Vardaan
Class 7 Maths • Chapter 11

Perimeter and Area

Vardaan Learning Institute • Detailed Notes with Practice Questions

📐 1. Perimeter and Area — Formula Reference Cards

a Square

P = 4a

A = a²

l × b Rectangle

P = 2(l+b)

A = l × b

hb Triangle

P = a+b+c

A = ½ × b × h

hb Parallelogram

P = 2(a+b)

A = b × h

r Circle

C = 2πr

A = πr²

d₁d₂ Rhombus

P = 4a

A = ½ × d₁ × d₂

π (Pi) — The Circle Constant

π = circumference / diameter = C/d ≈ 22/7 ≈ 3.14159...
• Circumference C = 2πr = πd
• Area A = πr² (where r = radius, d = diameter = 2r)
• π is an irrational number (non-terminating, non-repeating decimal)
Semicircle and Quarter Circle:
• Semicircle area = πr²/2 | Perimeter of semicircle = πr + 2r = r(π+2)
• Quarter circle area = πr²/4 | Perimeter of quarter circle = πr/2 + 2r = r(π/2+2)

📏 3. Paths Around and Inside Shapes

Path around a rectangle (uniform width w):
Outer rectangle: Length = l+2w, Breadth = b+2w
Area of path = Area of outer rectangle − Area of inner rectangle
= (l+2w)(b+2w) − l×b

Path inside a rectangle:
Inner rectangle: Length = l−2w, Breadth = b−2w
Area of path = l×b − (l−2w)(b−2w)

Unit Conversions (Important!)

Length Area Equivalents
1 m = 100 cm 1 m² = 10,000 cm²
1 km = 1000 m 1 km² = 1,000,000 m² = 10⁶ m²
1 cm = 10 mm 1 cm² = 100 mm²
1 hectare (ha) = 10,000 m²
✏️ Practice Questions — Perimeter and Area (30 Questions)

Section A — Direct Formulas Easy

Q1. Find area & perimeter of rectangle 12m × 8m
Q2. Find area and perimeter of square with side 9cm
Q3. Area of triangle with base=14cm, height=10cm
Q4. Circumference of circle with r=7cm (π=22/7)
Q5. Area of circle with r=14cm (π=22/7)
Q6. Area of parallelogram: base=15cm, height=9cm
Q7. Area of rhombus with diagonals 10cm and 16cm
Q8. Perimeter of triangle with sides 5, 7, 8 cm

Section B — Applications Medium

Q9. A room is 10m × 8m. Cost of tiling = ₹50/m². Find total cost.
Q10. A square field has perimeter 240m. Find its area.
Q11. A circular pond has radius 21m. Find fencing cost at ₹30/m.
Q12. A rectangular garden 20m × 15m has a 2m-wide path inside it. Find path area.
Q13. Area of a triangle = 84 cm², base = 14 cm. Find height.
Q14. A wheel has radius 35cm. How far does it travel in 100 revolutions?
Q15. A parallelogram has base 18cm and area 126 cm². Find height.
Q16. Convert: 5.6 m² to cm². Also, 3.5 hectares to m².

Section C — Path and Combined Figures Medium

Q17. A rectangular field 60m × 40m has a uniform path of 5m around it outside. Find area of path.
Q18. A square park (side 50m) has a 3m path inside. Find area of path only.
Q19. Two triangles share the same base 10cm. Heights are 8cm and 12cm. Find ratio of their areas.
Q20. A semicircle has diameter = 28cm. Find its area and perimeter. (π=22/7)
Q21. A figure is made of a rectangle 10cm × 4cm with a semicircle on one shorter side. Find total area.
Q22. Four circles of radius 3.5cm each are cut from a piece of cardboard of area 400 cm². Find remaining area.

Section D — Challenge Hard

  1. Q23. A right triangle has legs 24cm and 10cm. Find the area using both: (i) ½ × base × height method, (ii) Heron's formula with s = (24+10+26)/2 = 30. Confirm they match.
  2. Q24. The ratio of areas of two circles is 4:9. What is the ratio of their radii? Their circumferences?
  3. Q25. A field is in the shape of a trapezium with parallel sides 120m and 80m, height 60m. Find its area. (Use: Area of trapezium = ½(a+b)×h)
  4. Q26. If one side of a square is increased by 50%, by what % does the area increase?
  5. Q27. A rhombus has perimeter 52cm and one diagonal 24cm. Find the other diagonal and area.
  6. Q28. A wheel of a car rotates 500 times to cover 1.1 km. Find the radius of the wheel. (π=22/7)
  7. Q29. A rectangular swimming pool 40m × 20m has a running track of uniform width around it. If track area = 448 m², find track width.
  8. Q30. ABCD is a parallelogram where AB=base=15cm, height=8cm. E is midpoint of AB. Find area of △ADE where D is opposite vertex.
✅ Key Answers: Q1:A=96m², P=40m | Q4:C=44cm | Q5:A=616cm² | Q7:80cm² | Q9:₹4000 | Q10:3600m² | Q13:h=12cm | Q14:220m | Q17:700m² | Q20:A=308cm² | Q24:Radii=2:3, Circumf=2:3 | Q25:6000m² | Q26:125% increase | Q27:other d=10cm, A=120cm² | Q28:r=35cm

📝 Quick Revision

  1. Square: P=4a, A=a² | Rectangle: P=2(l+b), A=l×b
  2. Triangle: A = ½ × base × height | Circle: C=2πr, A=πr²
  3. Parallelogram: A = base × height (NOT side × side!)
  4. Rhombus: A = ½ × d₁ × d₂
  5. π ≈ 22/7 or 3.14. Always take the value given in the question.
  6. Path area = Outer rectangle area − Inner rectangle area
  7. 1 m² = 10,000 cm² | 1 hectare = 10,000 m²