NCERT Solutions Class 8 Maths Chapter 14 — Mensuration (Area & Volume)

Solution:

Step 1: The area of a trapezium is given by A = 1/2 × h × (sum of parallel sides). [A = 1/2 × h × (a + b)]

Step 2: The term (a+b)/2 represents the average length of the parallel sides. If we imagine stretching the shorter parallel side to match the longer, or vice versa, the 'effective' base length becomes the average. Multiplying this average base by the height 'h' indeed gives the area, just like a rectangle with that average base and height 'h'.

Step 3: Thus, Ravi's explanation correctly captures the essence of how the formula works conceptually.

Solution:

Step 1: A regular hexagon can be divided into simpler shapes. If divided into two identical trapeziums and one rectangle, you'd need the dimensions for each of these shapes.

Step 2: The side length of the hexagon would give you the parallel sides of the trapeziums and one side of the rectangle. To find the area of the trapeziums, you would also need their height (the perpendicular distance between their parallel sides).

Step 3: This height is typically derived from the hexagon's geometry, often related to its apothem or the overall width. Therefore, the side length and the height of the trapeziums are essential.

Solution:

Step 1: Surface area is the total area of all the faces (surfaces) of a three-dimensional object. For a cube, it has 6 identical square faces.

Step 2: Statement A is false: When a cube is cut into two cuboids, new surfaces are exposed (the cut faces), increasing the total surface area.

Step 3: Statement B is true: By definition, the total surface area of a cube is the sum of the areas of its six square faces. If 'a' is the side length, TSA = 6a².

Step 4: Statement C is false: Surface area is measured in square units (e.g., cm²), while volume is in cubic units (e.g., cm³); they cannot be equal.

Step 5: Statement D is false: If the side length 'a' is doubled to '2a', the new surface area becomes 6(2a)² = 6(4a²) = 24a², which is four times the original surface area (6a²), not double.

Solution:

Step 1: The quadrilateral ABCD can be divided into two triangles by the diagonal AC: triangle ABC and triangle ADC.

Step 2: Area of triangle ABC = 1/2 × base × height = 1/2 × AC × (perpendicular from B to AC) = 1/2 × 24 m × 8 m = 96 m².

Step 3: Area of triangle ADC = 1/2 × base × height = 1/2 × AC × (perpendicular from D to AC) = 1/2 × 24 m × 13 m = 156 m².

Step 4: Total area of the field = Area of triangle ABC + Area of triangle ADC = 96 m² + 156 m² = 252 m².

Solution:

Step 1: Original volume of the cuboid (V₁) = length × breadth × height = 10 cm × 5 cm × 4 cm = 200 cm³.

Step 2: New length = 2 × original length = 2 × 10 cm = 20 cm.

Step 3: New height = original height / 2 = 4 cm / 2 = 2 cm.

Step 4: New breadth remains the same = 5 cm.

Step 5: New volume of the cuboid (V₂) = new length × new breadth × new height = 20 cm × 5 cm × 2 cm = 200 cm³.

Step 6: Wait, let me re-calculate: New length = 20 cm, New breadth = 5 cm, New height = 2 cm. V₂ = 20 × 5 × 2 = 200 cm³.

Step 7: My calculation for V₂ was 200 cm³. Original V₁ = 200 cm³. This means the volume remains the same.

Step 8: Let me re-check my calculations and the question. Original L=10, B=5, H=4. V1 = 10*5*4 = 200. New L=2*10=20. New H=4/2=2. New B=5. V2 = 20*5*2 = 200. So the volume remains the same. The correct answer should be D. My previous correct answer was wrong.

Step 9: Let's re-evaluate. Original V = l × b × h. New V' = (2l) × b × (h/2) = 2 × l × b × h / 2 = l × b × h. So, the new volume is equal to the original volume.

Step 10: Original volume (V_original) = 10 cm × 5 cm × 4 cm = 200 cm³.

Step 11: New length (L') = 2 × 10 cm = 20 cm.

Step 12: New height (H') = 4 cm / 2 = 2 cm.

Step 13: New breadth (B') = 5 cm (unchanged).

Step 14: New volume (V_new) = L' × B' × H' = 20 cm × 5 cm × 2 cm = 200 cm³.

Step 15: Comparing V_new and V_original, we find that V_new = V_original. Therefore, the new volume remains the same as the original volume.