Mensuration: CBSE Class 8 Maths NCERT Guide

Welcome to the world of Mensuration! Ever wondered how much paint you need for a wall, how much water a tank can hold, or how much carpet is needed for a room? Mensuration is the branch of mathematics that helps us answer these questions. It's all about measuring geometric figures.

In Class 7, you learned about the perimeter and area of simple shapes like squares, rectangles, and circles. Now, in Class 8, we will go deeper. You will master how to find the area of more complex 2D shapes like trapeziums and rhombuses. We will then step into the third dimension to explore 3D shapes like cubes, cuboids, and cylinders. You will learn to calculate not just the space they occupy (Volume) but also the area of their surfaces (Surface Area). This chapter is very practical and important for your exams, so let's build a strong foundation together!

Area of 2D Shapes: Trapezium and Rhombus

Let's start by looking at some special quadrilaterals. A trapezium is a quadrilateral with at least one pair of parallel sides. Imagine a table top or a cross-section of a canal; they are often in the shape of a trapezium. To find its area, we can't just multiply length and breadth. The formula is: Area = 1/2 × (sum of parallel sides) × height. Here, the 'height' is the perpendicular distance between the two parallel sides.

Next, we have the rhombus. It's a quadrilateral where all four sides are equal, like a tilted square. Its diagonals bisect each other at right angles (90°). This special property gives us a simple way to calculate its area using the lengths of its diagonals (d1 and d2): Area = 1/2 × (d1 × d2). You can also find the area of any general quadrilateral by splitting it into two triangles using a diagonal and then adding their areas.

Worked Examples: Surface Area of 3D Shapes

  • Problem: Find the Total Surface Area (TSA) of a cuboid with length 8 cm, breadth 5 cm, and height 4 cm. Solution: Step 1: Identify the formula. The formula for the Total Surface Area of a cuboid is TSA = 2(lb + bh + hl). Step 2: Substitute the values. l = 8 cm, b = 5 cm, h = 4 cm. TSA = 2((8 × 5) + (5 × 4) + (4 × 8)) Step 3: Calculate. TSA = 2(40 + 20 + 32) TSA = 2(92) TSA = 184 Final Answer: The Total Surface Area of the cuboid is 184 cm².
  • Problem: Find the Curved Surface Area (CSA) of a cylinder with a radius of 7 cm and a height of 10 cm. (Use π = 22/7) Solution: Step 1: Identify the formula. The formula for the Curved Surface Area of a cylinder is CSA = 2πrh. Step 2: Substitute the values. r = 7 cm, h = 10 cm, π = 22/7. CSA = 2 × (22/7) × 7 × 10 Step 3: Calculate. CSA = 2 × 22 × 10 CSA = 440 Final Answer: The Curved Surface Area of the cylinder is 440 cm².

How to Calculate Volume

  1. Understanding Volume — Volume is the amount of 3D space an object occupies. Think of it as the object's capacity. The basic idea for most regular shapes is to find the area of the base and multiply it by the height. This is like stacking up layers of the base until you reach the total height.
  2. Volume of a Cuboid — The base of a cuboid is a rectangle with area = length × breadth (l × b). To find the volume, we multiply this base area by the height (h). So, the formula is Volume = l × b × h.
  3. Volume of a Cube — A cube is a special cuboid where length = breadth = height = side (a). So, the formula becomes Volume = a × a × a = a³.
  4. Volume of a Cylinder — The base of a cylinder is a circle with area = πr². To find the volume, we multiply this base area by the height (h). So, the formula is Volume = πr²h.

Important Exam Tips for Mensuration

1. Units are Crucial: Always pay attention to units. Area is measured in square units (cm², m²) and volume is measured in cubic units (cm³, m³). Forgetting or writing the wrong unit can cost you marks.

2. LSA vs. TSA: Read the question carefully to see if it asks for Lateral/Curved Surface Area (LSA/CSA) or Total Surface Area (TSA). LSA/CSA refers to the area of the 'walls' only, while TSA includes the top and bottom surfaces as well.

3. Radius vs. Diameter: A common trick in questions is to provide the diameter. Remember to always divide the diameter by 2 to get the radius before using it in any cylinder or circle formulas.

Practice Questions with Solutions

  • Q: The parallel sides of a trapezium are 14 cm and 6 cm, and its height is 5 cm. What is its area? A: Step 1: Write down the formula for the area of a trapezium. Area = 1/2 × (sum of parallel sides) × height. Step 2: Substitute the given values into the formula. Sum of parallel sides = 14 + 6 = 20 cm. Height = 5 cm. Area = 1/2 × 20 × 5. Step 3: Calculate the final area. Area = 10 × 5 = 50 cm². Final answer: The area of the trapezium is 50 cm².
  • Q: A road roller has a diameter of 84 cm and its length is 1 m. It takes 750 complete revolutions to level a road. Find the area of the road. (Use π = 22/7) A: Step 1: Find the radius and convert units. Diameter = 84 cm, so radius (r) = 84 / 2 = 42 cm. The length of the roller is its height (h) = 1 m = 100 cm. Step 2: Calculate the area covered in one revolution, which is the Curved Surface Area (CSA) of the cylinder. CSA = 2πrh = 2 × (22/7) × 42 × 100 = 2 × 22 × 6 × 100 = 26400 cm². Step 3: Calculate the total area of the road for 750 revolutions. Total Area = CSA × 750 = 26400 × 750 = 19,800,000 cm². Step 4: Convert the area to square meters (1 m² = 10000 cm²). Total Area = 19,800,000 / 10000 = 1980 m². Final answer: The area of the road is 1980 m².
  • Q: Find the volume of a cube whose Total Surface Area is 600 cm². A: Step 1: Find the length of the side (a) from the TSA. The formula for TSA of a cube is 6a². So, 6a² = 600. Step 2: Solve for 'a'. a² = 600 / 6 = 100. So, a = √100 = 10 cm. Step 3: Now, calculate the volume using the side length. The formula for the volume of a cube is a³. Volume = 10³ = 10 × 10 × 10 = 1000 cm³. Final answer: The volume of the cube is 1000 cm³.
  • Q: The diagonals of a rhombus are 7.5 cm and 12 cm. Find its area. A: Step 1: Write down the formula for the area of a rhombus using its diagonals. Area = 1/2 × (product of diagonals) = 1/2 × d1 × d2. Step 2: Substitute the given values of the diagonals. d1 = 7.5 cm, d2 = 12 cm. Area = 1/2 × 7.5 × 12. Step 3: Calculate the final area. Area = 7.5 × 6 = 45 cm². Final answer: The area of the rhombus is 45 cm².

Frequently Asked Questions

What is the main difference between Surface Area and Volume?

Surface Area is the total area of the outer surfaces of a 3D object, measured in square units (like cm²). It's about 'covering' an object. Volume is the space inside a 3D object, measured in cubic units (like cm³). It's about 'filling' an object.

When should I use Lateral Surface Area (LSA) instead of Total Surface Area (TSA)?

Use LSA when you only need the area of the vertical faces or 'walls' of an object. For example, calculating the cost of painting the four walls of a room. Use TSA when you need to include the top and bottom surfaces as well, like finding the material to make a closed box.

How do you convert cubic centimeters (cm³) to Litres (L)?

This is a very important conversion for volume and capacity problems. Remember that 1000 cubic centimeters is equal to 1 Litre (1000 cm³ = 1 L). Also, 1 cubic meter (m³) is equal to 1000 Litres.