Complete Solutions and Summary of Mensuration – NCERT Class 8 Mathematics Chapter 9
Comprehensive explanations, examples, and exercises on area and perimeter of polygons, surface area, lateral area, volume of cube, cuboid, and cylinder, as well as practical applications from NCERT Class 8 Mathematics Chapter 9.
Updated: 1 day ago
Mensuration
Chapter 9: Mathematics
Complete Study Guide with Interactive Learning
Chapter Overview
What You'll Learn
Perimeter and Area
Understanding perimeter as boundary distance and area as covered region for plane figures like triangles, rectangles, circles.
Area of Polygons
Splitting quadrilaterals and polygons into triangles and trapeziums to calculate areas.
Surface Area of Solids
Calculating total and lateral surface areas for cubes, cuboids, and cylinders.
Volume of Solids
Finding volumes for cubes, cuboids, and cylinders using formulas and unit cubes.
Historical Context
This chapter builds on basic plane figures to explore complex polygons and introduces 3D solids like cubes, cuboids, and cylinders. It emphasizes practical applications, such as calculating areas for fields or volumes for containers, with examples like trapezium-shaped fields and rhombus tiles.
Key Highlights
Key formulas include area of trapezium \( \frac{1}{2} h (a + b) \), rhombus \( \frac{1}{2} d_1 d_2 \), and cylinder volume \( \pi r^2 h \). Activities like dividing polygons and forming nets reinforce concepts.
Comprehensive Chapter Summary
1. Introduction to Mensuration
The chapter introduces perimeter as the distance around a closed plane figure and area as the region it covers. It reviews calculations for triangles, rectangles, circles, and borders in rectangles. It extends to quadrilaterals, polygons, and solids like cubes, cuboids, cylinders, with formulas for surface area and volume.
2. Area of a Polygon
Splitting Polygons
Quadrilaterals are split into triangles; polygons like pentagons into triangles and trapeziums using diagonals or perpendiculars. Example: Pentagon area = sum of triangle and trapezium areas.
Formula for trapezium: \( \frac{1}{2} h (a + b) \).
Examples and Try These
Activities divide polygons into parts. Example 1: Trapezium field area 480 m², height 15 m, one side 20 m, find other side (44 m).
Example 2: Rhombus area 240 cm², one diagonal 16 cm, find other (30 cm). Formula: \( \frac{1}{2} d_1 d_2 \).
Hexagon Division
Example 3: Hexagon side 5 cm divided into trapeziums or triangles/rectangle, area 64 cm² both ways.
3. Solid Shapes
Cuboid, Cube, Cylinder
Solids have faces: Cuboid (6 rectangular), Cube (6 squares), Cylinder (2 circular + curved). Activities identify congruent faces.
Right Circular Cylinder
Parallel congruent circular faces, perpendicular axis. Non-right cylinders exist but focus on right ones.
Activities
Cut boxes to observe faces; discuss why some shapes aren't cylinders.
4. Surface Area of Solids
Cuboid
Total: \( 2(lb + bh + hl) \). Lateral: \( 2h(l + b) \). Example: Room whitewashing.
Cube
Total: \( 6l^2 \). Activities form nets, calculate areas.
Cylinder
Curved: \( 2\pi r h \). Total: \( 2\pi r(r + h) \). Unrolling to rectangle.
5. Volume of Solids
Cuboid and Cube
Cuboid: \( l \times b \times h \). Cube: \( l^3 \). Use unit cubes.
Cylinder
Volume: \( \pi r^2 h \). Relation to capacity (1 L = 1000 cm³).
6. Examples and Applications
Examples calculate heights, volumes, costs for aquariums, rooms, pillars, tanks.
Key Concepts and Definitions
Perimeter
Distance around a closed figure.
Area
Region covered by a figure.
Trapezium Area
\( \frac{1}{2} h (a + b) \)
Rhombus Area
\( \frac{1}{2} d_1 d_2 \)
Cuboid Surface Area
\( 2(lb + bh + hl) \)
Cylinder Volume
\( \pi r^2 h \)
Important Facts and Figures
Questions and Answers from Chapter
Short Questions
Q1. What is the area of a trapezium with parallel sides 1 m and 1.2 m, height 0.8 m?
Q2. Find the length of the other parallel side of a trapezium with area 34 cm², one side 10 cm, height 4 cm.
Q3. Find the area of a rhombus with diagonals 7.5 cm and 12 cm.
Q4. Find the side of a cube with surface area 600 cm².
Q5. Find the height of a cuboid with volume 900 cm³ and base area 180 cm².
Q6. Find the height of a cylinder with volume 1.54 m³ and base diameter 140 cm.
Q7. Find the area of a rhombus with side 5 cm and altitude 4.8 cm.
Q8. Find the length of the other diagonal of a rhombus with side 5 cm, one diagonal 8 cm.
Q9. Find the area of a trapezium field with fence 120 m, sides 48 m, 17 m, 40 m (AB perpendicular).
Q10. Find the area of a quadrilateral field with diagonal 24 m, perpendiculars 8 m and 13 m.
Q11. Find the metal sheet required for a closed cylindrical tank with radius 7 m, height 3 m.
Q12. Find the perimeter of a rectangular sheet from a hollow cylinder lateral area 4224 cm², width 33 cm.
Q13. Find the area leveled by a road roller (diameter 84 cm, length 1 m) in 750 revolutions.
Q14. Find the label area for a milk powder cylinder (diameter 14 cm, height 20 cm, label 2 cm from top/bottom).
Q15. If each edge of a cube is doubled, how many times does the surface area increase?
Medium Questions
Q1. Find the area of a trapezium field with area 10500 m², height 100 m, one side along river twice the road side.
Q2. Find the total cost of polishing a floor with 3000 rhombus tiles (diagonals 45 cm, 30 cm) at ₹4/m².
Q3. For two cuboidal boxes (60 cm × 50 cm × 50 cm and 50 cm × 60 cm × 50 cm), which requires less material?
Q4. How many meters of 96 cm wide tarpaulin for 100 suitcases (80 cm × 48 cm × 24 cm)?
Q5. Rukhsar painted a cabinet 1 m × 2 m × 1.5 m except bottom. Find surface area covered.
Q6. Daniel paints a hall 15 m × 10 m × 7 m with 100 m² per can. How many cans?
Q7. Compare two figures: cylinder (r=7 cm, h=7 cm) and cube (side=7 cm). Which has larger lateral area?
Q8. How many 6 cm cubes in a cuboid 60 cm × 54 cm × 30 cm?
Q9. Milk tank cylinder r=1.5 m, length 7 m. Quantity in liters?
Q10. If cube edge doubled, how many times volume increases?
Q11. Reservoir volume 108 m³ fills at 60 L/min. Hours to fill?
Q12. Find volume of cylinder from paper 11 cm × 4 cm folded to height 4 cm.
Q13. Compare volumes of cylinders A (d=7 cm, h=14 cm) and B (d=14 cm, h=7 cm).
Q14. Godown 60 m × 40 m × 30 m, boxes 0.8 m³. How many boxes?
Q15. Cylinder from paper 14 cm width, radius 20 cm. Volume?
Long Questions
Q1. Explain how to find the area of a polygon by splitting it into triangles and trapeziums, with an example from the chapter.
Q2. Describe the surface area calculation for a cuboid, including lateral area, with an example.
Q3. Explain volume and capacity, with relations between units and an example.
Q4. Discuss finding area of a rhombus floor with tiles and cost calculation.
Q5. Analyze painting cost for cylindrical pillars.
Q6. Explain cylinder height from total surface area.
Q7. Discuss cuboid whitewashing cost including ceiling.
Q8. How to find boxes in godown?
Q9. Explain cylinder from rolled paper volume.
Q10. Compare cylinder volumes and surfaces.
Q11. Discuss octagonal surface area.
Q12. Explain pentagonal park area methods.
Q13. Picture frame sections area.
Q14. Discuss road roller area.
Q15. Hollow cylinder sheet perimeter.
Interactive Knowledge Quiz
Test your understanding of Mensuration
Quick Revision Notes
Trapezium Area
- \( \frac{1}{2} h (a + b) \)
- For fields, borders
Rhombus
- \( \frac{1}{2} d_1 d_2 \)
- Side × altitude
Cuboid SA/Vol
- 2(lb+bh+hl)
- l × b × h
Cylinder
- 2πr(r+h)
- πr²h
Exam Strategy Tips
- Memorize formulas
- Practice splitting shapes
- Use units correctly
- Solve examples
- Check calculations
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