Mensuration Word Problems
Mensuration word problems require you to apply formulas for area, surface area, and volume to real-life situations. These problems involve painting walls, filling tanks, wrapping boxes, and calculating material needed for construction.
The key to solving these problems is identifying the correct shape (cube, cuboid, cylinder, trapezium, etc.) and selecting the right formula.
This topic covers mixed word problems from the Class 8 Mensuration chapter, combining all formulas you have learned.
What is Mensuration Word Problems?
Key formulas for Class 8 Mensuration:
- Area of trapezium = ½ × (sum of parallel sides) × height
- Area of rhombus = ½ × d₁ × d₂
- Area of general quadrilateral = ½ × diagonal × (sum of perpendiculars)
- Surface area of cube = 6a²
- Surface area of cuboid = 2(lb + bh + lh)
- Lateral surface area of cuboid = 2h(l + b)
- Surface area of cylinder = 2πr(r + h)
- Volume of cube = a³
- Volume of cuboid = l × b × h
- Volume of cylinder = πr²h
Solved Examples
Example 1: Example 1: Painting a room
Problem: A room is 8 m long, 6 m wide, and 3.5 m high. It has one door (2 m × 1 m) and two windows (1.5 m × 1 m each). Find the cost of painting the walls at Rs 25 per m².
Solution:
- Lateral surface area = 2h(l + b) = 2 × 3.5 × (8 + 6) = 7 × 14 = 98 m²
- Door area = 2 × 1 = 2 m²
- Window area = 2 × (1.5 × 1) = 3 m²
- Area to paint = 98 − 2 − 3 = 93 m²
- Cost = 93 × 25 = Rs 2,325
Answer: Cost of painting = Rs 2,325.
Example 2: Example 2: Filling a cylindrical tank
Problem: A cylindrical tank has radius 1.4 m and height 2 m. How many litres of water can it hold? (1 m³ = 1000 litres)
Solution:
- Volume = πr²h = 22/7 × 1.4² × 2
- = 22/7 × 1.96 × 2
- = 22/7 × 3.92
- = 12.32 m³
- = 12.32 × 1000 = 12,320 litres
Answer: The tank holds 12,320 litres.
Example 3: Example 3: Wrapping a cuboid box
Problem: A gift box is 30 cm × 20 cm × 10 cm. Find the area of wrapping paper needed (ignore overlaps).
Solution:
- Total surface area = 2(lb + bh + lh)
- = 2(30×20 + 20×10 + 30×10)
- = 2(600 + 200 + 300)
- = 2 × 1100 = 2200 cm²
Answer: Paper needed = 2,200 cm².
Example 4: Example 4: Trapezium-shaped field
Problem: A field shaped like a trapezium has parallel sides 25 m and 15 m, and the distance between them is 10 m. Find its area.
Solution:
- Area = ½ × (a + b) × h
- = ½ × (25 + 15) × 10
- = ½ × 40 × 10 = 200 m²
Answer: Area = 200 m².
Example 5: Example 5: Metal sheet for a cube
Problem: How much metal sheet (in m²) is needed to make 20 cubes, each of side 0.5 m?
Solution:
- Surface area of one cube = 6a² = 6 × (0.5)² = 6 × 0.25 = 1.5 m²
- For 20 cubes = 20 × 1.5 = 30 m²
Answer: Metal sheet needed = 30 m².
Example 6: Example 6: Rhombus-shaped tile
Problem: A tile is shaped like a rhombus with diagonals 12 cm and 8 cm. Find the area of 50 such tiles.
Solution:
- Area of one tile = ½ × d₁ × d₂ = ½ × 12 × 8 = 48 cm²
- Area of 50 tiles = 50 × 48 = 2,400 cm²
Answer: Total area = 2,400 cm².
Example 7: Example 7: Water in a cuboid tank
Problem: A cuboidal tank is 2 m long, 1.5 m wide, and 1 m deep. Water flows in at 0.5 litres per second. How long to fill the tank?
Solution:
- Volume = 2 × 1.5 × 1 = 3 m³ = 3000 litres
- Time = 3000 ÷ 0.5 = 6000 seconds
- = 6000/60 = 100 minutes = 1 hour 40 minutes
Answer: Time to fill = 1 hour 40 minutes.
Example 8: Example 8: Lateral surface of a cylinder
Problem: A cylindrical pillar has diameter 1.4 m and height 5 m. Find the cost of painting its curved surface at Rs 20 per m².
Solution:
- Radius = 1.4/2 = 0.7 m
- Curved surface area = 2πrh = 2 × 22/7 × 0.7 × 5 = 22 m²
- Cost = 22 × 20 = Rs 440
Answer: Cost = Rs 440.
Example 9: Example 9: Comparing volumes
Problem: Which has more volume: a cube of side 10 cm or a cylinder of radius 7 cm and height 7 cm?
Solution:
- Volume of cube = 10³ = 1000 cm³
- Volume of cylinder = πr²h = 22/7 × 7² × 7 = 22/7 × 343 = 1078 cm³
Cylinder has more volume.
Answer: The cylinder has more volume (1078 cm³ > 1000 cm³).
Example 10: Example 10: General quadrilateral
Problem: A quadrilateral has a diagonal of 16 cm. The perpendiculars from the other two vertices to this diagonal are 5 cm and 7 cm. Find the area.
Solution:
- Area = ½ × diagonal × (h₁ + h₂)
- = ½ × 16 × (5 + 7)
- = ½ × 16 × 12 = 96 cm²
Answer: Area = 96 cm².
Real-World Applications
Where these problems appear:
- Construction: Calculating paint, tiles, flooring material needed.
- Water storage: Tank capacities for homes, factories, and agriculture.
- Packaging: Material needed for boxes, cartons, and cylinders.
- Farming: Area of irregularly shaped fields (trapezium, quadrilateral).
- Manufacturing: Metal sheet calculations for containers.
Key Points to Remember
- Identify the shape first, then select the correct formula.
- For painting problems, use lateral/curved surface area and subtract doors and windows.
- For filling problems, find volume and convert units (1 m³ = 1000 litres).
- For wrapping problems, use total surface area.
- 1 m³ = 1000 litres = 1,000,000 cm³.
- 1 litre = 1000 cm³.
- Use π = 22/7 for most calculations (or 3.14 if specified).
- Read carefully: is it total surface area or lateral surface area?
Practice Problems
- A cuboid tank is 4 m × 3 m × 2.5 m. How many litres of water can it hold?
- A trapezium-shaped park has parallel sides 40 m and 30 m, and height 12 m. Find its area.
- A cylindrical pipe has radius 3.5 cm and length 2 m. Find the volume of water it can hold.
- Find the cost of painting 4 walls of a room (5 m × 4 m × 3 m) at Rs 30/m², with 2 doors of 2 m × 1 m.
- A cube has surface area 600 cm². Find its volume.
- How many litres of water can fill a cylinder of diameter 2.8 m and height 3 m?
Frequently Asked Questions
Q1. When do I use lateral surface area vs total surface area?
Use lateral (curved) surface area for problems like painting walls or wrapping the sides only. Use total surface area when the top and bottom are also covered.
Q2. How do I convert m³ to litres?
1 m³ = 1000 litres. Multiply the volume in m³ by 1000.
Q3. What if the problem gives dimensions in different units?
Convert all dimensions to the same unit before calculating. For example, convert cm to m or m to cm first.
Q4. How do I find the cost from surface area?
Cost = Surface area × rate per unit area. Make sure the units match (both in m² or both in cm²).
Q5. What formula is used for the area of a general quadrilateral?
Area = ½ × diagonal × (sum of perpendicular heights from the other two vertices to the diagonal).
Q6. Should I use π = 22/7 or 3.14?
Use whichever is specified in the problem. If not specified, use 22/7 for NCERT problems (it's easier for calculations with 7 or 14 in the numbers).
