Perimeter and Area
Overview
Perimeter and Area form the backbone of mensuration at the primary level and appear consistently in WB TET Paper I Mathematics. These concepts test a candidate's ability to apply formulas to plane figures and solve word problems involving fencing, flooring, painting and similar real-life contexts.
For WB TET, you must master the basic formulas for square, rectangle and triangle, understand the distinction between perimeter (boundary length) and area (surface covered), and convert between units when required. Questions often combine these concepts with cost calculations — for example, finding the cost of fencing a rectangular plot or tiling a square room.
Conceptual clarity matters more than rote memorisation. Examiners frequently test whether candidates can identify which formula applies to a given situation and whether they can handle composite figures or missing-dimension problems.
Key Concepts
- Perimeter is the total length of the boundary of a closed plane figure. It is measured in linear units (cm, m, km).
- Area is the amount of surface enclosed within the boundary. It is measured in square units (cm², m², km²).
- For any figure, perimeter and area are independent properties — two figures can have the same perimeter but different areas, and vice versa.
- Square: All four sides equal; all angles 90°. Perimeter depends on one measurement (side), area depends on side squared.
- Rectangle: Opposite sides equal; all angles 90°. Perimeter uses length and breadth; area is their product.
- Triangle: Three-sided polygon. Perimeter is the sum of all three sides; area requires base and corresponding height.
- Unit consistency is crucial. Always ensure all measurements are in the same unit before applying formulas.
- Composite figures can be broken into simpler shapes. Find individual areas/perimeters and combine appropriately.
Formulas / Key Facts
Square (side = a)
| Property | Formula |
|---|---|
| Perimeter | P = 4a |
| Area | A = a² |
| Diagonal | d = a√2 |
Rectangle (length = l, breadth = b)
| Property | Formula |
|---|---|
| Perimeter | P = 2(l + b) |
| Area | A = l × b |
| Diagonal | d = √(l² + b²) |
Triangle (sides a, b, c; base = b; height = h)
| Property | Formula |
|---|---|
| Perimeter | P = a + b + c |
| Area (using base and height) | A = ½ × base × height |
| Area (Heron's formula) | A = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2 |
Useful Conversions
- 1 m = 100 cm
- 1 km = 1000 m
- 1 m² = 10,000 cm²
- 1 hectare = 10,000 m²
Quick Facts
- If the side of a square is doubled, area becomes 4 times; perimeter becomes 2 times.
- For a fixed perimeter, a square encloses the maximum area among all rectangles.
- Right-angled triangle: Area = ½ × product of the two perpendicular sides.
Worked Examples
Example 1: Rectangle — Fencing Cost
Problem: A rectangular garden is 25 m long and 15 m wide. Find the cost of fencing it at ₹18 per metre.
Solution:
- Perimeter = 2(l + b) = 2(25 + 15) = 2 × 40 = 80 m
- Cost = Perimeter × Rate = 80 × 18 = ₹1440
Answer: ₹1440
Example 2: Square — Finding Side from Area
Problem: The area of a square plot is 625 m². Find its perimeter.
Solution:
- Area = a², so a² = 625
- a = √625 = 25 m
- Perimeter = 4a = 4 × 25 = 100 m
Answer: 100 m
Example 3: Triangle — Area Calculation
Problem: A triangular signboard has a base of 80 cm and a height of 50 cm. Find its area in m².
Solution:
- Convert to metres: base = 0.8 m, height = 0.5 m
- Area = ½ × base × height = ½ × 0.8 × 0.5 = 0.2 m²
Answer: 0.2 m²
Example 4: Composite Figure
Problem: A rectangular lawn 20 m × 12 m has a square flower bed of side 4 m inside it. Find the area of the lawn excluding the flower bed.
Solution:
- Area of rectangle = 20 × 12 = 240 m²
- Area of square = 4² = 16 m²
- Remaining area = 240 − 16 = 224 m²
Answer: 224 m²
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Confusing perimeter and area formulas — writing area as 2(l + b). | Remember: Perimeter is boundary (addition-based), area is surface (multiplication-based). |
| Forgetting to square units when reporting area — writing "20 m" instead of "20 m²". | Always attach "²" (squared) to area units. |
| Mixing units — adding cm and m directly without conversion. | Convert all measurements to the same unit first. |
| Using Heron's formula when base and height are given — wasting time. | Use ½ × base × height when height is available; Heron's is for when only sides are known. |
| Doubling perimeter when side is doubled instead of recognising that area quadruples. | Perimeter scales linearly; area scales with the square of the linear factor. |
Quick Reference
- Square: P = 4a, A = a²
- Rectangle: P = 2(l + b), A = l × b
- Triangle: P = sum of sides, A = ½ × base × height
- Perimeter → linear units (m, cm); Area → square units (m², cm²)
- Cost problems: Total cost = Perimeter (or Area) × Rate per unit
- Composite figures: Break into simpler shapes, calculate separately, then add or subtract as needed.