📑 On this page
⚡ The short answer
Perimeter is the edge (fence, frame), in m. Area is the surface (tiles, paint), in m². Rectangle: l × b and 2(l + b); triangle: ½ × base × height; trapezium: ½(a + b) × h. The big trap: a perimeter alone never fixes the area.
🎬 The 30-second story
Forty metres of bamboo
1
“Society gave us 40 m of bamboo fencing. My garden: 18 m by 2 m. Elegant.”
🎋
Aarav
Terrace garden project.
2
“Same 40 m. Mine is 10 m by 10 m.”
📐
Zoya
Two gardens, identical fence.
3
“Yours grows 36 m² of plants. Hers grows 100 m². Same bamboo!”
🧮
Meher
Area = l × b for both.
Tara's maths: The fence fixes only l + b = 20. The area can be anything up to 100 m², and the square gives the most. One measurement doesn't tell you the other.
What is the difference between area and perimeter?
Walk round a cricket ground and count your steps: that's the perimeter, a length in m. Cover it with 1 m × 1 m grass mats and count the mats: that's the area, in m². Different questions, different units.
📊 The data
40 m of fence, five rectangular gardens (area in m²)
19 m × 1 m
19 m²
18 m × 2 m · Aarav's
36 m²
15 m × 5 m
75 m²
12 m × 8 m
96 m²
10 m × 10 m · Zoya's square
100 m²
Every row has perimeter 40 m. Closer sides, bigger area.
Flip it: three rectangles of 36 m² need 74 m of fence (36 × 1), 26 m (9 × 4) or just 24 m (6 × 6). For any area, the square needs the least edge.
It all starts with one fact: a 5 m × 3 m floor holds 5 rows of 3 one-metre squares, so a rectangle's area is l × b. Every other straight-edged shape is a rectangle halved, or cut and slid.
🧩 At a glance
Every shape is a rectangle in disguise
◺Triangle
Half the rectangle drawn round it: ½ × base × height.
▱Parallelogram
Slide the slanted end across: a rectangle. base × height.
⏢Trapezium
Two copies make a parallelogram of base a + b. Halve it: ½(a + b) × h.
◇Rhombus
Half the box round its diagonals: ½ × d₁ × d₂.
📌 Height means perpendicular height
In ½ × base × height, the height is at 90° to the base. A slanted side is longer, so using it inflates the area, and that wrong number sits in the options.
What are the area and perimeter formulas for every 2D shape?
📐 The figure
The three shapes CLAT uses most
Rectangle
Area l × b · Perimeter 2(l + b)
Triangle
Area ½ × b × h · Perimeter a + b + c
Trapezium
Area ½(a + b) × h · Perimeter: all 4 sides
Figures are not to scale.
| Shape | Area | Perimeter | Remember |
|---|---|---|---|
| Rectangle | l × b | 2(l + b) | diagonal √(l² + b²) |
| Square (side a) | a², or d²/2 | 4a | diagonal a√2 |
| Triangle | ½ × b × h | a + b + c | three sides only? Heron |
| Equilateral (side a) | (√3/4) × a² | 3a | height (√3/2) × a |
| Parallelogram | base × height | 2(a + b) | not side × side |
| Rhombus | ½ × d1 × d2 | 4a | diagonals cross at 90° |
| Trapezium | ½(a + b) × h | sum of 4 sides | a, b = parallel sides |
| Circle | πr² | 2πr | see the circles page |
→ scroll to see more
Area and perimeter questions solved, from easy to tricky
✍️ Worked example
Warm-up
⏱ 45 sec
Glass for a study table
Meher's table top is 1.2 m by 75 cm. Glass costs ₹800 per m²; steel edging round it costs ₹60 per metre. Total bill?
- 1Same unit first: 75 cm = 0.75 m.
- 2Area = 1.2 × 0.75 = 0.9 m²; glass = 0.9 × 800 = ₹720.
- 3Perimeter = 2(1.2 + 0.75) = 3.9 m; edging = 3.9 × 60 = ₹234.
- 4Total = 720 + 234 = ₹954.
Answer: ₹954 (glass ₹720 + edging ₹234)
⚡ The exam-hall shortcut
Glass covers (area); edging runs round (perimeter). Write each rate's unit, ₹/m² or ₹/m, and you can't swap them.
✍️ Worked example
CLAT level
⏱ 60 sec
A jogging path inside the park
A park is 30 m by 20 m. A 2 m wide jogging path runs along the inside of its boundary. Paving costs ₹15 per m². Total cost?
- 1The path takes 2 m off both ends of each side: lawn = 26 × 16 = 416 m².
- 2Path = whole park − lawn = 600 − 416 = 184 m².
- 3Cost = 184 × 15 = ₹2,760.
Answer: ₹2,760 (path area 184 m²)
⚡ The exam-hall shortcut
Inside path of width w: 2w(L + B) − 4w² = 200 − 16 = 184. The −4w² removes corners counted twice.
✍️ Worked example
CLAT level
⏱ 60 sec
The rhombus rangoli
A rhombus rangoli has a perimeter of 52 cm and one diagonal of 24 cm. Find its area.
- 1All four sides are equal: side = 52 ÷ 4 = 13 cm.
- 2The diagonals bisect at 90°, making right triangles with hypotenuse 13 and a leg of 12. The other leg is √(169 − 144) = 5, so the second diagonal is 10 cm.
- 3Area = ½ × 24 × 10 = 120 cm².
Answer: 120 cm²
⚡ The exam-hall shortcut
Spot the 5–12–13 triplet and skip the square root entirely.
What are the fastest shortcuts for area and perimeter?
| Situation | Shortcut | Example |
|---|---|---|
| Biggest rectangle for a fixed perimeter P | the square: (P/4)² | P = 40 → 100 |
| Square from its diagonal d | d²/2 | d = 10 → 50 |
| Rectangle from perimeter P and diagonal d | [(P/2)² − d²] ÷ 2 | P = 34, d = 13 → 60 |
| Path w wide outside L × B | 2w(L + B) + 4w² | 12 × 8, w = 1 → 44 |
| Path w wide inside L × B | 2w(L + B) − 4w² | 30 × 20, w = 2 → 184 |
| Two crossroads w wide through the middle | w(L + B) − w² | 30 × 20, w = 2 → 96 |
→ scroll to see more
🎛️ Try it yourself — change any number
Stretch Aarav's garden
Area
36
Perimeter
40
Diagonal
18.11
Area = l × b; Perimeter = 2(l + b); Diagonal = √(l² + b²)
It opens on 18 × 2: perimeter 40, area 36. Try 15 × 5, 12 × 8, then 10 × 10. The perimeter stays 40; the area climbs to 100.
💡 Estimate before you multiply
Fixed perimeter caps the area: l + b = 30 means area ≤ 15² = 225. A triangle is exactly half the rectangle on its base and height. Options above these ceilings go first.
How do area and perimeter hide inside a CLAT caselet?
Since 2020, CLAT describes a plot or a campus in one paragraph and asks three or four questions off it. Area becomes whole minus pieces. List each piece's area, then answer.
📊 The data
A school sports plot, piece by piece (area in m²)
Whole plot · 64 m × 45 m
2,880 m²
Basketball court · 28 m × 15 m
420 m²
Badminton hall · square, side 18 m
324 m²
Herb garden · right triangle, legs 12 m and 9 m
54 m²
Lawn left · 2,880 − 420 − 324 − 54
2,082 m²
The garden is a triangle: half of 12 × 9.
🧪 CLAT-style practice (not a past question)
Composite plot: subtract the pieces
A school's sports plot is a rectangle 64 m by 45 m. On it stand a basketball court 28 m by 15 m, a square badminton hall of side 18 m and a herb garden shaped like a right-angled triangle with perpendicular sides of 12 m and 9 m. The rest is lawn, to be turfed at ₹25 per m². The outer boundary will be fenced at ₹120 per metre.
What will turfing the lawn cost?
A₹50,700
B₹52,050
C₹60,150
D₹72,000
Reveal answer + reasoning
Answer: B
Plot = 64 × 45 = 2,880 m². Subtract the court (420), the hall (18² = 324) and the garden (½ × 12 × 9 = 54): lawn = 2,082 m². Turf = 2,082 × 25 = ₹52,050. The fencing rate is a decoy. About 70 seconds.
Why the others fail
- A — Takes the garden as 12 × 9 = 108, forgetting the ½ (lawn 2,028).
- C — Forgets to subtract the badminton hall (lawn 2,406).
- D — Turfs the whole plot, 2,880 m².
Which area and perimeter traps cost the most marks?
🧭 Decision flow
Fence or floor? Pick the measurement
❓ Does the job run along the edge? Fence, frame, border, lace.
Yes →
Perimeter, in m. Multiply by a ₹/m rate.
No ↓
❓ Does it cover the surface? Tiles, paint, turf, carpet.
Yes →
Area, in m². Multiply by a ₹/m² rate.
No ↓
❓ Does it fill a space? Water, soil, grain.
Yes →
Volume, in m³.
No →
Re-read the stem: the rate's unit tells you.
⚠️ Perimeter alone = cannot be determined
'A rectangle has a perimeter of 30 m. Find its area.' It could be 14 × 1 (14 m²) or 7.5 × 7.5 (56.25 m²). You need a ratio, a side or a diagonal too. Nothing more given? Pick 'cannot be determined'.
- ✓Mixed units. 1 m² = 10,000 cm², not 100. Convert lengths before multiplying.
- ✓Shared edges. Where two shapes join, that side is inside: drop it from the perimeter.
- ✓Inside vs outside path. Inside shrinks the lawn by 2w each way; outside grows it. Sketch it.
How did CLAT ask area and perimeter?
📝 Asked in CLAT 2019
· Q43
Perimeter and a ratio to area
The perimeter of a rectangle is 60 cms. If its length is twice its breadth, then its area is:
A200 cm²
B180 cm²
C160 cm²
D220 cm²
Reveal answer + reasoning
Answer: A
Our solution (2019 has no official key). l + b = 60 ÷ 2 = 30. With l = 2b, 3b = 30, so b = 10, l = 20 and area = 200 cm². Check: 2(20 + 10) = 60. About 30 seconds.The ratio is what makes it solvable; without it, 'cannot be determined'.
Why the others fail
- B — Sides adding to 30 with area 180 would be about 21.7 and 8.3, not in a 2 : 1 ratio.
- C — Area 160 needs sides of about 23.1 and 6.9: wrong ratio again.
- D — Area 220 needs about 17.2 and 12.8, close to a square, not 2 : 1.
📐 The figure
The 2018 field, split into two pieces
Trapezium field
Area ½(25 + 10) × h
Slanted sides 14 m and 13 m.
The cut-off triangle
Heron: s = 21, area 84
15 = 25 − 10.
Not to scale. From a top corner, draw a line parallel to the far slanted side.
📝 Asked in CLAT 2018
· Q147
Trapezium from four sides
The parallel sides of a trapezium shaped field are 25 m and 10 m and non parallel sides are 14 m and 13 m. What is the area (in m²) of the field?
A204
B196
C156
D144
Reveal answer + reasoning
Answer: B
The height isn't given, so make it. From a top corner, draw a line parallel to the opposite slanted side: the field splits into a parallelogram 10 m wide and a triangle with sides 15, 14, 13.Heron: s = 21, area = √(21 × 6 × 7 × 8) = 84 m². Height = 2 × 84 ÷ 15 = 11.2 m.
Field = ½ × 35 × 11.2 = 196 m², option B. Check: 10 × 11.2 + 84 = 196. About 90 seconds.
Why the others fail
- A — 84 + 10 × 12: a guessed height of 12 for the parallelogram part.
- C — Needs a height under 9 m (156 ÷ 17.5); these sides force 11.2.
- D — Needs a height of about 8.2 m: impossible here.
Which area and perimeter myths should you drop?
💥 Myth vs fact
Four myths, four lost marks
🙅 Myth
A bigger perimeter means a bigger area.
Tap to bust it →
✅ Fact
18 × 2 has perimeter 40 and area 36; a 7 × 7 square has perimeter 28 and area 49.
🙅 Myth
A parallelogram's area is one side × the other.
Tap to bust it →
✅ Fact
It's base × perpendicular height. Sides 10 and 6 with a height of 5 give 50, not 60.
🙅 Myth
1 m² = 100 cm².
Tap to bust it →
✅ Fact
Square the conversion: 100 × 100 = 10,000 cm².
🙅 Myth
Know the perimeter and you know the area.
Tap to bust it →
✅ Fact
It leaves the area open, anywhere up to the square's (P/4)². You need a second fact.
Can you crack these five area and perimeter questions?
🎯 Quick quiz
Five shapes, two measurements
0 / 5
Rohan fences his square vegetable patch with 64 m of wire.
Q1. What is its area?
Side = 64 ÷ 4 = 16; area = 256 m². 1,024 squares 32, half the perimeter.A rectangular poster is 96 cm² in area and 12 cm long.
Q2. How much ribbon edges it?
Breadth = 8; perimeter = 2(12 + 8) = 40 cm. 20 forgets to double.A parallelogram stage has a 15 m base, 10 m slanted sides and an 8 m perpendicular height.
Q3. What is its floor area?
Base × height = 120 m². 150 multiplies the sides; 60 halves it like a triangle.Meher's rectangular room has a perimeter of 30 m.
Q4. What is its floor area?
Only l + b = 15 is known: 10 × 5 = 50, 9 × 6 = 54, the square 56.25. Cannot be determined.A square garden has a diagonal of 20 m.
Q5. What is its area?
d²/2 = 400 ÷ 2 = 200 m². 400 treats the diagonal as a side.🎯 Area and perimeter in a nutshell
- Edge → perimeter (m); surface → area (m²).
- Every formula is a rectangle halved or slid: ½bh, bh, ½(a + b)h, ½d₁d₂.
- Height is perpendicular, never the slanted side.
- Perimeter alone never fixes area; the square gives the most.
- Composite plots: whole minus pieces.
📎 Formula card
- Rectangle: A = l × b; P = 2(l + b); diagonal √(l² + b²).
- Square: A = a² = d²/2; P = 4a.
- Triangle: ½ × b × h; Heron √[s(s − a)(s − b)(s − c)], s = half the perimeter. Equilateral: (√3/4)a².
- Parallelogram: b × h. Rhombus: ½d1d2. Trapezium: ½(a + b) × h.
- Paths: outside 2w(L + B) + 4w²; inside 2w(L + B) − 4w²; crossroads w(L + B) − w².
- Max area for perimeter P: (P/4)². From P and diagonal: [(P/2)² − d²] ÷ 2.
- Units: 1 m² = 10,000 cm²; 1 hectare = 10,000 m².
Frequently asked questions
What is the difference between area and perimeter?
Perimeter is the length of a shape's boundary, in m or cm. Area is the surface it covers, in m² or cm². Fencing and framing use perimeter; tiling, painting and turfing use area. A 10 m by 10 m garden has a perimeter of 40 m and an area of 100 m².
Can you find the area of a rectangle from its perimeter alone?
No. A perimeter fixes only length + breadth, and many rectangles share that. A 30 m perimeter fits 10 × 5 (50 m²) and 7.5 × 7.5 (56.25 m²). You also need a ratio, one side or the diagonal. In CLAT, if none is given, look for 'cannot be determined'.
How do you find the area of a triangle when only the three sides are given?
Use Heron's formula. Find the semi-perimeter s = (a + b + c) ÷ 2, then area = √[s(s − a)(s − b)(s − c)]. For sides 13, 14 and 15, s = 21 and the area is √(21 × 8 × 7 × 6) = 84. Multiply inside the root first, then take one square root.
Which shape has the largest area for a given perimeter?
Among rectangles, the square: 40 m of fence encloses at most 10 × 10 = 100 m². Among all shapes, the circle wins: the same 40 m bent into a circle encloses about 127.3 m². So for a fixed fence, the more square the rectangle, the bigger the area.
Keep going
📚
Part of the chapter
Mensuration & Geometry for CLAT Quantitative Techniques
See how this topic fits the whole of Mensuration & Geometry →
Connected topics in other chapters
Circles, Semicircles and Sectors
The curved shapes: πr², 2πr and the semicircle's missing diameter.
Triangles and Pythagoras
Heights, diagonals and Heron's formula in full.
Tiling and How Many Fit
Once you have the floor area, count the tiles, chairs and cartons.
Scaling: Area and Volume Changes
Double every side and the area goes up four times.
Dividing in a Given Ratio
Perimeter plus a length-to-breadth ratio is a ratio split in disguise.