Radius = 'r'
- Volume = (4/3)πr³
- Surface Area = 4πr²

Value of π (Pi) ≈ 22/7 or 3.14

Section 2: Solved CLAT-Style Examples

Example 1: Capacity (Volume) and Cost Calculation for a Cuboidal Tank (Caselet)

Passage Context: "A city's municipal law department is overseeing the installation of a new rectangular water tank for a public school. The inner dimensions of the tank are: length = 5 meters, width = 4 meters, and height = 3 meters. The tank needs to be filled with water, and the cost of water is ₹5 per cubic meter. The inner surface of the tank (excluding the top) needs to be waterproofed at a cost of ₹10 per square meter."

Question A: "What is the total capacity of the tank in liters?" (1 cubic meter = 1000 liters)

Question B: "What is the total cost of waterproofing the inner surface of the tank?"

Detailed Solution A (Capacity in Liters):
1. Calculate Volume of the Cuboid: Volume = l × w × h = 5 m × 4 m × 3 m = 60 cubic meters.
2. Convert Volume to Liters: Capacity = 60 cubic meters × 1000 liters/cubic meter = 60,000 liters.
Answer A: The total capacity of the tank is 60,000 liters.

Detailed Solution B (Cost of Waterproofing):
1. Calculate Area to be Waterproofed (5 faces: bottom + 4 walls): Area = Area of bottom + Area of 4 walls Area of bottom = l × w = 5 × 4 = 20 sq m. Area of 4 walls (LSA) = 2h(l + w) = 2 × 3 × (5 + 4) = 6 × 9 = 54 sq m. Total Area = 20 + 54 = 74 sq m.
2. Calculate Total Cost of Waterproofing: Total Cost = Total Area × Cost per sq meter = 74 sq m × ₹10/sq m = ₹740.
Answer B: The total cost of waterproofing the inner surface of the tank is ₹740.

Example 2: Volume Comparison (Cylinder and Cone)

Passage Context: "A law firm is designing a new cylindrical water cooler with a radius of 0.35 meters and a height of 1 meter. For a special event, they also order conical paper cups with a radius of 3.5 cm and a height of 10 cm."

Question: "How many conical paper cups can be completely filled from a full water cooler?" (Use π = 22/7)

Detailed Solution:
1. Convert Units to be Consistent (e.g., all to meters or all to cm): Water Cooler: Radius (R) = 0.35 m, Height (H) = 1 m. Paper Cup: Radius (r) = 3.5 cm = 0.035 m, Height (h) = 10 cm = 0.1 m.
2. Calculate Volume of Water Cooler (Cylinder): Volume_Cylinder = πR²H = (22/7) * (0.35)² * 1 = (22/7) * 0.1225 * 1 = 22 * 0.0175 = 0.385 cubic meters.
3. Calculate Volume of One Paper Cup (Cone): Volume_Cone = (1/3)πr²h = (1/3) * (22/7) * (0.035)² * 0.1 = (1/3) * (22/7) * 0.001225 * 0.1 = (1/3) * 22 * 0.000175 * 0.1 = (1/3) * 0.000385 = 0.00012833 cubic meters (approx).
4. Calculate Number of Cups: Number of Cups = Volume of Cooler / Volume of One Cup = 0.385 / 0.00012833 ≈ 3000.
Answer: Approximately 3000 conical paper cups can be filled from the water cooler.