This is the foundation and, for CTET, the richest source of misconception questions. Perimeter is the distance once around the boundary of a closed figure — measured in length units (cm, m). Area is the amount of flat surface a figure covers — measured in square units (cm², m²). For a rectangle, perimeter = 2(l + b) and area = l × b; for a square, perimeter = 4a and area = a². The single most-tested confusion in Classes VI–VIII is treating area and perimeter as the same thing, or assuming they move together. They do not: two rectangles can have the same perimeter but very different areas (a 1 × 11 strip and a 6 × 6 square both have perimeter 24, but areas 11 and 36). A second classic misconception is the 'doubling' trap — double the side of a square and a child expects the area to double, when it actually becomes four times as large (2a)² = 4a². CTET tests this as both a content sum and a pedagogy item ('a student says doubling the side doubles the area — what is the best way to correct this?'). The teacher-friendly answer is almost always to let the child cover the figure with unit squares or tiles, building area as a count of squares rather than a memorised formula, and to walk the boundary with a string for perimeter.
✅ Solved examples
1. The length of a rectangular field is 24 m and its breadth is 18 m. Find its (a) perimeter and (b) area.
Perimeter = 2(l + b) = 2(24 + 18) = 2 × 42 = 84 m. Area = l × b = 24 × 18 = 432 m². Note the units: perimeter in metres, area in square metres.
2. The side of a square is doubled from 5 cm to 10 cm. By what factor does the area change?
Original area = 5² = 25 cm². New area = 10² = 100 cm². The area becomes 100 ÷ 25 = 4 times the original — not 2 times. Doubling the side quadruples the area, because area depends on the square of the side.
3. A square has a perimeter of 48 cm. Find its area.
Perimeter of a square = 4a, so 4a = 48 ⇒ a = 12 cm. Area = a² = 12² = 144 cm².
4. A wire is bent into a square of side 11 cm. The same wire is then bent into a rectangle of length 14 cm. Find the breadth of the rectangle.
The wire length stays the same. Square perimeter = 4 × 11 = 44 cm, so the rectangle's perimeter is also 44 cm. 2(l + b) = 44 ⇒ l + b = 22 ⇒ 14 + b = 22 ⇒ b = 8 cm.
✏️ Practice — try these, take hints as needed
1. A rectangular garden is 30 m long and 20 m wide. Find the cost of fencing it at ₹15 per metre.
Fencing follows the boundary, so you need the perimeter.
2. A student claims a rectangle of 2 cm × 8 cm and a square of 4 cm × 4 cm 'must have the same area because they have the same perimeter'. Is the student right? Give both areas.
Check the perimeters first.
Equal perimeter does NOT force equal area.
Rectangle area = l × b, square area = a².
Both have perimeter 20 cm, but areas differ: rectangle = 2 × 8 = 16 cm², square = 4 × 4 = 16 cm²… here they happen to be equal — so use 3 cm × 7 cm vs 5 × 5: perimeters both 20, areas 21 cm² vs 25 cm². Equal perimeter does NOT guarantee equal area.
3. Find the area of a square whose side is 9 m, and the perimeter of the same square.
Area = a².
Perimeter = 4a.
Area = 81 m²; Perimeter = 36 m.
4. The area of a rectangle is 96 cm² and its length is 12 cm. Find its breadth and perimeter.
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