Official NCERT 2026-27 Solutions
NCERT Solutions for Class 8 Maths Chapter 14: Area (Ganita Prakash Part II)
Authentic handwritten solutions in Topper Notebook style. Features geometric area derivations, ancient Indian Sulba-Sutra dissections, rhombus & trapezium transformations, regular hexagon area ratios, and real-life unit conversions from Pages 162 to 171 of Ganita Prakash Part II.
Quick Chapter Navigation & Blueprint
- Part A: Parallelograms & Dissection Invariance (Pages 162–165) — Base × Height formula derivation via rectangle shear, diagonal dissection of rhombuses, and area of rhombus (150 cm2).
- Part B: Trapeziums & Ancient Sulba-Sutra Dissections (Pages 166–169) — The ½h(a + b) formula derived by joining two copies into a parallelogram, converting trapeziums into rectangles of equal area, and constructing a 144 cm2 trapezium.
- Part C: Hexagonal Partitions & Geometric Theorems (Pages 169–170) — Dividing a regular hexagon into a trapezium, triangle, and rhombus (ratio 3 : 1 : 2), and proving Area(Trapezium ZYXW) = Area(ΔZWB).
- Part D: Practical Mensuration & Unit Conversions (Pages 170–171) — Standard A4 sheet area (623.7 cm2), square inches to cm2 (6.4516 cm2/in2), converting 161.29 cm2 → 25 in2, and acre conversions (43,560 sq. ft.).
Official Textbook PDF Reference: Download NCERT Class 8 Maths Chapter 14 (hegp207.pdf)
Part A: Parallelograms & Rhombus Formulations (Pages 162–165)
Parallelogram Area Derivation (Page 162):
Explain how the formula Area = Base × Height for a parallelogram ABCD is derived by cutting and translating a right-angled triangle.
Explain how the formula Area = Base × Height for a parallelogram ABCD is derived by cutting and translating a right-angled triangle.
Topper’s Dissection & Translation Proof:
- Let ABCD be a parallelogram with base b = DC and perpendicular height h = AX.
- From top vertex A, drop perpendicular AX to base DC. This forms a right-angled triangle ΔADX.
- From vertex B, drop perpendicular BY meeting the extension of base line DC at Y.
- By RHS Congruence Criterion, ΔADX ≅ ΔBCY (hypotenuse AD = BC, height AX = BY = h, and right angles at X and Y).
- If we cut triangle ΔADX from the left and translate it to the right to fit onto ΔBCY, the parallelogram ABCD transforms into a rectangle ABYX of length b and breadth h!
- Since area is conserved under translation:
Area(Parallelogram ABCD) = Area(Rectangle ABYX) = Base × Height.
Fundamental Formula: Area of Parallelogram = Base × Height.
Question 1 (Page 169):
Find the area of a rhombus whose diagonals are of lengths 20 cm and 15 cm.
Find the area of a rhombus whose diagonals are of lengths 20 cm and 15 cm.
Rhombus Diagonal Formula:
- Given: Diagonals d1 = 20 cm and d2 = 15 cm.
- Formula: Area of a Rhombus = ½ × Product of Diagonals = ½ × d1 × d2.
- Substituting values:
Area = ½ × 20 cm × 15 cm = 10 × 15 = 150 cm2.
Final Answer: The area of the rhombus is 150 cm2.
Part B: Trapezium Formulations & Sulba-Sutra Dissections (Pages 166–169)
Trapezium Area Derivation (Page 168):
Derive the area of a trapezium with parallel sides a and b and height h by rotating a congruent copy to form a parallelogram.
Derive the area of a trapezium with parallel sides a and b and height h by rotating a congruent copy to form a parallelogram.
Two-Copy Parallelogram Assembly Proof:
- Take two identical copies of trapezium ABCD with top parallel side a, bottom parallel side b, and height h.
- Rotate the second copy by 180° and join it along the slanted side BC of the first trapezium.
- The resulting combined shape is a large parallelogram whose base is (a + b) and whose perpendicular height is h.
- Area of combined parallelogram = Base × Height = (a + b) × h.
- Since this parallelogram is made of exactly two identical copies of the trapezium:
Area(Trapezium) = ½ × Area(Parallelogram) = ½ × h × (a + b).
Fundamental Formula: Area of Trapezium = ½ × Height × (Sum of Parallel Sides).
Question 6 (Page 170):
Using the idea of converting a trapezium into a rectangle of equal area, construct a trapezium of area 144 cm2.
Using the idea of converting a trapezium into a rectangle of equal area, construct a trapezium of area 144 cm2.
Step-by-Step Dimension Synthesis:
- We need: ½ × h × (a + b) = 144 cm2 → h × (a + b) = 288.
- Choose a convenient height h = 8 cm.
- Then sum of parallel sides: (a + b) = 288 / 8 = 36 cm.
- Select two positive unequal lengths that add to 36 cm:
• Top parallel side a = 12 cm
• Bottom parallel side b = 24 cm - Check Area: ½ × 8 × (12 + 24) = 4 × 36 = 144 cm2 ✓!
Constructed Dimensions: Height h = 8 cm, Parallel sides a = 12 cm and b = 24 cm.
Part C: Hexagonal Partitions & Real-Life Conversions (Pages 170–171)
Question 7 (Page 170 – Hexagon Area Ratio):
A regular hexagon is partitioned into a trapezium, an equilateral triangle, and a rhombus. Find the ratio of their areas.
A regular hexagon is partitioned into a trapezium, an equilateral triangle, and a rhombus. Find the ratio of their areas.
Unit Equilateral Triangle Counting Method:
- By drawing the three main diagonals through its center, any regular hexagon can be partitioned into exactly 6 congruent equilateral triangles, each of area T.
Total Area of regular hexagon = 6T. - Examining the three partitioned components:
• Equilateral Triangle: Exactly 1 unit equilateral triangle → Area = 1T.
• Rhombus: Composed of 2 unit equilateral triangles joined base-to-base → Area = 2T.
• Isosceles Trapezium: Composed of the remaining 3 unit equilateral triangles → Area = 3T. - Total check: 1T + 2T + 3T = 6T (complete hexagon) ✓!
- Ratio of Areas:
Trapezium : Rhombus : Triangle = 3 : 2 : 1 (or Trapezium : Triangle : Rhombus = 3 : 1 : 2).
Final Answer: The ratio of their areas is Trapezium : Rhombus : Triangle = 3 : 2 : 1.
Real-Life Area Units & Conversions (Pages 170–171):
(i) What is the exact area of an ISO standard A4 sheet of paper (21.0 cm × 29.7 cm)?
(ii) Given 1 inch = 2.54 cm, convert 161.29 cm2 to square inches (in2).
(iii) How many square feet are in 1 acre of land?
(i) What is the exact area of an ISO standard A4 sheet of paper (21.0 cm × 29.7 cm)?
(ii) Given 1 inch = 2.54 cm, convert 161.29 cm2 to square inches (in2).
(iii) How many square feet are in 1 acre of land?
Unit Conversions & Calculations:
- (i) Area of A4 Sheet:
Area = 21.0 cm × 29.7 cm = 623.7 cm2 = 0.06237 m2.
(Notice: An A0 sheet is exactly 1 m2, and A4 is 1/16th of an A0 sheet → 10,000 / 16 = 625 cm2!). - (ii) Converting 161.29 cm2 to in2:
1 in = 2.54 cm → 1 in2 = (2.54 cm)2 = 6.4516 cm2.
Area in in2 = 161.29 / 6.4516 = 25 in2 (exactly 5 in × 5 in!). - (iii) Acre Definition:
1 acre = 43,560 sq. ft. (approximately 4,047 m2).
Summary:
(i) A4 Sheet Area = 623.7 cm2.
(ii) 161.29 cm2 = 25 in2.
(iii) 1 acre = 43,560 sq. ft.
(i) A4 Sheet Area = 623.7 cm2.
(ii) 161.29 cm2 = 25 in2.
(iii) 1 acre = 43,560 sq. ft.