Perimeter & Area — Class 7 IMO

Area of Parallelogram & Triangle

The area of a parallelogram is base × height. The area of a triangle is ½ × base × height. The height is always measured at right angles to the base.

Example 1: Find the area of a parallelogram with base 8 cm and height 5 cm.

8 × 5 = 40 sq cm.

Example 2: Find the area of a triangle with base 10 cm and height 6 cm.

½ × 10 × 6 = 30 sq cm.

Quick recap

Parallelogram area = base × height.
Triangle area = ½ × base × height.

✓ Quick check

What is the area of a triangle with base 12 cm and height 5 cm?

½ × 12 × 5 = 30 sq cm.

What is the area of a parallelogram with base 7 cm and height 4 cm?

7 × 4 = 28 sq cm.

Circumference & Area of a Circle

A circle is the set of all points equidistant from a fixed center point.

Term	Definition	Formula
Radius (r)	Distance from center to edge	r = d / 2
Diameter (d)	Distance across circle through center	d = 2r
Chord	Line segment joining any two points on circle	—
Circumference (C)	Distance around the circle	C = 2πr = πd
Area (A)	Space enclosed by the circle	A = πr²

Value of pi: π ≈ 3.14159... or 22/7 (approx). Use 22/7 when radius is a multiple of 7.

Semicircle: Area = πr²/2, Perimeter = πr + 2r

Quarter circle: Area = πr²/4, Perimeter = πr/2 + 2r

CIRCLE PARTS DIAGRAM:

              P
             /|
            / |r (radius)
           /  |
Q---------O---------R
   (diameter = 2r)

  O = center
  OP = radius (r)
  QR = diameter (d = 2r)
  PS = chord (does not pass through center)

FORMULAS:
  Circumference = 2 * pi * r  (or pi * d)
  Area          = pi * r^2
  Semicircle area = pi*r^2 / 2
  Quarter circle area = pi*r^2 / 4

Example 1: Find the circumference of a circle of radius 7 cm (π = 22/7).

2 × (22/7) × 7 = 44 cm.

Example 2: Find the area of a circle of radius 7 cm (π = 22/7).

(22/7) × 7 × 7 = 154 sq cm.

Example 3: Circle radius = 7cm. Find diameter, circumference, area. (use pi = 22/7)

d = 2 x 7 = 14cm. C = 2 x 22/7 x 7 = 44cm. A = 22/7 x 7 x 7 = 154cm^2.

Example 4: Circle circumference = 62.8cm. Find radius and area. (pi = 3.14)

2 x pi x r = 62.8 => r = 62.8 / 6.28 = 10cm. A = 3.14 x 100 = 314cm^2.

Example 5: Semicircle radius = 14cm. Find perimeter and area. (pi = 22/7)

Perimeter = pi x r + 2r = 44 + 28 = 72cm. Area = pi x r^2 / 2 = 616/2 = 308cm^2.

Quick recap

Circumference = 2πr = πd.
Area of a circle = πr²; use π ≈ 22/7.
radius (r) = half of diameter: r = d/2
Circumference = 2 x pi x r
Area = pi x r^2
Semicircle area = pi x r^2 / 2
Use pi = 22/7 when radius is a multiple of 7; otherwise use 3.14

✓ Quick check

A circle has radius 7 cm. What is its area (π = 22/7)?

(22/7) × 49 = 154 sq cm.

A circle has diameter 7 cm. What is its circumference (π = 22/7)?

π × d = (22/7) × 7 = 22 cm.

Unit Conversion & Word Problems

Areas convert in squares: 1 m² = 10000 cm² (because 1 m = 100 cm and 100 × 100 = 10000). Word problems may involve a path or ring — find the larger area and subtract the smaller one.

Example 1: Convert 2 m² into cm².

2 × 10000 = 20000 cm².

Example 2: Convert 30000 cm² into m².

30000 ÷ 10000 = 3 m².

Quick recap

1 m² = 10000 cm².
For a path or ring, subtract the inner area from the outer area.

✓ Quick check

1 m² = ___ cm²?

100 × 100 = 10,000 cm².

Convert 30000 cm² into m².

30000 ÷ 10000 = 3 m².