Perimeter, Area, Volume and Roots അളവുകൾ
Every polynomial in this chapter comes from a measurement: the perimeter of a rectangle, the area of a figure, the volume of a box, the length of a rope. Here is a quick, hands-on refresher of all the formulas the chapter uses – plus square roots, reciprocals and percentages, which help us spot expressions that are not polynomials.
- Letters for numbers (page 1) and squares (page 2).
1. Perimeter and area of a rectangle
A rectangle (ചതുരം) with length l and width w: the perimeter (ചുറ്റളവ്), the distance all the way round, is l + w + l + w = 2(l + w). The area (പരപ്പളവ്), the number of unit squares inside, is l × w. For a square (സമചതുരം) of side s: perimeter 4s, area s².
Set l = 3, w = 2: perimeter 10, area 6. Now make both 1 bigger (4 and 3): perimeter 14, area 12. This is exactly how the chapter begins!
2. Volume of a block
A rectangular block (ചതുരക്കട്ട) with edges l, w, h holds l × w unit cubes in each layer and h layers, so its volume (വ്യാപ്തം) is l × w × h cubic units. An open box made by folding up a card is a block too.
3. Right triangles and Pythagoras
In a right triangle the longest side, opposite the right angle, is the hypotenuse (കർണം). Its square equals the sum of the squares of the other two sides: c² = a² + b², so c = √(a² + b²). The diagonal (വികർണം) of a rectangle is the hypotenuse of the right triangle formed by two sides.
If both legs are a, the hypotenuse is √(a² + a²) = √(2a²) = a√2 (chapter “New Numbers”). Legs 4 → hypotenuse 4√2 ≈ 5.66. Its area is ½ × a × a = ½a². Set a = b = 4 in the lab above.
4. Square roots, reciprocals and percentages
Square root (വർഗമൂലം): √x is the side of a square of area x; √9 = 3, √2 ≈ 1.414. Reciprocal (വ്യുൽക്രമം): 1/x is the number that multiplies x to give 1; the reciprocal of 4 is ¼, of ½ is 2. Rectangles of area 25 with one side x have other side 25/x – a reciprocal appears.
As x grows, the other side 25/x shrinks. The perimeter 2x + 50/x is smallest for the square, x = 5. This expression divides by x – keep it in mind on page 8.
Percentage: “out of 100”. 3 litres of acid in a 10-litre mixture is 3/10 × 100 = 30% acid. Pour in x more litres of acid and the acid share becomes (3 + x)/(10 + x) × 100 percent.
Painters work out wall area to buy paint, packers work out box volume, electricians use Pythagoras for cable along a diagonal, and chemists use percentages for mixtures. The same formulas, with a letter in place of a changing measurement, give the polynomials of this chapter.