Measuring Space: Perimeter and Area – Class 9 Mathematics notes, important questions and MCQs

Chapter overview

Perimeter is the length around a figure and area is the space inside it. Heron's formula gives a triangle's area from its three sides.

Common figures

• Rectangle: P = 2(l + b), A = l × b. Square: P = 4a, A = a².
• Triangle: A = ½ × base × height. Parallelogram: A = base × height.
• Circle: C = 2πr, A = πr².

Heron's formula

• s = (a + b + c)/2 (semi-perimeter).
• Area = √[s(s − a)(s − b)(s − c)].
• Equilateral triangle: A = (√3/4)a².

Quadrilaterals by splitting

• Split a quadrilateral along a diagonal into two triangles and add their areas.
• Rhombus: A = ½ d₁ d₂. Trapezium: A = ½(a + b)h.

Key terms

Perimeter

total length of the boundary

Area

amount of surface enclosed

Semi-perimeter

half the perimeter of a triangle

Formulas

• Heron: A = √[s(s − a)(s − b)(s − c)]
• Equilateral: A = (√3/4)a²
• Trapezium: A = ½(a + b)h

Important questions with answers

Quick revision

• s = half of a + b + c.
• Split complex shapes into triangles and rectangles.
• Area of rhombus = half the product of diagonals.

Exam tips

• Keep units the same; area units are squared (cm², m²).
• Use Heron when the height is not given.

Practice MCQs

1. Area of a rectangle 8 cm by 5 cm is

   1. 13 cm²
   2. 26 cm²
   3. 40 cm²
   4. 80 cm²
   Show answer

   Answer: (c) 40 cm². 8 × 5 = 40.

2. Semi-perimeter of a triangle with sides 3, 4, 5 cm is

   1. 12 cm
   2. 5 cm
   3. 6 cm
   4. 7.5 cm
   Show answer

   Answer: (c) 6 cm. 12/2 = 6.

3. Perimeter of a square of side 9 cm is

   1. 18 cm
   2. 36 cm
   3. 81 cm
   4. 27 cm
   Show answer

   Answer: (b) 36 cm. 4 × 9.

4. Area of a triangle with sides 3, 4, 5 cm is

   1. 12 cm²
   2. 6 cm²
   3. 10 cm²
   4. 7.5 cm²
   Show answer

   Answer: (b) 6 cm². √(6 × 3 × 2 × 1) = 6.

5. Area of a rhombus with diagonals 10 cm and 8 cm is

   1. 80 cm²
   2. 18 cm²
   3. 40 cm²
   4. 20 cm²
   Show answer

   Answer: (c) 40 cm². ½ × 10 × 8 = 40.

6. Area of an equilateral triangle of side 4 cm is

   1. 8√3 cm²
   2. 4√3 cm²
   3. 16 cm²
   4. 2√3 cm²
   Show answer

   Answer: (b) 4√3 cm². (√3/4) × 16.

7. Area of a triangle with sides 13, 14, 15 cm is

   1. 105 cm²
   2. 96 cm²
   3. 72 cm²
   4. 84 cm²
   Show answer

   Answer: (d) 84 cm². s = 21; √(21 × 8 × 7 × 6) = √7056 = 84.

8. A triangle has perimeter 32 cm and two sides 8 cm and 11 cm. Its area is

   1. 8√30 cm²
   2. 16√30 cm²
   3. 44 cm²
   4. 4√30 cm²
   Show answer

   Answer: (a) 8√30 cm². Third side 13, s = 16: √(16 × 8 × 5 × 3) = √1920 = 8√30.