CBSE Class 8 Maths Chapter 7 Area

In earlier classes, you learned how to find the area of basic 2D shapes like rectangles, squares, and triangles. In Chapter 7, "Area" (often categorized under Mensuration), we take a step further. You will learn how to calculate the area of more complex and irregular shapes like trapeziums, general quadrilaterals, rhombuses, and multi-sided polygons.

The core secret to mastering this chapter is decomposition—the art of breaking down any weird, irregular, or complex shape into simpler, familiar shapes (like triangles and rectangles) to easily calculate the total area.

CBSE Class 8 Maths Chapter 7 Area

Questions & Answers

Q1. What is a trapezium, and what is the formula to calculate its area?

Answer: A trapezium is a quadrilateral that has exactly one pair of parallel sides.

    • The Formula: The area is calculated by taking the average of the two parallel sides and multiplying it by the perpendicular height (distance between the parallel sides).

    • Algebraic Form:

      $$Area = \frac{1}{2} \times (a + b) \times h$$

      (Where $a$ and $b$ are the lengths of the parallel sides, and $h$ is the perpendicular height).

Q2. How do you find the area of a general quadrilateral if it is not a special shape like a square or rectangle?

Answer: To find the area of a general quadrilateral, we split it into two triangles by drawing one of its diagonals.

  • The Method: Draw a diagonal (let's call its length $d$). Drop perpendicular lines (heights) from the remaining two opposite vertices onto this diagonal. Let's call these heights $h_1$ and $h_2$.

  • The Formula:

    $$Area = \frac{1}{2} \times d \times (h_1 + h_2)$$

  • Why it works: You are simply adding the areas of the two triangles: $(\frac{1}{2} \times d \times h_1) + (\frac{1}{2} \times d \times h_2)$.

Q3. What is the special formula for calculating the area of a rhombus using its diagonals?

Answer: While a rhombus is a parallelogram (and you can use base $\times$ height), there is a much faster formula if you know the lengths of its diagonals.

  • The Property: The diagonals of a rhombus bisect each other at perfect right angles (90 degrees), effectively dividing the rhombus into four identical right-angled triangles.

  • The Formula:

    $$Area = \frac{1}{2} \times d_1 \times d_2$$

    (Where $d_1$ and $d_2$ are the full lengths of the two diagonals).

Q4. A trapezium has parallel sides of 10 cm and 15 cm. The perpendicular distance between them is 6 cm. Find its area.

Answer: We directly apply the formula for the area of a trapezium.

  • Given: $a = 10$ cm, $b = 15$ cm, and $h = 6$ cm.

  • Formula: $Area = \frac{1}{2} \times (a + b) \times h$

  • Calculation Step 1: $= \frac{1}{2} \times (10 + 15) \times 6$

  • Calculation Step 2: $= \frac{1}{2} \times 25 \times 6$

  • Calculation Step 3: $= 25 \times 3$

  • Final Answer: The area of the trapezium is 75 sq cm.

Q5. The diagonals of a rhombus are 8 cm and 12 cm. Find its area.

Answer: Since the lengths of the diagonals are given, we use the specific diagonal formula for a rhombus.

  • Given: $d_1 = 8$ cm and $d_2 = 12$ cm.

  • Formula: $Area = \frac{1}{2} \times d_1 \times d_2$

  • Calculation Step 1: $= \frac{1}{2} \times 8 \times 12$

  • Calculation Step 2: $= 4 \times 12$

  • Final Answer: The area of the rhombus is 48 sq cm.

Q6. What is "Triangulation" and how is it used to find the area of an irregular polygon?

Answer: Triangulation is a method used to find the area of irregular polygons (like a 5-sided pentagon or an irregular field).

  • The Process: You draw non-intersecting diagonals inside the polygon to divide the entire shape into a series of smaller triangles.

  • Calculation: You then measure the base and height of each individual triangle, calculate their areas separately using the formula $\frac{1}{2} \times base \times height$, and add all those areas together to get the total area of the polygon.

Q7. A general quadrilateral has a diagonal of 20 cm. The perpendiculars dropped on it from the opposite vertices are 8 cm and 5 cm. Find its area.

Answer: We use the formula for a general quadrilateral broken into two triangles.

  • Given: Diagonal $d = 20$ cm, $h_1 = 8$ cm, and $h_2 = 5$ cm.

  • Formula: $Area = \frac{1}{2} \times d \times (h_1 + h_2)$

  • Calculation Step 1: $= \frac{1}{2} \times 20 \times (8 + 5)$

  • Calculation Step 2: $= 10 \times 13$

  • Final Answer: The area of the quadrilateral is 130 sq cm.

Q8. How can you find the area of a regular hexagon?

Answer: A regular hexagon has 6 equal sides and equal angles. You can calculate its area by decomposing it in two easy ways:

  • Method 1 (Equilateral Triangles): Draw diagonals from the center to all 6 vertices. This splits the hexagon into exactly 6 identical equilateral triangles. Find the area of one triangle and multiply by 6.

  • Method 2 (Trapeziums): Draw a single straight diagonal across the exact center. This splits the hexagon into two identical trapeziums. Find the area of one trapezium and multiply by 2.

Q9. If the area of a rhombus is 240 sq cm and one of its diagonals is 16 cm, find the length of the other diagonal.

Answer: We use the area formula backward to find the missing variable.

  • Given: $Area = 240$, $d_1 = 16$. Let the unknown diagonal be $d_2$.

  • Formula: $Area = \frac{1}{2} \times d_1 \times d_2$

  • Substitution: $240 = \frac{1}{2} \times 16 \times d_2$

  • Calculation Step 1: $240 = 8 \times d_2$

  • Calculation Step 2: $d_2 = \frac{240}{8}$

  • Final Answer: The length of the other diagonal is 30 cm.

Q10. Explain how to find the area of a complex polygonal field using a "Field Book" method.

Answer: In real-life land measurement (surveying), the shape of a field is often highly irregular.

  • The Method: A surveyor draws one long straight line through the center of the field, called the baseline or main diagonal. From the different corners of the field, they measure the perpendicular distance to this baseline (these are called offsets).

  • Calculation: These offsets divide the entire field into a sequence of right-angled triangles and trapeziums aligned along the baseline. By calculating the area of each distinct triangle and trapezium sequentially and adding them up, the total area of the irregular field is accurately found.

Frequently Asked Questions (FAQs)

1. Do I need to memorize separate formulas for squares and rectangles in this chapter?

No, squares and rectangles are just special types of quadrilaterals. However, knowing that a square's area is $side^2$ and a rectangle's area is $length \times width$ makes calculations faster. If you forget, you can always split them into two triangles and the rule will still work.

2. Can I use the rhombus diagonal formula for a square?

Yes. A square is technically a special rhombus where all angles are 90 degrees and both diagonals are perfectly equal. If a square has diagonals of length $d$, its area can be calculated as $\frac{1}{2} \times d \times d$.

3. What is the most common mistake students make in Chapter 7?

Forgetting the $\frac{1}{2}$ in the formulas. Whether it is a triangle, a trapezium, a rhombus, or a general quadrilateral, the area formula usually involves dividing by 2. Always double-check your final steps to ensure you didn't miss it.

4. How do I know where to draw the diagonal in an irregular polygon?

You can draw the diagonal between any two non-adjacent vertices. The total area will remain exactly the same no matter which diagonal you choose. Choose the diagonal that makes calculating the perpendicular heights easiest based on the given question values.