EXPLANATION AND ANALYSIS
🌍 INTRODUCTION — MEASURING SURFACE, NOT BOUNDARY
🧠 In everyday life, we often need to know how much surface is covered, not just the boundary around it.
📘 For example:
how much carpet is needed for a room
how much land is available for farming
how much paper is required to wrap a book
📌 All these situations involve area.
🎯 This lesson focuses on:
understanding what area means
comparing areas of different shapes
developing a clear sense of surface measurement
Area helps us measure space covered, not distance around.
📐 WHAT IS AREA?
🧠 Area is the measure of the surface enclosed by a closed figure.
📘 Important understanding:
area deals with two dimensions
it depends on length and breadth together
🔵 A larger boundary does not always mean larger area
🟡 Shapes with different boundaries can have equal areas
📌 Area answers the question:
👉 How much surface is inside the shape?
📏 AREA VS PERIMETER — DO NOT CONFUSE
🧠 Area and perimeter are often confused, but they measure different things.
📘 Difference in idea:
perimeter → length of boundary
area → surface enclosed
🔵 perimeter is one-dimensional
🟡 area is two-dimensional
📌 Two figures may have:
same perimeter but different area
same area but different perimeter
Understanding this difference is essential.
🟦 AREA AS COUNTING SQUARE UNITS
🧠 Area is measured by counting equal-sized squares.
📘 These squares are called square units.
🔵 square centimetre
🟡 square metre
🟣 square kilometre
📌 A shape’s area depends on:
number of square units covering it
no gaps and no overlaps
🧠 This idea helps us understand area visually and logically.
📐 COMPARING AREAS OF SHAPES
🧠 To compare areas, we do not rely on shape appearance.
📘 Instead, we:
cover shapes with square units
count the number of units
🔵 a long thin shape may have less area
🟡 a compact shape may have more area
📌 Comparison should be based on measurement, not guesswork.
🔄 SAME AREA, DIFFERENT SHAPES
🧠 Different shapes can have the same area.
📘 For example:
a rectangle
a square
an irregular shape
🔵 They may look different
🟡 but cover the same surface
📌 Area depends on coverage, not shape form.
This idea is important in design and construction.
📏 AREA OF RECTANGLES AND SQUARES (IDEA LEVEL)
🧠 Rectangles and squares are commonly used shapes.
📘 Their area depends on:
length
breadth
🔵 increasing one side increases area
🟡 decreasing one side reduces area
📌 Though formulas exist, the idea is:
area comes from combining two dimensions
Understanding the idea is more important than memorising formulas.
🟫 AREA AND GRIDS
🧠 Grids help us measure area easily.
📘 A grid divides a surface into equal squares.
🔵 each square represents one unit area
🟡 counting squares gives total area
📌 Grids are useful for:
irregular shapes
estimation
comparison
🧠 Grids connect geometry with practical measurement.
🔢 AREA OF IRREGULAR SHAPES
🧠 Not all shapes are perfect rectangles or squares.
📘 Irregular shapes appear in:
fields
lakes
plots of land
🔵 To find their area:
divide them into regular shapes
estimate using square grids
📌 Area measurement is flexible and adaptable.
📊 ESTIMATING AREA
🧠 Sometimes exact area is not required.
📘 Estimation helps when:
shapes are uneven
precise measurement is difficult
🔵 count full squares
🟡 estimate partial squares
📌 Estimation gives a reasonable idea of area.
This skill is useful in real-world situations.
🌍 AREA IN DAILY LIFE
🧠 Area is used in many daily activities.
🔵 measuring rooms
🟡 buying land
🟣 painting walls
🟠 laying tiles
🔴 designing parks
📘 Area helps in planning, budgeting, and decision-making.
🧠 AREA AND CONSERVATION OF SPACE
🧠 Area can remain constant even when shapes change.
📘 If we rearrange parts:
the area remains same
only shape changes
🔵 cutting and rearranging shapes preserves area
🟡 this idea is used in puzzles and design
📌 Area is about quantity of space, not arrangement.
🔍 THINKING LOGICALLY ABOUT AREA
🧠 Area is not just calculation; it involves reasoning.
📘 We must think about:
coverage
overlap
gaps
🔵 counting blindly may give wrong answers
🟡 logic ensures correctness
📌 Area measurement requires attention and understanding.
⚠️ COMMON MISTAKES TO AVOID
🔴 confusing area with perimeter
🟡 counting boundary instead of surface
🟣 ignoring partial units
🟠 assuming larger shape means larger area
✔️ Always focus on surface coverage.
🌟 IMPORTANCE OF THIS LESSON
🏆 builds strong measurement sense
🧠 improves spatial thinking
⚡ prepares for mensuration
📘 useful in science and geography
🌱 connects maths with real life
Area is a foundational concept for higher mathematics.
🧾 SUMMARY
🔵 area measures surface covered
🟡 it is measured in square units
🟣 area and perimeter are different
🟠 shapes can have equal area
🔴 grids help measure area
🟢 estimation is useful
🔁 QUICK RECAP
🔵 area means surface inside
🟡 measured using square units
🟣 comparison needs counting
🟠 same area, different shapes
🔴 logic is important
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TEXTBOOK QUESTIONS
🔒 ❓ 1. Find the area of a rhombus whose diagonals are 20 cm and 15 cm.
📌 ✅ Answer:
🟢 Step 1: Identify the Formula
⬥ Area of a rhombus = (Diagonal 1 × Diagonal 2) ÷ 2
⬥ A = (d_1 × d_2) ÷ 2
🔵 Step 2: Substitute Values
⬥ d_1 = 20 cm
⬥ d_2 = 15 cm
⬥ A = (20 × 15) ÷ 2
🟡 Step 3: Calculate
⬥ 20 × 15 = 300
⬥ 300 ÷ 2 = 150
🔴 Result:
⬥ Area = 150 cm²
🔒 ❓ 2. Give a method to convert a rectangle into a rhombus of equal area using dissection.
📌 ✅ Answer:
🟢 Step 1: Understanding Dissection
⬥ Dissection means cutting a shape into pieces and rearranging them to form a new shape without losing or adding any area.
🔵 Step 2: The Method
⬥ Concept: A rectangle is a parallelogram with 90° angles. To make it a rhombus (equal sides), we need to change the angles while keeping the base and height product the same.
⬥ Action: Cut a right-angled triangle from one side of the rectangle.
⬥ Move: Paste this triangle onto the opposite side.
🟡 Step 3: Adjustment
⬥ By sliding the cut angle, you can adjust the slant length until all four sides are equal, forming a rhombus of the exact same area.
🔒 ❓ 3. Find the areas of the following figures:
Figure (i): Right Trapezium
(Top = 10 ft, Height = 16 ft, Bottom projection = 7 ft)
📌 ✅ Answer:
🟢 Step 1: Identify Dimensions
⬥ Top side (a) = 10 ft.
⬥ Bottom side (b) = Top + Projection = 10 + 7 = 17 ft.
⬥ Height (h) = 16 ft.
🔵 Step 2: Calculate Area
⬥ Area = 0.5 × (10 + 17) × 16
⬥ Area = 0.5 × 27 × 16 = 27 × 8
⬥ Area = 216 sq ft.
Figure (ii): Trapezium
(Parallel sides = 24 m and 36 m, Height = 14 m)
📌 ✅ Answer:
🟢 Step 1: Identify Dimensions
⬥ a = 24 m, b = 36 m, h = 14 m.
🔵 Step 2: Calculate Area
⬥ Area = 0.5 × (24 + 36) × 14
⬥ Area = 0.5 × 60 × 14 = 30 × 14
⬥ Area = 420 sq m.
Figure (iii): Trapezium (Vertical)
(Parallel sides = 14 in and 6 in, Height = 10 in)
📌 ✅ Answer:
🟢 Step 1: Identify Dimensions
⬥ a = 14 in, b = 6 in, h = 10 in.
🔵 Step 2: Calculate Area
⬥ Area = 0.5 × (14 + 6) × 10
⬥ Area = 0.5 × 20 × 10 = 10 × 10
⬥ Area = 100 sq in.
Figure (iv): Isosceles Trapezium
(Top = 12 ft, Bottom = 18 ft, Height = 8 ft)
📌 ✅ Answer:
🟢 Step 1: Identify Dimensions
⬥ a = 12 ft, b = 18 ft, h = 8 ft.
🔵 Step 2: Calculate Area
⬥ Area = 0.5 × (12 + 18) × 8
⬥ Area = 0.5 × 30 × 8 = 15 × 8
⬥ Area = 120 sq ft.
🔒 ❓ 4. [Sulba-Sutras] Give a method to convert an isosceles trapezium to a rectangle using dissection.
📌 ✅ Answer:
🟢 Step 1: Visualize the Shape
⬥ An isosceles trapezium has symmetric slanted sides.
🔵 Step 2: The Dissection Method
⬥ Draw a vertical line from one top corner to the base, creating a right-angled triangle.
⬥ Cut: Cut this right-angled triangle off.
⬥ Move: Move the triangle to the opposite side of the trapezium.
⬥ Paste: Attach it to fill the empty slant.
🟡 Result:
⬥ The new shape is a perfect rectangle with the same area as the original trapezium.
🔒 ❓ 5. Here is one of the ways to convert trapezium ABCD into a rectangle EFGH of equal area. Given the trapezium ABCD, how do we find the vertices of the rectangle EFGH?
[Hint: If DeltaAHI cong DeltaDGI and DeltaBEJ cong DeltaCFJ, then the trapezium and rectangle have equal areas.]
📌 ✅ Answer:
🟢 Step 1: Identify Midpoints
⬥ Locate point I, the midpoint of the non-parallel side AD.
⬥ Locate point J, the midpoint of the non-parallel side BC.
🔵 Step 2: Draw Vertical Lines
⬥ Draw a vertical line passing through I perpendicular to the base. This forms side HG.
⬥ Draw a vertical line passing through J perpendicular to the base. This forms side EF.
🟡 Step 3: Locate Vertices
⬥ Vertices G and F lie on the base line DC.
⬥ Vertices H and E lie on the extended top line AB.
🔴 Conclusion:
⬥ The triangles rotated around midpoints I and J fill the gaps perfectly to form the rectangle.
🔒 ❓ 6. Using the idea of converting a trapezium into a rectangle of equal area, and vice versa, construct a trapezium of area 144 cm².
📌 ✅ Answer:
🟢 Step 1: Start with a Rectangle
⬥ Select dimensions for a rectangle with area 144 cm².
⬥ Example: Base = 16 cm, Height = 9 cm. (16 × 9 = 144).
🔵 Step 2: Convert to Trapezium
⬥ Keep the height at 9 cm.
⬥ Change the parallel sides (a and b) such that their average is still 16.
⬥ Formula: (a + b) ÷ 2 = 16, so a + b = 32.
🟡 Step 3: Select Sides
⬥ Let side a = 12 cm and side b = 20 cm. (12 + 20 = 32).
🔴 Verification:
⬥ Area = 0.5 × (12 + 20) × 9
⬥ Area = 0.5 × 32 × 9 = 16 × 9 = 144 cm².
🔒 ❓ 7. A regular hexagon is divided into a trapezium, an equilateral triangle, and a rhombus, as shown. Find the ratio of their areas.
📌 ✅ Answer:
🟢 Step 1: Analyze Unit Shapes
⬥ A regular hexagon consists of 6 congruent equilateral triangles.
🔵 Step 2: Determine Values
⬥ Equilateral Triangle: occupies 1 unit.
⬥ Rhombus: consists of 2 equilateral triangles joined together.
⬥ Trapezium: consists of the remaining 3 equilateral triangles (half the hexagon).
🟡 Step 3: Form Ratio
⬥ Order: Trapezium : Triangle : Rhombus
⬥ Ratio = 3 : 1 : 2
🔒 ❓ 8. ZYXW is a trapezium with ZY || WX. A is the midpoint of XY. Show that the area of the trapezium ZYXW is equal to the area of DeltaZWB.
📌 ✅ Answer:
🟢 Step 1: Identify Congruency
⬥ Compare triangle DeltaYAZ (top cut-off) and triangle DeltaXAB (bottom added).
⬥ Angle AYZ = Angle AXB (Alternate interior angles).
⬥ Side YA = Side XA (Midpoint).
⬥ Angle YAZ = Angle XAB (Vertically opposite).
🔵 Step 2: Apply Logic
⬥ By Angle-Side-Angle (ASA), DeltaYAZ cong DeltaXAB.
⬥ This means their areas are identical.
🟡 Step 3: Rearrange Area
⬥ Area(Trapezium) = Quadrilateral ZWXA + DeltaYAZ.
⬥ Area(Triangle ZWB) = Quadrilateral ZWXA + DeltaXAB.
⬥ Since DeltaYAZ = DeltaXAB, the areas are equal.
🔴 Conclusion:
⬥ The area of the trapezium ZYXW is exactly equal to the area of the triangle DeltaZWB.
——————————————————————————————————————————————————————————————————————————–
OTHER IMPORTANT QUESTIONS
🟦 Part A — MCQs (10)
🔒 ❓ Q1. A rhombus has diagonals 18 cm and 12 cm. Its area is:
🟢1️⃣ 108 cm²
🔵2️⃣ 216 cm²
🟡3️⃣ 324 cm²
🟣4️⃣ 432 cm²
✔️ Answer: 🟢1️⃣ 108 cm²
📌 ✅ Answer:
🔹 Area of rhombus = 1/2 × d₁ × d₂
🔹 = 1/2 × 18 × 12
🔹 = 1/2 × 216
🔹 = 108 cm²
🔒 ❓ Q2. A trapezium has parallel sides 24 m and 36 m, and height 14 m. Its area is:
🟢1️⃣ 336 m²
🔵2️⃣ 420 m²
🟡3️⃣ 840 m²
🟣4️⃣ 168 m²
✔️ Answer: 🔵2️⃣ 420 m²
📌 ✅ Answer:
🔹 Area of trapezium = 1/2 × h × (a + b)
🔹 = 1/2 × 14 × (24 + 36)
🔹 = 1/2 × 14 × 60
🔹 = 7 × 60
🔹 = 420 m²
🔒 ❓ Q3. If 1 in = 2.54 cm, then 1 in² equals:
🟢1️⃣ 2.54 cm²
🔵2️⃣ 5.08 cm²
🟡3️⃣ 6.4516 cm²
🟣4️⃣ 12.7 cm²
✔️ Answer: 🟡3️⃣ 6.4516 cm²
📌 ✅ Answer:
🔹 1 in² = (2.54 cm) × (2.54 cm)
🔹 = 2.54² cm²
🔹 = 6.4516 cm²
🔒 ❓ Q4. A parallelogram has base 15 cm and height 9 cm. Its area is:
🟢1️⃣ 24 cm²
🔵2️⃣ 135 cm²
🟡3️⃣ 270 cm²
🟣4️⃣ 67.5 cm²
✔️ Answer: 🔵2️⃣ 135 cm²
📌 ✅ Answer:
🔹 Area of parallelogram = base × height
🔹 = 15 × 9
🔹 = 135 cm²
🔒 ❓ Q5. A triangle has base 20 cm and height 13 cm. Its area is:
🟢1️⃣ 130 cm²
🔵2️⃣ 260 cm²
🟡3️⃣ 33 cm²
🟣4️⃣ 40 cm²
✔️ Answer: 🟢1️⃣ 130 cm²
📌 ✅ Answer:
🔹 Area of triangle = 1/2 × base × height
🔹 = 1/2 × 20 × 13
🔹 = 10 × 13
🔹 = 130 cm²
🔒 ❓ Q6. A rectangle is converted into a parallelogram by slanting one side without changing base and height. The area:
🟢1️⃣ increases
🔵2️⃣ decreases
🟡3️⃣ remains the same
🟣4️⃣ becomes half
✔️ Answer: 🟡3️⃣ remains the same
📌 ✅ Answer:
🔹 Area depends on base and height only.
🔹 Base same and height same.
🔹 So area remains the same.
🔒 ❓ Q7. A rhombus and a rectangle have equal area. If the rhombus diagonals are 16 cm and 10 cm, the rectangle area is:
🟢1️⃣ 80 cm²
🔵2️⃣ 160 cm²
🟡3️⃣ 320 cm²
🟣4️⃣ 26 cm²
✔️ Answer: 🟢1️⃣ 80 cm²
📌 ✅ Answer:
🔹 Area of rhombus = 1/2 × 16 × 10
🔹 = 80 cm²
🔹 Rectangle has equal area.
🔹 So rectangle area = 80 cm²
🔒 ❓ Q8. A trapezium has area 192 cm², height 12 cm, and one parallel side 10 cm. The other parallel side is:
🟢1️⃣ 12 cm
🔵2️⃣ 14 cm
🟡3️⃣ 22 cm
🟣4️⃣ 6 cm
✔️ Answer: 🟡3️⃣ 22 cm
📌 ✅ Answer:
🔹 Area = 1/2 × h × (a + b)
🔹 192 = 1/2 × 12 × (10 + b)
🔹 192 = 6 × (10 + b)
🔹 192/6 = 10 + b
🔹 32 = 10 + b
🔹 b = 22 cm
🔒 ❓ Q9. The area of an A4 sheet (21 cm × 29.7 cm) is:
🟢1️⃣ 623.7 cm²
🔵2️⃣ 50.7 cm²
🟡3️⃣ 987 cm²
🟣4️⃣ 6237 cm²
✔️ Answer: 🟢1️⃣ 623.7 cm²
📌 ✅ Answer:
🔹 Area = length × breadth
🔹 = 21 × 29.7
🔹 = 623.7 cm²
🔒 ❓ Q10. Which formula is correct for trapezium area?
🟢1️⃣ base × height
🔵2️⃣ 1/2 × (sum of parallel sides) × height
🟡3️⃣ 1/2 × diagonal₁ × diagonal₂
🟣4️⃣ 2 × (length + breadth)
✔️ Answer: 🔵2️⃣
📌 ✅ Answer:
🔹 Trapezium has two parallel sides.
🔹 Area = 1/2 × h × (a + b).
🟩 Part B — SAQs (10)
🔒 ❓ Q11. Find the area of a rhombus whose diagonals are 20 cm and 15 cm.
📌 ✅ Answer:
🔹 Area = 1/2 × d₁ × d₂
🔹 Area = 1/2 × 20 × 15
🔹 Area = 1/2 × 300
🔹 Area = 150 cm²
🔒 ❓ Q12. A trapezium has parallel sides 18 cm and 26 cm, and height 9 cm. Find its area.
📌 ✅ Answer:
🔹 Area = 1/2 × h × (a + b)
🔹 Area = 1/2 × 9 × (18 + 26)
🔹 Area = 1/2 × 9 × 44
🔹 Area = 4.5 × 44
🔹 Area = 198 cm²
🔒 ❓ Q13. Convert 161.29 cm² to in². (Use 1 in² = 6.4516 cm²)
📌 ✅ Answer:
🔹 1 in² = 6.4516 cm²
🔹 in² = 161.29 ÷ 6.4516
🔹 in² = 25
🔹 So, 161.29 cm² = 25 in²
🔒 ❓ Q14. A parallelogram has area 252 cm² and base 21 cm. Find its height.
📌 ✅ Answer:
🔹 Area = base × height
🔹 252 = 21 × height
🔹 height = 252 ÷ 21
🔹 height = 12 cm
🔒 ❓ Q15. A triangle and a parallelogram have the same base 16 cm and the same height 9 cm. Compare their areas.
📌 ✅ Answer:
🔹 Area(triangle) = 1/2 × base × height
🔹 Area(triangle) = 1/2 × 16 × 9
🔹 Area(triangle) = 72 cm²
🔹 Area(parallelogram) = base × height
🔹 Area(parallelogram) = 16 × 9
🔹 Area(parallelogram) = 144 cm²
🔹 Ratio (triangle : parallelogram) = 72 : 144
🔹 Ratio = 1 : 2
🔒 ❓ Q16. A trapezium has area 300 m² and height 12 m. The sum of the parallel sides is?
📌 ✅ Answer:
🔹 Area = 1/2 × h × (a + b)
🔹 300 = 1/2 × 12 × (a + b)
🔹 300 = 6 × (a + b)
🔹 (a + b) = 300 ÷ 6
🔹 (a + b) = 50 m
🔒 ❓ Q17. The area of a rectangle is 180 cm². It is cut and rearranged into a rhombus. What can you say about the rhombus area? Why?
📌 ✅ Answer:
🔹 Cutting and rearranging pieces does not change total area.
🔹 No part is removed or overlapped.
🔹 So area remains the same.
🔹 Rhombus area = 180 cm²
🔒 ❓ Q18. A rhombus has diagonals in the ratio 5 : 3. If its area is 135 cm², find the diagonals.
📌 ✅ Answer:
🔹 Let diagonals be 5x and 3x
🔹 Area = 1/2 × 5x × 3x
🔹 135 = 1/2 × 15x²
🔹 135 = 7.5x²
🔹 x² = 135 ÷ 7.5
🔹 x² = 18
🔹 x = √18
🔹 x = 3√2
🔹 d₁ = 5x = 15√2 cm
🔹 d₂ = 3x = 9√2 cm
🔒 ❓ Q19. A trapezium has parallel sides 12 cm and 20 cm. Its area is 128 cm². Find its height.
📌 ✅ Answer:
🔹 Area = 1/2 × h × (a + b)
🔹 128 = 1/2 × h × (12 + 20)
🔹 128 = 1/2 × h × 32
🔹 128 = 16h
🔹 h = 128 ÷ 16
🔹 h = 8 cm
🔒 ❓ Q20. If the base and height of a parallelogram are both doubled, how does its area change?
📌 ✅ Answer:
🔹 Original area = b × h
🔹 New base = 2b
🔹 New height = 2h
🔹 New area = (2b) × (2h)
🔹 New area = 4bh
🔹 So area becomes 4 times.
🟥 Part C — DAQs (10)
🔒 ❓ Q21. A trapezium has parallel sides 18 ft and 30 ft, and height 8 ft.
(i) Find its area.
(ii) If it is converted into a rectangle of equal area with height 8 ft, find the rectangle’s length.
📌 ✅ Answer:
🔹 Area(trapezium) = 1/2 × h × (a + b)
🔹 Area = 1/2 × 8 × (18 + 30)
🔹 Area = 4 × 48
🔹 Area = 192 ft²
🔹 Rectangle area = length × height
🔹 192 = length × 8
🔹 length = 192 ÷ 8
🔹 length = 24 ft
🔒 ❓ Q22. A rhombus has area 216 cm² and one diagonal 24 cm. Find the other diagonal.
📌 ✅ Answer:
🔹 Area = 1/2 × d₁ × d₂
🔹 216 = 1/2 × 24 × d₂
🔹 216 = 12 × d₂
🔹 d₂ = 216 ÷ 12
🔹 d₂ = 18 cm
🔒 ❓ Q23. A parallelogram has base 17 cm. Its height is 4 cm more than its base. Find its area.
📌 ✅ Answer:
🔹 Base = 17 cm
🔹 Height = 17 + 4
🔹 Height = 21 cm
🔹 Area = base × height
🔹 Area = 17 × 21
🔹 Area = 357 cm²
🔒 ❓ Q24. Convert 10 in² to cm² and then convert 64.516 cm² to in². (Use 1 in² = 6.4516 cm²)
📌 ✅ Answer:
🔹 10 in² = 10 × 6.4516 cm²
🔹 10 in² = 64.516 cm²
🔹 64.516 cm² in in² = 64.516 ÷ 6.4516
🔹 = 10 in²
🔒 ❓ Q25. A triangular plot has base 28 m and height 15 m. A rectangular plot has the same area and width 12 m. Find the rectangle’s length.
📌 ✅ Answer:
🔹 Area(triangle) = 1/2 × base × height
🔹 Area = 1/2 × 28 × 15
🔹 Area = 14 × 15
🔹 Area = 210 m²
🔹 Area(rectangle) = length × width
🔹 210 = length × 12
🔹 length = 210 ÷ 12
🔹 length = 17.5 m
🔒 ❓ Q26. A trapezium has parallel sides (x + 6) cm and (2x − 4) cm, height 10 cm, and area 180 cm². Find x and both parallel sides.
📌 ✅ Answer:
🔹 Area = 1/2 × h × (a + b)
🔹 180 = 1/2 × 10 × [(x + 6) + (2x − 4)]
🔹 180 = 5 × (3x + 2)
🔹 180/5 = 3x + 2
🔹 36 = 3x + 2
🔹 3x = 34
🔹 x = 34/3
🔹 a = x + 6 = 34/3 + 6
🔹 a = 34/3 + 18/3
🔹 a = 52/3 cm
🔹 b = 2x − 4 = 2(34/3) − 4
🔹 b = 68/3 − 12/3
🔹 b = 56/3 cm
🔒 ❓ Q27. A4 sheet area check:
(i) Find area in cm² of 21 cm × 29.7 cm.
(ii) If 1 cm² = 100 mm², find the area in mm².
📌 ✅ Answer:
🔹 Area = 21 × 29.7
🔹 Area = 623.7 cm²
🔹 1 cm² = 100 mm²
🔹 623.7 cm² = 623.7 × 100 mm²
🔹 = 62,370 mm²
🔒 ❓ Q28. A rhombus is cut along both diagonals into 4 triangles. Explain (with reasoning) why the area formula 1/2 × d₁ × d₂ is true.
📌 ✅ Answer:
🔹 Diagonals of a rhombus bisect each other at right angles.
🔹 So diagonals divide the rhombus into 4 right triangles.
🔹 Each triangle has legs d₁/2 and d₂/2.
🔹 Area of 1 triangle = 1/2 × (d₁/2) × (d₂/2)
🔹 = 1/2 × d₁d₂/4
🔹 = d₁d₂/8
🔹 Total area = 4 × (d₁d₂/8)
🔹 Total area = d₁d₂/2
🔹 Area = 1/2 × d₁ × d₂
🔒 ❓ Q29. A school hall is 18 m long and 12 m wide.
(i) Find its area in m².
(ii) Convert it to cm². (Use 1 m = 100 cm)
📌 ✅ Answer:
🔹 Area = 18 × 12
🔹 Area = 216 m²
🔹 1 m = 100 cm
🔹 1 m² = (100 cm)²
🔹 1 m² = 10,000 cm²
🔹 216 m² = 216 × 10,000 cm²
🔹 216 m² = 2,160,000 cm²
🔒 ❓ Q30. A trapezium is converted into a rectangle of equal area by dissection.
Explain one correct method step-by-step using the idea “cut off a triangle and shift it” (no diagram needed), and justify why the areas remain equal.
📌 ✅ Answer:
🔹 Take the trapezium with parallel sides a and b, and height h.
🔹 Identify the extra length (b − a) on the longer base side.
🔹 From one non-parallel side, cut a right triangle whose base equals (b − a)/2 and height equals h.
🔹 Cut an equal triangle from the other side in the same way.
🔹 Shift these cut triangles to the opposite sides so that the top shorter base extends equally on both sides.
🔹 After shifting, the top and bottom edges become equal in length.
🔹 The new shape becomes a rectangle (or parallelogram that can be straightened to rectangle).
🔹 No piece is added or removed.
🔹 Pieces only move position.
🔹 So total area remains unchanged.
🔹 Therefore, trapezium and the formed rectangle have equal area.
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