Area

The formulas are short; the marks are lost on reading the question.

• Rectangle: area l × b, perimeter 2(l + b)
• Square: area a², perimeter 4a, diagonal a√2
• Triangle: area ½ × base × height
• Circle: area πr², circumference 2πr (take π = 22/7 when the numbers are multiples of 7)

Four question types:

1. Direct substitution into one of the above.
2. Cost problems. Paving, carpeting, painting → per square metre, so use the area. Fencing, a wire around a field → per metre, so use the perimeter. The word tells you which.
3. Percentage change in area. Take both sides as 100. If the length rises by p% and the breadth falls by q%, the new area is (100 + p)(100 − q) against 10000. Area depends on the product, so the percentages never simply subtract.
4. A path around a garden. Outer rectangle minus inner. For a path of width w outside: (l + 2w)(b + 2w) − lb. The w is doubled because the path runs along both sides.

The single most common mistake is answering with the perimeter when the area was wanted, or the reverse. Check the unit: an area is in m², a length in m, and a cost in rupees. If the option list is in m² and you computed a perimeter, you will not even find your answer there.

Note that the shortcut 2w(l + b) for the path is wrong — it misses the four corner squares, worth 4w².

rectangle l×b · square a² · triangle ½bh · circle πr², circumference 2πr · path outside: (l+2w)(b+2w) − lb · area in m², length in m

A rectangular garden is 20 m by 15 m, with a path 2 m wide all around it on the outside. Find the area of the path. → Outer rectangle = (20 + 4) × (15 + 4) = 24 × 19 = 456 m². Inner = 20 × 15 = 300 m². Path = 456 − 300 = 156 m². Sanity check: the strip formula 2 × 2 × (20 + 15) = 140 m² is short by exactly the four corner squares, 4 × (2 × 2) = 16 m², and 140 + 16 = 156.