Perimeter and Area
Perimeter and Area
Mensuration is examined heavily in Part-III, and the two-dimensional half gives 2 to 3 questions. Formulas alone are not enough here, because the paper prefers questions where a dimension changes and you must work out what happens to the area.
The Formula Sheet
| Figure | Area | Perimeter |
|---|---|---|
| Square | side² | 4 × side |
| Rectangle | l × b | 2(l + b) |
| Triangle | ½ × base × height | Sum of the sides |
| Equilateral triangle | (√3/4) × side² | 3 × side |
| Parallelogram | base × height | 2(a + b) |
| Rhombus | ½ × d₁ × d₂ | 4 × side |
| Trapezium | ½ × (sum of parallel sides) × height | Sum of the sides |
| Circle | πr² | 2πr |
Use π as 22/7 whenever the radius is a multiple of 7, and 3.14 otherwise. Examiners choose radii of 7, 14 or 21 precisely so the fraction cancels.
✗ Circumference of a circle is πr² | ✓ Circumference is 2πr; πr² is the area
Heron's Formula
Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c) ÷ 2
Use it when all three sides are known but no height is given. For sides 13, 14 and 15, s is 21 and the area works out to 84 square units. Recognising when the height is missing is the whole skill.
What Happens When Dimensions Change
This is the question type that separates Part-III from Part-II mensuration.
- If every side of a two-dimensional figure is multiplied by k, the area is multiplied by k².
- Doubling the radius makes the area four times, not twice.
- Increasing the side of a square by 10 per cent increases the area by 21 per cent, from the successive-change formula.
- Increasing length by 20 per cent and reducing breadth by 20 per cent gives a 4 per cent decrease in area, not no change.
Each of these has an obvious-looking wrong answer that is always among the options.
Paths, Borders and Combined Figures
A path around a garden, or a border inside a picture frame, is handled the same way each time: find the outer area, find the inner area, subtract.
A rectangular garden 20 m by 15 m has a 2 m wide path outside it. The outer rectangle is 24 by 19, so the path area is 456 − 300 = 156 square metres.
Note the 2 m added on both sides, making each dimension grow by 4, not 2. Adding the width only once is the standard error, and the resulting value is planted in the options.
For combined figures, split the shape into a rectangle plus a triangle or a semicircle, compute each part and add. Never try to find a single formula for an irregular shape.
सभी भुजाएँ k गुनी हों तो क्षेत्रफल k² गुना होता है।
TRE pointer: Straight formula questions are the minority here. The paper prefers the percentage-change-in-area type and the path-around-a-rectangle type, both of which have a tempting wrong answer built in. Heron's formula is worth memorising because it appears whenever three sides are given with no height. With −1/3 negative marking, take the extra ten seconds to check whether a path lies inside or outside the figure.
60-Second Recap
- Area of a rhombus is half the product of the diagonals; trapezium is half the sum of parallel sides times height.
- Circle: area πr², circumference 2πr. Use 22/7 when r is a multiple of 7.
- Heron's formula when three sides are known and no height is given.
- Scaling all sides by k multiplies the area by k².
- A 20% rise in length with a 20% fall in breadth gives a 4% fall in area.
- For an outside path, each dimension grows by twice the path width.
Frequently Asked Questions
If the radius of a circle is doubled, what happens to its area?
It becomes four times as large, because area depends on the square of the radius. The circumference, which depends on the radius directly, merely doubles. Questions often ask about both in the same statement to see whether you distinguish the linear and squared relationships.
When should I use Heron's formula?
When all three sides of a triangle are known but no height is given. Compute the semi-perimeter, then take the square root of s times each of the three differences. If a height is available, the simpler half base times height formula is faster and less error-prone.
How do I find the area of a path around a garden?
Find the area of the larger rectangle including the path, then subtract the area of the garden. Remember that a path of width two metres adds two metres on each side, so each dimension grows by four, not two. That doubling is the usual point of error.
If length increases 20 per cent and breadth decreases 20 per cent, does the area stay the same?
No, it falls by four per cent. Use the successive change formula: twenty minus twenty minus four hundred over a hundred gives minus four. The intuitive answer of no change is always offered as an option and is the most commonly chosen wrong one.