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ProSyllabus Journal CBSE Class 8 Annual Assessment

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 2 weeks agoNCERT · Class 8 · Mathematics · Mensuration · Geometry · Area · Volume · Surface Area · Chapter 9

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Complete Solutions and Summary of Mensuration – NCERT Class 8 Mathematics Chapter 9

Mensuration - Complete Study Guide

Chapter Overview

2(lb + bh + hl) Cuboid Surface Area

6l² Cube Surface Area

2πr(r + h) Cylinder Surface Area

l × b × h Cuboid Volume

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.

Mensuration

Chapter Overview

What You'll Learn

Perimeter and Area

Area of Polygons

Surface Area of Solids

Volume of Solids

Historical Context

Key Highlights

Comprehensive Chapter Summary

1. Introduction to Mensuration

2. Area of a Polygon

Splitting Polygons

Examples and Try These

Hexagon Division

3. Solid Shapes

Cuboid, Cube, Cylinder

Right Circular Cylinder

Activities

4. Surface Area of Solids

Cuboid

Cube

Cylinder

5. Volume of Solids

Cuboid and Cube

Cylinder

6. Examples and Applications

Key Concepts and Definitions

Perimeter

Area

Trapezium Area

Rhombus Area

Cuboid Surface Area

Cylinder Volume

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.