NCERT Solutions & Concepts for Class 8 Maths Mensuration Ex 11.2

Understanding the Formulas: Trapezium and Rhombus

To solve the problems in mensuration ex 11 2 class 8 ncert, we need to master two major formulas:

1. Area of a Trapezium: A trapezium is a quadrilateral with one pair of parallel sides. If the lengths of the parallel sides are $a$ and $b$, and the perpendicular distance (height) between them is $h$, then:

$\text{Area of Trapezium} = \frac{1}{2} \times (a + b) \times h$
This formula works because you can split any trapezium into two triangles and a rectangle, or simply two triangles with the same height.

1. Area of a Rhombus: A rhombus is a special parallelogram where all four sides are equal, and its diagonals intersect at right angles ($90^\circ$). If the lengths of the two diagonals are $d_1$ and $d_2$, then:

$\text{Area of Rhombus} = \frac{1}{2} \times d_1 \times d_2$
Alternatively, since a rhombus is also a parallelogram, its area can also be calculated as $\text{base} \times \text{altitude}$ if those measurements are given instead.

Step-by-Step Approach to Solve Area Problems

1. Identify the Shape and Given Values — Read the problem statement carefully. Determine if the quadrilateral is a trapezium, a rhombus, or a general irregular polygon. Note down all given dimensions such as side lengths, heights, or diagonal lengths.
2. Unify the Units of Measurement — Ensure all dimensions are in the same unit (e.g., all in centimeters or all in meters). If they differ, convert them before applying any formula. For example, $1\text{ m} = 100\text{ cm}$.
3. Select and Apply the Correct Formula — Substitute the identified dimensions into the correct mathematical formula. For trapeziums, use $\frac{1}{2}(a+b)h$. For rhombuses, use $\frac{1}{2} d_1 d_2$.
4. Simplify and State the Final Unit — Perform the arithmetic calculation carefully. Always write your final answer with the appropriate unit of area, such as $\text{cm}^2$ or $\text{m}^2$.

Exam Tips & Common Mistakes to Avoid

• Height vs. Slant Side: In a trapezium, always use the perpendicular height ($h$) and not the non-parallel slant sides for calculating the area.
• Diagonal Confusion: In a rhombus, the diagonals are represented by $d_1$ and $d_2$. Do not mistake the side length of the rhombus for a diagonal.
• Unit Square: Remember that area is always measured in square units (e.g., $\text{m}^2$, $\text{cm}^2$). Do not forget to write the exponent $2$ in your exam sheet!
• Splitting Shapes: For general quadrilaterals, draw a diagonal to split it into two triangles. Find the area of both triangles using the formula $\text{Area} = \frac{1}{2} \times \text{diagonal} \times (h_1 + h_2)$ and add them together.

Practice Questions with Solutions

• Q: The shape of the top surface of a table is a trapezium. Find its area if its parallel sides are $1\text{ m}$ and $1.2\text{ m}$ and the perpendicular distance between them is $0.8\text{ m}$. A: Step 1: Identify the given values. Parallel sides of the trapezium are $a = 1\text{ m}$ and $b = 1.2\text{ m}$. Perpendicular height $h = 0.8\text{ m}$. Step 2: Apply the formula for the area of a trapezium. $\text{Area} = \frac{1}{2} \times (a + b) \times h$ Step 3: Substitute the values into the formula. $\text{Area} = \frac{1}{2} \times (1 + 1.2) \times 0.8$ $\text{Area} = \frac{1}{2} \times 2.2 \times 0.8$ $\text{Area} = 1.1 \times 0.8 = 0.88\text{ m}^2$ Final answer: The area of the top surface of the table is $0.88\text{ m}^2$.
• Q: The area of a trapezium is $34\text{ cm}^2$ and the length of one of the parallel sides is $10\text{ cm}$ and its height is $4\text{ cm}$. Find the length of the other parallel side. A: Step 1: Identify the given values. Area of trapezium $= 34\text{ cm}^2$. One parallel side $a = 10\text{ cm}$. Height $h = 4\text{ cm}$. Let the other parallel side be $b$. Step 2: Set up the equation using the formula. $\text{Area} = \frac{1}{2} \times (a + b) \times h$ $34 = \frac{1}{2} \times (10 + b) \times 4$\n Step 3: Solve for $b$. $34 = (10 + b) \times 2$ $17 = 10 + b$ $b = 17 - 10 = 7\text{ cm}$ Final answer: The length of the other parallel side is $7\text{ cm}$.
• Q: Find the area of a rhombus whose diagonals are of lengths $10\text{ cm}$ and $8.2\text{ cm}$. A: Step 1: Identify the given values. Diagonal $d_1 = 10\text{ cm}$. Diagonal $d_2 = 8.2\text{ cm}$. Step 2: Apply the formula for the area of a rhombus. $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ Step 3: Calculate the area. $\text{Area} = \frac{1}{2} \times 10 \times 8.2$ $\text{Area} = 5 \times 8.2 = 41\text{ cm}^2$ Final answer: The area of the rhombus is $41\text{ cm}^2$.
• Q: The floor of a building consists of 3000 tiles which are rhombus shaped and each of its diagonals are $45\text{ cm}$ and $30\text{ cm}$ in length. Find the total cost of polishing the floor, if the cost per $\text{m}^2$ is $Rs.\ 4$. A: Step 1: Find the area of one rhombus tile. $d_1 = 45\text{ cm}, \quad d_2 = 30\text{ cm}$ $\text{Area of 1 tile} = \frac{1}{2} \times 45 \times 30 = 675\text{ cm}^2$ Step 2: Find the total area of 3000 tiles. $\text{Total Area} = 3000 \times 675\text{ cm}^2 = 2,025,000\text{ cm}^2$ Step 3: Convert the total area from $\text{cm}^2$ to $\text{m}^2$. Since $1\text{ m}^2 = 10,000\text{ cm}^2$: $\text{Total Area in m}^2 = \frac{2,025,000}{10,000} = 202.5\text{ m}^2$ Step 4: Calculate the total cost. $\text{Total Cost} = 202.5 \times 4 = Rs.\ 810$ Final answer: The total cost of polishing the floor is $Rs.\ 810$.

Frequently Asked Questions