This MCQ module is based on: Polygons, Parallelograms, Rhombus and Trapezium
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Polygons, Parallelograms, Rhombus and Trapezium
🎓 Class 8
Mathematics
CBSE
Theory
Ch 7 — Understanding Quadrilaterals
⏱ ~35 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Polygons, Parallelograms, Rhombus and Trapezium
Targeting Class 8 level in General Mathematics, with Basic difficulty.
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7.3 Area of any Polygon — Divide, then Add
How do we find the area of an irregular quadrilateral? ABCD, or a pentagon, or any general polygon? Every polygon can be cut into triangles along its diagonals. Since we already know the area of a triangle, we can simply add up the triangle pieces.
Any polygon can be divided into triangles by drawing diagonals from a single vertex.
Method
To find the area of a quadrilateral ABCD with diagonal AC = d, drop perpendiculars BM and DN from B and D to AC. Then
\[ \text{Area(ABCD)} = \tfrac{1}{2}\,d\,(h_1+h_2), \] where \(h_1 = BM\) and \(h_2 = DN\).
Worked Example: Diagonal and Perpendiculars
In quadrilateral ABCD, AC = 22 cm, BM = 3 cm and DN = 3 cm, with BM ⊥ AC and DN ⊥ AC. Find the area.
Area(ABCD) = Area(ΔABC) + Area(ΔACD) = ½ × 22 × 3 + ½ × 22 × 3 = 33 + 33 = 66 cm².
7.4 Area of a Parallelogram
A parallelogram? is a quadrilateral whose opposite sides are parallel (and hence equal). To find its area we'll convert it into a rectangle of equal area — this technique is called dissection?.
Dissection: cut the right-triangle ΔAXD off the parallelogram and glue it to the other end — we get a rectangle ABYX with equal area.
Formula — Parallelogram
\[ \text{Area of a parallelogram} \;=\; \text{base}\;\times\;\text{height} \]
Here the height is the perpendicular distance between the two parallel lines containing the chosen base — not the slanted side.
Why the dissection works. Construct AX ⊥ BC (X on line BC). Then \(\triangle AXB\) is a right triangle. Slide \(\triangle AXB\) across to the other side so AB sits over DC. The two triangles \(\triangle AXB\) and the newly placed one are congruent by AAS (one right angle + equal AB = DC + equal vertical sides), so rearranging preserves total area. The resulting figure ABYX is a rectangle with base BC and height AX.
🔵 Can any side of a parallelogram serve as the base? Yes — but then the height must be the perpendicular distance measured to that chosen base. The product base × height remains the same either way.
Worked Examples — Parallelograms
P1. A parallelogram has base 7 cm and the perpendicular height to that base is 4 cm. Find its area.
Area = base × height = 7 × 4 = 28 cm².
P2. A parallelogram has base 6 cm and slanted side 5 cm, with height to the 6 cm base equal to 4.8 cm. Find its area. Also, find the corresponding height if the 5 cm side is used as base.
Area = 6 × 4.8 = 28.8 cm². Using 5 cm as base: 5 × h = 28.8 ⇒ h = 5.76 cm.
P3. Of two parallelograms with equal side lengths 5 cm and 4 cm, one is a rectangle and one has a slant. Which has the greater area?
The rectangle. For the rectangle, the height equals the other side (4 cm). For the slanted parallelogram the height is less than 4 (a leg of a right triangle with hypotenuse 4). So rectangle area = 20 cm² > slanted parallelogram area.
Activity: Paper-Cut Parallelogram → Rectangle
L3 Apply
Materials: Coloured paper, ruler, scissors, pencil.
Predict: If you cut a right-triangle off one end of a parallelogram and paste it on the other end, what new shape do you get?
- Draw a parallelogram with base 10 cm and slant side 6 cm tilted at some angle.
- Drop a perpendicular from one top vertex to the base and cut along it.
- Slide the right-triangle piece across to the other end so the matching edges align.
- Measure the resulting rectangle. Verify that (length × width) equals (base × height) of the original.
You get a rectangle whose length equals the original base and whose width equals the perpendicular height. The paper pieces are conserved, so area is preserved — giving Area of parallelogram = base × height.
7.5 Area of a Rhombus — From Diagonals
A rhombus? is a parallelogram with all four sides equal — so the parallelogram formula (base × height) still works. But a rhombus has an extra symmetry: its two diagonals are perpendicular and bisect each other. Let us use this to find a cleaner formula in terms of the diagonals.
Dissection idea. Cut a rhombus along one of its diagonals; rearrange the two halves so they form a rectangle whose length = (one diagonal) and whose width = (half the other diagonal). The total area is preserved.
Dissection: halves of the rhombus rearrange into a rectangle of sides BD and ½·AC.
Formula — Rhombus
\[ \text{Area of a rhombus} \;=\; \tfrac{1}{2}\,\times\,d_1\,\times\,d_2 \]
where \(d_1\) and \(d_2\) are the lengths of the two diagonals.
Alternative derivation — sum of two triangles
Since the diagonals of a rhombus are perpendicular, each diagonal splits it into two triangles whose heights equal half the other diagonal:
\[ \text{Area}(\triangle ADB) = \tfrac{1}{2}\cdot BD\cdot \tfrac{AC}{2}, \qquad \text{Area}(\triangle BDC) = \tfrac{1}{2}\cdot BD\cdot \tfrac{AC}{2} \] \[ \text{Area(ABCD)} = \text{Area}(\triangle ADB) + \text{Area}(\triangle BDC) = \tfrac{1}{2}\,AC\cdot BD. \]R1. A rhombus has diagonals of length 20 cm and 15 cm. Find its area.
Area = ½ × 20 × 15 = 150 cm².
7.6 Area of a Trapezium
A trapezium? has exactly one pair of parallel sides. Call the parallel side-lengths \(a\) and \(b\) and the perpendicular distance between them \(h\). We'll build the formula in two elegant ways.
Approach 1: Split into a Rectangle and Two Right-triangles
Trapezium WXYZ — two perpendiculars from the shorter parallel side split it into a rectangle (in the middle) and two right triangles (on the sides).
Let the widths of the two right triangles be \(x\) and \(y\), so \(x + y = b - a\). Then:
\[ \text{Area} = \underbrace{a\cdot h}_{\text{rectangle}} + \underbrace{\tfrac{1}{2}xh}_{\text{left tri}} + \underbrace{\tfrac{1}{2}yh}_{\text{right tri}} = ah + \tfrac{1}{2}h(x+y) = ah + \tfrac{1}{2}h(b-a) = \tfrac{1}{2}h(a+b). \]Approach 2: Two Copies of the Trapezium = One Parallelogram
Rotate a second copy of the trapezium 180° and paste it along one of the slanted sides. The two copies fit together into a parallelogram whose base is \(a + b\) and whose height is the same \(h\). So:
\[ 2\cdot \text{Area(trap)} = (a+b)\cdot h \quad\Longrightarrow\quad \text{Area(trap)} = \tfrac{1}{2}\,h\,(a+b). \]Formula — Trapezium
\[ \text{Area of a trapezium} \;=\; \tfrac{1}{2}\,\times\,\text{height}\,\times\,(\text{sum of parallel sides}) \]
Worked Examples — Trapezium
T1. A trapezium has parallel sides 16 m and 7 m with perpendicular height 10 m. Find its area.
Area = ½ × 10 × (16 + 7) = 5 × 23 = 115 m².
T2. A trapezium-shaped field has parallel roads 24 m and 36 m apart by 14 m perpendicular distance. Find its area.
Area = ½ × 14 × (24 + 36) = 7 × 60 = 420 m².
7.7 Areas in Real Life — Units and Conversions
An A4 sheet of paper is about 21 cm × 29.7 cm. Its area is \(21 \times 29.7 = 623.7\) cm². A table-top at home might be 5 ft × 2 ft — mixing inch-based units with metric ones.
Useful Conversions
1 in = 2.54 cm ⇒ 1 in² = 6.4516 cm².1 ft = 12 in ⇒ 1 ft² = 144 in².
1 m = 100 cm ⇒ 1 m² = 10 000 cm².
For land: 1 acre ≈ 43 560 ft²; 1 hectare = 10 000 m².
U1. Express 5 in and 7.4 in in centimetres.
5 in = 5 × 2.54 = 12.7 cm. 7.4 in = 7.4 × 2.54 = 18.796 cm.
U2. Convert 161.29 cm² to in².
\( \dfrac{161.29}{6.4516} = 25 \) ⇒ 25 in².
Competency-Based Questions
Scenario: A farmer owns a trapezium-shaped field whose two parallel boundaries run along a canal (36 m) and a road (24 m); the perpendicular distance between the canal and the road is 14 m. Separately, inside the field lies a rhombus-shaped tile pattern whose diagonals measure 5 m and 4 m.
Q1. What is the area of the trapezium-shaped field?
L3 Apply(c) 420 m². Area = ½ × 14 × (36 + 24) = 7 × 60 = 420 m².
Q2. Compare: which is larger — the trapezium field above, or a rectangular field with the same perimeter as the trapezium's parallel-sides sum (60 m) with breadth 14 m? Show calculations.
L4 AnalyseRectangle area = 60 × 14 = 840 m². Trapezium = 420 m². The rectangle is exactly twice the trapezium, because the trapezium formula averages the two parallel sides to 30 m, halving the result.
Q3. The tile-pattern rhombus has diagonals 5 m and 4 m. A rival claims "a square tile with the same diagonal length (5 m) always covers more area than this rhombus." Evaluate.
L5 EvaluateRhombus area = ½ × 5 × 4 = 10 m². Square with diagonal 5 m has side 5/√2 and area = (5/√2)² = 12.5 m². Claim true — because a square is the rhombus that maximises area for a given diagonal (when both diagonals are equal to the given length).
Q4. Design a composite plot (mixing rectangles, triangles, a rhombus and a trapezium) whose total area is exactly 1000 m². Sketch it with labelled dimensions, and show how the total sums to 1000.
L6 CreateOne design: rectangle 20 × 25 = 500, triangle base 20 height 20 = 200, rhombus diagonals 20 × 10 = 100 m², trapezium ½·10·(18+22) = 200 m². Total = 500 + 200 + 100 + 200 = 1000 m². Many other designs are valid.
Assertion–Reason Questions
Assertion (A): The area of a parallelogram with sides 6 cm and 5 cm is at most 30 cm².
Reason (R): The height of the parallelogram cannot exceed the length of the slanting side.
Reason (R): The height of the parallelogram cannot exceed the length of the slanting side.
Answer: (a) — Maximum occurs when the parallelogram is a rectangle (angle 90°): area = 6 × 5 = 30. Any slant reduces the height below 5. R correctly explains A.
Assertion (A): A rhombus with diagonals 10 cm and 8 cm has area 40 cm².
Reason (R): The diagonals of a rhombus bisect each other at right angles.
Reason (R): The diagonals of a rhombus bisect each other at right angles.
Answer: (a) — Area = ½ × 10 × 8 = 40. The perpendicular bisecting property of the diagonals is exactly the property that yields the ½·d₁·d₂ formula.
Assertion (A): Two trapeziums with equal areas must have equal heights.
Reason (R): Trapezium area = ½ × height × (sum of parallel sides).
Reason (R): Trapezium area = ½ × height × (sum of parallel sides).
Answer: (d) — A is false: a trapezium with h=4, a+b=10 and one with h=5, a+b=8 both have area 20. R is true and actually shows why A is false (equal areas come from a product, not one factor alone).
Frequently Asked Questions
What is a polygon?
A polygon is a closed plane figure bounded by three or more straight line segments. Polygons are classified by the number of sides: triangle (3), quadrilateral (4), pentagon (5) and so on.
How is a rhombus different from a square?
Both have four equal sides, but a rhombus has angles that are not necessarily 90 degrees. A square is a special rhombus where all angles are right angles.
What is unique about the diagonals of a parallelogram?
The diagonals of a parallelogram bisect each other, meaning each diagonal is divided into two equal halves at the point of intersection.
What makes a trapezium special?
A trapezium has exactly one pair of parallel sides. Its area is computed as one-half times the sum of the parallel sides times the perpendicular distance between them.
What is the sum of interior angles of a polygon with n sides?
It equals (n minus 2) times 180 degrees. So a quadrilateral's interior angles sum to 360 degrees and a pentagon's to 540 degrees.
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