Class 8 Ganita Prakash Part 2 Chapter 7 Area: Complete NCERT Solutions, Triangle Parallelogram Rhombus Trapezium Formulas & Real-Life Applications

Introduction

“Area” is Chapter 7 of Ganita Prakash Part-II (Class 8). It shows how to measure the space inside shapes using unit squares and develops powerful formulas for triangles, parallelograms, rhombus and trapeziums. You will learn dissection methods from ancient Śulba-Sūtras, convert shapes of equal area, compare perimeter vs area, and apply everything to real-life objects like A4 sheets, classrooms and land measurement. Every concept is built step-by-step so you can solve any area problem confidently.

Basic Knowledge Required

• Area of rectangle = length × width
• Perimeter vs area difference
• Midpoint and median in triangles
• Unit conversion (cm, in, ft, m, km, acre)
• Basic dissection and rearrangement of shapes

Key Definitions

• Area: Number of unit squares that exactly cover a region (can be fractional).
• Height (altitude): Perpendicular distance from base to opposite vertex/side.
• Dissection: Cutting a shape into pieces and rearranging to form another shape of equal area.
• Trapezium: Quadrilateral with exactly one pair of parallel sides.

Important Formulas Used

1. Rectangle: $\text{Area}=\text{length}\times \text{width}$
2. Triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
3. Parallelogram: $\text{Area}=\text{base}\times \text{height}$
4. Rhombus: $= \frac{1}{2} \times d_1 \times d_2$
5. Trapezium: $\text{Area} = \frac{1}{2} \times \text{height} \times (\text{sum of parallel sides})$

7.1 Rectangle and Squares

Why perimeter cannot measure area Same perimeter can give different areas (e.g., long thin rectangle vs square).

Figure it Out 1 (Page 150) (i) Missing sides calculated from given areas: top = 7 in, right = 2 in (using 28 in², 21 in², 35 in²). (ii) Right side = 11 in – 4 in = 7 in (from 29 in² and total 50 in²).

Figure it Out 2 – Path around park (Page 151) (i) Need outer length, width & inner park size. Formula: Area of path = Area(outer) – Area(inner). (ii) Width given → break into 4 rectangles + 4 corner squares. Formula: $2w(L + B) + 4w^2$. (iii) Area remains same (outer moves but inner fixed → difference constant).

Figure it Out 3 – Cross-path (Page 151) Need width of each arm. Formula: 2 × length × width – overlapping square.

Spiral tube (Page 152) Area = 20×20 – 15×15 + 10×10 – 5×5 = 400 – 225 + 100 – 25 = 250 sq units. Straight tube length = 250 / 5 = 50 units.

Square doubled (Page 152) Region 1 increases 3 times, Region 2 & 3 increase 4 times (scale factor 4 for area).

Figure it Out 6 – Rearrange square (Page 152) Pieces form larger square with central hole (area preserved).

Triangles

Area of any triangle = $\frac{1}{2} \times \text{base} \times \text{height}$

Figure it Out 1 (Page 157) (i) $\frac{1}{2} \times 4 \times 3$ cm² (ii) $\frac{1}{2} \times 5 \times 3.2 =$ cm² (iii) $\frac{1}{2} \times 3 \times 4$ cm²

Altitude BY (Page 158) Area same → $\frac{1}{2} \times 6 \times BY = \frac{1}{2} \times 8 \times 4$ → BY = $\frac{16}{3}$ ≈ 5.33 units.

Isosceles ΔSUB (Page 158) Area ΔSUB = 2 × 24 = 48 sq units (SE common height).

Śulba-Sūtras conversions Rectangle → triangle: cut diagonally. Triangle → rectangle: cut median, rearrange halves.

Three squares problem (Page 158) (i) Blue = 49 sq units (red = blue). (ii) Each square = 30 sq units (total 180).

Midpoints M,N (Page 159) Area ΔXMN = $\frac{1}{4}$ of ΔXYZ (join NY, parallel lines halve area twice).

Area of any Polygon

Break into triangles using diagonals from one vertex.

Figure it Out (Page 160)

1. Quadrilateral: $\frac{1}{2} \times 22 \times 3 + \frac{1}{2} \times 22 \times 3$ cm².
2. Shaded = 10×6 – $\frac{1}{2} \times 10 \times 10$ + adjustments = 40 cm².
3. Regular hexagon: side + apothem (or 6 triangles).
4. Blue = $\frac{1}{2}$ of rectangle.
5. Draw one diagonal, cut half triangle.

Parallelogram

Area = base × height (dissection to rectangle).

Figure it Out (Page 162–163)

1. All same area (same base & height); perimeters different.
2. (i) 4×7 = 28 cm² (ii) $\frac{1}{2} \times 3 \times 5$ wait no – full base 5, height 3 = 15 cm² (iii) 5×4.8 = 24 cm² (iv) 4.4×2 = 8.8 cm²
3. QN = 7.6 cm (area same).
4. Rectangle larger (height full vs slanted).
5. Double triangle → two copies side-by-side.

Rhombus

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

Figure it Out (Page 169)

1. $\frac{1}{2} \times 20 \times 15$ cm². 2–5. Dissection methods as per Śulba-Sūtras.
   

Trapezium

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

Figure it Out All trapeziums solved by breaking or formula (details match page 166–169 examples).

Areas in Real Life

A4 sheet = 21 × 29.7 = 623.7 cm². 1 in² = 6.4516 cm², etc. (all conversions solved).

15 Most Asked FAQs – Area (Click to Expand)

1. Formula for area of triangle?

Answer: \( \frac{1}{2} \times \) base \( \times \) height. Holds for all triangles (difference of two rectangle triangles proves it).

2. How to find area of parallelogram?

Answer: Base × height (dissect to rectangle: cut perpendicular, rearrange triangle).

3. Rhombus area formula?

Answer: \( \frac{1}{2} \times d_1 \times d_2 \) (diagonals perpendicular bisectors → two rectangles).

4. Trapezium area formula?

Answer: \( \frac{1}{2} \times h \times (a + b) \) (rectangle + two triangles or two copies make parallelogram).

5. Area of spiral tube (width 5)?

Answer: 250 sq units (20²-15²+10²-5²); straight length = 50 units.

6. A4 sheet area?

Answer: 21 × 29.7 = 623.7 cm².

7. 1 in² in cm²?

Answer: 6.4516 cm² (2.54²).

8. Midpoints M,N in triangle XYZ → fraction?

Answer: \( \frac{1}{4} \) (parallel lines halve twice).

9. Same perimeter different area example?

Answer: 7×4 rectangle (28) vs 8×3 (24) – perimeter same 22 cm.

10. Regular hexagon area method?

Answer: 6 equilateral triangles or side + apothem.

11. Convert rectangle to triangle Śulba way?

Answer: Draw diagonal – two congruent triangles.

12. Rhombus diagonals 20 & 15 → area?

Answer: 150 cm².

13. Path around park formula?

Answer: Outer area – inner area (unchanged if outer moves).

14. Square side doubled – region increase?

Answer: Region 1 ×3, 2 & 3 ×4 (area scale factor).

15. 1 acre in ft²?

Answer: 43,560 ft².