Area and Perimeter of Quadrilaterals: Rectangle, Square, Rhombus, Parallelogram, Trapezium
Area and Perimeter of Quadrilaterals: Rectangle, Square, Rhombus, Parallelogram, Trapezium
Quadrilateral questions test whether you know which formula uses which measurement: a rhombus by its diagonals, a parallelogram by base and height, a trapezium by the average of its parallel sides. NTPC also asks a rectangle from its diagonal and area, squares compared by their diagonals, and paths around or through a field.
1. Formula Box
| Figure | Area | Perimeter / other |
|---|---|---|
| Rectangle (l, b) | l × b | 2(l + b); diagonal √(l² + b²) |
| Square (side a, diagonal d) | a² = d²/2 | 4a; d = a√2 |
| Rhombus (diagonals d₁, d₂) | (1/2) d₁ d₂ | Side = √((d₁/2)² + (d₂/2)²); perimeter 4 × side |
| Parallelogram (sides a, b; height h on base b) | b × h | 2(a + b); area also = a × its height |
| Trapezium (parallel sides a, b, height h) | (1/2)(a + b) h | |
| Path of width w OUTSIDE an l × b field | (l + 2w)(b + 2w) − lb | = 2w(l + b + 2w) |
| Path of width w INSIDE | lb − (l − 2w)(b − 2w) | = 2w(l + b − 2w) |
| Two crossroads of width w through the middle | w(l + b) − w² | Subtract the central square once |
Useful identity: for a rectangle, (l + b)² = diagonal² + 2 × area, so the perimeter follows from the diagonal and the area.
✗ Squares with diagonals in ratio 3 : 2 have areas in ratio 3 : 2 | ✓ Area ∝ diagonal², so 9 : 4
✗ Crossroads 3 m wide on a 60 × 40 park: 3 × 60 + 3 × 40 | ✓ The 3 × 3 crossing is counted twice: subtract 9
हिंदी नोट: समचतुर्भुज का क्षेत्रफल विकर्णों से (1/2)d₁d₂, समांतर चतुर्भुज का आधार × ऊँचाई और समलंब का (1/2)(समांतर भुजाओं का योग) × ऊँचाई होता है।
Exam Pointer: Verified NTPC patterns: rectangle perimeter from its diagonal and area (May 2022 CBT-2) and ratio of square areas from their diagonals (April 2016). Rhombus, parallelogram, trapezium and path questions had no verified NTPC shift in our check and are tagged syllabus-based.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Rectangle: diagonal, area and perimeter
[PYQ: NTPC CBT-2 10-May-2022 Shift-1]
EXAM LEVEL
Q. A rectangle is 24 cm long and 7 cm wide. Find its diagonal, area and perimeter.
Diagonal = √(576 + 49) = 25 cm (7-24-25 triplet). Area = 168 cm². Perimeter = 62 cm.
Answer: 25 cm, 168 cm², 62 cm
EXAMATLAS LEVEL
Q. The diagonal of a rectangle is 17 cm and its area is 120 cm². Find its perimeter.
(l + b)² = 17² + 2 × 120 = 289 + 240 = 529, so l + b = 23 and the perimeter = 46 cm. (The sides are 15 and 8.)
Answer: 46 cm
Pattern 2: Square: from diagonal or perimeter, and comparing squares
[PYQ: NTPC CBT-1 27-Apr-2016 Shift-2]
EXAM LEVEL
Q. The diagonal of a square is 12√2 cm. Find its area.
Area = d²/2 = 288/2 = 144 cm² (side 12 cm).
Answer: 144 cm²
EXAMATLAS LEVEL
Q. The diagonals of two squares are in the ratio 3 : 2. The perimeter of the larger square is 48 cm. Find the area of the smaller square.
Larger side = 12 cm, area 144 cm². Areas are in the ratio 9 : 4, so the smaller area = 144 × 4/9 = 64 cm².
Answer: 64 cm²
Pattern 3: Rhombus from its diagonals
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. The diagonals of a rhombus are 16 cm and 30 cm. Find its area and side.
Area = (1/2) × 16 × 30 = 240 cm². Half-diagonals 8 and 15, so side = 17 cm.
Answer: 240 cm²; 17 cm
EXAMATLAS LEVEL
Q. A rhombus has side 13 cm and one diagonal 24 cm. Find its area.
Half of the known diagonal = 12, so half of the other = √(169 − 144) = 5 and the other diagonal = 10 cm. Area = (1/2) × 24 × 10 = 120 cm².
Answer: 120 cm²
Pattern 4: Parallelogram and trapezium
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. A trapezium has parallel sides 14 cm and 22 cm, 9 cm apart. Find its area.
(1/2)(14 + 22) × 9 = 18 × 9 = 162 cm².
Answer: 162 cm²
EXAMATLAS LEVEL
Q. A parallelogram has sides 10 cm and 8 cm, and the distance between the longer sides is 6 cm. Find its area and the distance between the shorter sides.
Area = 10 × 6 = 60 cm². The same area = 8 × h, so h = 7.5 cm.
Answer: 60 cm²; 7.5 cm
Pattern 5: Paths, crossroads and costs
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. A 40 m × 30 m field has a 2.5 m wide path all round it outside. Find the area of the path.
Outer rectangle = 45 × 35 = 1,575 m². Path = 1,575 − 1,200 = 375 m².
Answer: 375 m²
EXAMATLAS LEVEL
Q. A 60 m × 40 m park has two crossroads, each 3 m wide, running through the middle parallel to the sides. Find the cost of paving them at ₹20 per m².
Area = 3 × 60 + 3 × 40 − 3 × 3 = 180 + 120 − 9 = 291 m². Cost = 291 × 20 = ₹5,820.
Answer: ₹5,820
60-Second Revision
- Rectangle: (l + b)² = d² + 2 × area.
- Square: area = d²/2; area ratio = (diagonal ratio)².
- Rhombus: (1/2)d₁d₂; side from half-diagonals.
- Parallelogram: base × its height; trapezium: average of parallel sides × height.
- Paths: outer minus inner; crossroads: subtract the overlap once.