Mensuration

Rectangles and Squares

Every one of you must have played the game carrom or chess. What is the shape of the carrom board or the chessboard? Isn’t it square?  Also, the tiles and most of the frames in our house are square. So let us learn more about them.

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Area of Rectangle and Square
Introduction to Mensuration
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Rectangles

The above figure is a rectangle. The rectangle is a plane shape with four sides. It is a 4 sided polygon with opposite sides parallel.

Properties of Rectangle

(Source: Quora)

  • Looking at the above figure we see the opposite sides are parallel and equal.
  • This means DA and CB are parallel to one another. What about the other side? If we look at another side we see that those sides are parallel to that means AB and DC are also parallel to one another. So we say that in this figure, the opposite lines are parallel to one another.
  • The sides DA and CB have the same length, so it clear that they are congruent. Also, side DC and AB are congruent to one another. Here in this figure, we have four angles. All the angles in a rectangle are 90°.
  • So we can write it as m∠A = m∠B = m∠C = m∠D = 90°. We can also see that the adjacent angles are supplementary.
  • That is 90° + 90° = 180°. The sum of all the interior angles is 90° + 90°+ 90° + 90° = 360°
  • The diagonals of the rectangle are also congruent to each other and they bisect each other at their point of intersection.
  • A rectangle can also be called as a quadrilateral as it has 4 sides.

Area of rectangle = length × breadth

Squares

Square is a quadrilateral in which all its sides have equal length and all the four corners are right angles.

square

Properties of Square

  • Opposite sides are parallel, with all sides being equal
  • A square has four lines of symmetry
  • The order of rotational symmetry is 4
  • The diagonals bisect each other at 90° or right angles
  • All sides are equal
  • Opposite sides are equal and parallel
  • All angles are equal to 90 degrees
  • The diagonals are equal
  • Diagonals bisect each other at right angles
  • Diagonals bisect the angles
  • Any two adjacent angles add up to 180 degrees
  • Each diagonal divides the square into two congruent isosceles right-angled triangles
  • The sum of the four exterior angles is 4 right angles
  • The sum of the four interior angles is 4 right angles

Solved Questions

Q1. Dexter has to divide his rectangular field into two parts from one corner to the other using fence. If the area of the field is  450m² and the length of the field is 36m then what will be the length of the fence needed?

Sol: Area of rectangle = length × breadth
Since the area of the field is  450m² and the length of the field is 36m the breadth would be 540/36 = 15m
The length of the fence needed is the length of the diagonal.
= √(15²) + (36²) = √225 + 1296
= √1521
= 39m

Q2. The length of a room exceeds the breadth by 22 metres. If both the length and the breadth are increased by 1 meter, then the area of the room is increased by 11 sq. m. Find the length and the breadth of the room.

  1. 3m and 2 m
  2. 2m and 7m
  3. 7m and 9m
  4. 6m and 4m

Solution: D. Let the breadth of the room is x meter. Then,

length of room = x+ 2 (given) and
area of room =  (x+2) x sq  meter

If length and breadth increased 1 meter,
length = (x+2) + 1 = x + 3 meter and breadth = x + 1 meter
Then area of new room = ( x + 3) (x + 1)  sq m

As per given in question
( x + 3) (x + 1) –  (x+2)x = 11
= x² + 4x + 3 – x² -2x  = 11
= 2x = 8
x = 4
So breadth of room = 4
And length of room = 4 + 2 = 6
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