Mensuration: Area & Perimeter // āĻŽāĻžāĻĒāĻā§āĻ (Mensuration): āĻā§āĻˇā§āĻ¤ā§ā§°āĻĢāĻ˛ āĻā§°ā§ āĻĒā§°āĻŋāĻ¸ā§āĻŽāĻž
Basic Concepts (āĻŽā§āĻ˛ āĻ§āĻžā§°āĻŖāĻž)
1. What is perimeter ?
Perimeter: Perimeter is the total length of the boundary of a closed figure. It is measured in units (cm, m, etc.)
āĻĒā§°āĻŋāĻ¸ā§āĻŽāĻž : āĻĒā§°āĻŋāĻ¸ā§āĻŽāĻž āĻšā§āĻā§ āĻā§āĻ¨ā§ āĻŦāĻ¨ā§āĻ§ āĻāĻā§āĻ¤āĻŋā§° āĻŽā§āĻ āĻ¸ā§āĻŽāĻžāĻ¨āĻžā§° āĻĻā§ā§°ā§āĻā§āĻ¯āĨ¤ āĻāĻ¯āĻŧāĻžāĻ cm, m āĻāĻĻāĻŋ āĻāĻāĻāĻ¤ āĻŽāĻžāĻĒ āĻā§°āĻž āĻšāĻ¯āĻŧāĨ¤
Formulas (Perimeter) : āĻĒā§°āĻŋāĻ¸ā§āĻŽāĻž
Square : (āĻŦā§°ā§āĻāĻā§āĻˇā§āĻ¤ā§ā§°)
Perimeter = 4 × side (4a) : - All sides are equal : āĻ¸āĻāĻ˛ā§ āĻŦāĻžāĻšā§ āĻ¸āĻŽāĻžāĻ¨
Rectangle (āĻāĻ¯āĻŧāĻ¤āĻā§āĻˇā§āĻ¤ā§ā§°)
Perimeter = 2(l + b), l = length (āĻĻā§ā§°ā§āĻā§āĻ¯), b = breadth (āĻĒā§ā§°āĻ¸ā§āĻĨ)
Triangle (āĻ¤ā§ā§°āĻŋāĻā§āĻ)
Perimeter = a + b + c, Sum of all three sides : āĻ¤āĻŋāĻ¨āĻŋāĻāĻž āĻŦāĻžāĻšā§ā§° āĻ¯ā§āĻāĻĢāĻ˛
2. What is Area ?
Area: Area is the region enclosed inside a figure. Measured in square units (cm², m²)
Area: āĻā§āĻˇā§āĻ¤ā§ā§°āĻĢāĻ˛ āĻšā§āĻā§ āĻā§āĻ¨ā§ āĻāĻā§āĻ¤āĻŋā§° āĻāĻŋāĻ¤ā§°ā§° āĻŽā§āĻ āĻ āĻā§āĻāĻ˛, āĻāĻ¯āĻŧāĻžāĻ āĻŦā§°ā§āĻ āĻāĻāĻāĻ¤ (cm², m²) āĻŽāĻžāĻĒ āĻā§°āĻž āĻšāĻ¯āĻŧ
Formulas (Area)
Square (āĻŦā§°ā§āĻāĻā§āĻˇā§āĻ¤ā§ā§°) : Area = side × side = a², Side (āĻŦāĻžāĻšā§) × Side (āĻŦāĻžāĻšā§)
Rectangle (āĻāĻ¯āĻŧāĻ¤āĻā§āĻˇā§āĻ¤ā§ā§°): Area = l × b, Length(āĻĻā§ā§°ā§āĻā§āĻ¯) × Breadth(āĻĒā§ā§°āĻ¸ā§āĻĨ)
Triangle (āĻ¤ā§ā§°āĻŋāĻā§āĻ): Area = ½ × b × h, b = base(āĻāĻŋāĻ¤ā§āĻ¤āĻŋ), h = height(āĻāĻā§āĻāĻ¤āĻž)
Quick Revision : (āĻĻā§ā§°ā§āĻ¤ āĻĒā§āĻ¨ā§°āĻžāĻŦā§āĻ¤ā§āĻ¤āĻŋ)
- Perimeter = Boundary length (āĻ¸ā§āĻŽāĻžā§° āĻĻā§ā§°ā§āĻā§āĻ¯)
- Area = Inside region (āĻāĻŋāĻ¤ā§°ā§° āĻ āĻā§āĻāĻ˛)
- Square → P = 4a, A = a²
- Rectangle → P = 2(l+b), A = l×b
- Triangle → P = a+b+c, A = ½×b×h
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Q1. The perimeter of a square is 40 cm. What is the length of one side ? āĻĒā§ā§°āĻļā§āĻ¨: āĻāĻāĻž āĻŦā§°ā§āĻāĻā§āĻˇā§āĻ¤ā§ā§°ā§° āĻĒā§°āĻŋāĻ¸ā§āĻŽāĻž 40 cmāĨ¤ āĻāĻāĻž āĻŦāĻžāĻšā§ā§° āĻĻā§ā§°ā§āĻā§āĻ¯ āĻāĻŋāĻŽāĻžāĻ¨ ?
Options: (a) 5 cm (b) 10 cm (c) 15 cm (d) 20 cm
Ans: (b) 10 cm
Explanation: Perimeter = 4 × side → side = 40 ÷ 4 = 10 cm : āĻŦā§āĻ¯āĻžāĻā§āĻ¯āĻž: Perimeter = 4 × āĻŦāĻžāĻšā§ → āĻŦāĻžāĻšā§ = 40 ÷ 4 = 10 cm
Q2. The area of a rectangle is 60 cm² and breadth is 5 cm. Find the length. āĻĒā§ā§°āĻļā§āĻ¨: āĻāĻāĻž āĻāĻ¯āĻŧāĻ¤āĻā§āĻˇā§āĻ¤ā§ā§°ā§° āĻā§āĻˇā§āĻ¤ā§ā§°āĻĢāĻ˛ 60 cm² āĻā§°ā§ āĻĒā§ā§°āĻ¸ā§āĻĨ 5 cmāĨ¤ āĻĻā§ā§°ā§āĻā§āĻ¯ āĻāĻŋāĻŽāĻžāĻ¨ ?
Options: (a) 10 cm (b) 12 cm (c) 15 cm (d) 20 cm
Ans: (b) 12 cm
Explanation: Area = l × b → l = 60 ÷ 5 = 12 cm āĻŦā§āĻ¯āĻžāĻā§āĻ¯āĻž: Area = l × b → l = 60 ÷ 5 = 12 cm
Q3. The perimeter of a rectangle is 50 cm and length is 15 cm. Find breadth. : āĻĒā§ā§°āĻļā§āĻ¨: āĻāĻāĻž āĻāĻ¯āĻŧāĻ¤āĻā§āĻˇā§āĻ¤ā§ā§°ā§° āĻĒā§°āĻŋāĻ¸ā§āĻŽāĻž 50 cm āĻā§°ā§ āĻĻā§ā§°ā§āĻā§āĻ¯ 15 cmāĨ¤ āĻĒā§ā§°āĻ¸ā§āĻĨ āĻāĻŋāĻŽāĻžāĻ¨ ?
Options: (a) 5 cm (b) 10 cm (c) 15 cm (d) 20 cm
Ans: (b) 10 cm
Explanation: 2(l + b) = 50 → l + b = 25 → b = 25 − 15 = 10 cm āĻŦā§āĻ¯āĻžāĻā§āĻ¯āĻž: 2(l + b) = 50 → l + b = 25 → b = 25 − 15 = 10 cm
Q4. The area of a square is 144 cm². Find its side. āĻĒā§ā§°āĻļā§āĻ¨: āĻāĻāĻž āĻŦā§°ā§āĻāĻā§āĻˇā§āĻ¤ā§ā§°ā§° āĻā§āĻˇā§āĻ¤ā§ā§°āĻĢāĻ˛ 144 cm²āĨ¤ āĻŦāĻžāĻšā§ āĻāĻŋāĻŽāĻžāĻ¨ ?
Options: (a) 10 cm (b) 12 cm (c) 14 cm (d) 16 cm
Ans: (b) 12 cm
Explanation: Area = a² → a = √144 = 12 cm : āĻŦā§āĻ¯āĻžāĻā§āĻ¯āĻž: Area = a² → a = √144 = 12 cm
Q5. Find the area of a triangle with base 10 cm and height 6 cm. : āĻĒā§ā§°āĻļā§āĻ¨: āĻāĻāĻž āĻ¤ā§ā§°āĻŋāĻā§āĻā§° āĻāĻŋāĻ¤ā§āĻ¤āĻŋ 10 cm āĻā§°ā§ āĻāĻā§āĻāĻ¤āĻž 6 cmāĨ¤ āĻā§āĻˇā§āĻ¤ā§ā§°āĻĢāĻ˛ āĻāĻŋāĻŽāĻžāĻ¨ ?
Options: (a) 20 cm² (b) 30 cm² (c) 40 cm² (d) 60 cm²
Ans: (b) 30 cm²
Explanation: Area = ½ × b × h = ½ × 10 × 6 = 30 cm² : āĻŦā§āĻ¯āĻžāĻā§āĻ¯āĻž: Area = ½ × b × h = ½ × 10 × 6 = 30 cm²
Q6. The perimeter of a triangle is 30 cm. Two sides are 10 cm and 8 cm. Find the third side. : āĻĒā§ā§°āĻļā§āĻ¨: āĻāĻāĻž āĻ¤ā§ā§°āĻŋāĻā§āĻā§° āĻĒā§°āĻŋāĻ¸ā§āĻŽāĻž 30 cmāĨ¤ āĻĻā§āĻāĻž āĻŦāĻžāĻšā§ 10 cm āĻā§°ā§ 8 cmāĨ¤ āĻ¤ā§āĻ¤ā§āĻ¯āĻŧ āĻŦāĻžāĻšā§ āĻāĻŋāĻŽāĻžāĻ¨ ?
Options: (a) 10 cm (b) 12 cm (c) 8 cm (d) 6 cm
Ans: (b) 12 cm
Explanation: Third side = 30 − (10 + 8) = 12 cm āĻŦā§āĻ¯āĻžāĻā§āĻ¯āĻž: āĻ¤ā§āĻ¤ā§āĻ¯āĻŧ āĻŦāĻžāĻšā§ = 30 − (10 + 8) = 12 cm
Q7. If the side of a square is doubled, the area becomes how many times ? : āĻĒā§ā§°āĻļā§āĻ¨: āĻāĻāĻž āĻŦā§°ā§āĻāĻā§āĻˇā§āĻ¤ā§ā§°ā§° āĻŦāĻžāĻšā§ āĻĻā§āĻā§āĻŖ āĻā§°āĻŋāĻ˛ā§ āĻā§āĻˇā§āĻ¤ā§ā§°āĻĢāĻ˛ āĻāĻŋāĻŽāĻžāĻ¨ āĻā§āĻŖ āĻšāĻ¯āĻŧ ?
Options: (a) 2 times (b) 3 times (c) 4 times (d) 6 times
Ans: (c) 4 times
Explanation: Area ∝ side² → (2a)² = 4a² → 4 times : āĻŦā§āĻ¯āĻžāĻā§āĻ¯āĻž: Area ∝ āĻŦāĻžāĻšā§² → (2a)² = 4a² → 4 āĻā§āĻŖ
Q8. The length and breadth of a rectangle are in ratio 3:2 and perimeter is 100 cm. Find length. : āĻĒā§ā§°āĻļā§āĻ¨: āĻāĻāĻž āĻāĻ¯āĻŧāĻ¤āĻā§āĻˇā§āĻ¤ā§ā§°ā§° āĻĻā§ā§°ā§āĻā§āĻ¯:āĻĒā§ā§°āĻ¸ā§āĻĨ = 3:2 āĻā§°ā§ āĻĒā§°āĻŋāĻ¸ā§āĻŽāĻž 100 cmāĨ¤ āĻĻā§ā§°ā§āĻā§āĻ¯ āĻāĻŋāĻŽāĻžāĻ¨ ?
Options: (a) 20 cm (b) 25 cm (c) 30 cm (d) 35 cm
Ans: (c) 30 cm
Explanation: Let l = 3x, b = 2x, 2(5x) = 100 → x = 10 → l = 30 cm
Q9. Find the perimeter of a square whose side is 9 cm. : āĻĒā§ā§°āĻļā§āĻ¨: 9 cm āĻŦāĻžāĻšā§ā§° āĻŦā§°ā§āĻāĻā§āĻˇā§āĻ¤ā§ā§°ā§° āĻĒā§°āĻŋāĻ¸ā§āĻŽāĻž āĻāĻŋāĻŽāĻžāĻ¨ ?
Options: (a) 18 cm (b) 27 cm (c) 36 cm (d) 45 cm
Ans: (c) 36 cm
Explanation: Perimeter = 4 × 9 = 36 cm : āĻŦā§āĻ¯āĻžāĻā§āĻ¯āĻž: Perimeter = 4 × 9 = 36 cm
Q10. Area of rectangle = 200 cm² and length = 20 cm. Find breadth. : āĻĒā§ā§°āĻļā§āĻ¨: āĻāĻ¯āĻŧāĻ¤āĻā§āĻˇā§āĻ¤ā§ā§°ā§° āĻā§āĻˇā§āĻ¤ā§ā§°āĻĢāĻ˛ 200 cm² āĻā§°ā§ āĻĻā§ā§°ā§āĻā§āĻ¯ 20 cmāĨ¤ āĻĒā§ā§°āĻ¸ā§āĻĨ āĻāĻŋāĻŽāĻžāĻ¨?
Options: (a) 5 cm (b) 10 cm (c) 15 cm (d) 20 cm
Ans: (b) 10 cm
Explanation: b = 200 ÷ 20 = 10 cm : āĻŦā§āĻ¯āĻžāĻā§āĻ¯āĻž: b = 200 ÷ 20 = 10 cm
Exam Tip (āĻā§ā§°ā§āĻ¤ā§āĻŦāĻĒā§āĻ°ā§āĻŖ āĻāĻŋāĻĒāĻ):
- Square → 4a, a²
- Rectangle → 2(l+b), l×b
- Triangle → a+b+c, ½bh