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Let us consider the following figures which can be formed by folding a wire or a string. In each of the following figures, if we start from the point S and move along the line segments to reach again at the same point S by making a complete round, then the long distance covered is equal to the length of the wire used to draw the figure. This distance is known as the perimeter of the closed figure. The idea of perimeter is very widely used in our day-to-day life.
For Example, a person preparing a track to conduct sports uses the idea of perimeter. Also, if a farmer wants to fence his field uses the ideas of perimeter. Thus, we can say that the perimeter is the distance along the line or curve forming a closed figure when we go around the figure once.
Perimeter:
The length of the boundary of a closed figure is known as its perimeter.
Perimeter of a Rectangle:
The sum of all the sides of a rectangle is known as its perimeter.
Consider a rectangle as shown in fig. Let l and b denote its the length and breadth respectively. If P denotes perimeter, then
P = AB + BC + CD + DA
=> P = (l + b + l + b)
=> P = 21 + 2b
=> P = 2(l + b)
=> P = 2 (Length + Breadth)
It follows from this formula that
Length = \(\frac{P}{2}\) – Breadth or, Length = \(\frac{1}{2}\)Perimeter – Breadth
Breadth = \(\frac{P}{2}\) – Length or, Breadth = \(\frac{1}{2}\)Perimeter – Length
Perimeter of a Square:
We know that a square is a rectangle whose length and breadth are equal. Therefore, all formulae that we have attained for the perimeter of a rectangle may be used to obtain perimeter of a square. However, since all the four sides are equal, so we may use the following simpler forms of the said formulae.
Perimeter of a square = 2 (length + breadth)
= 2 (length + length)
= 2(2 length)
= 4 x length
= 4 x side
Also, Side of a Square = \( \frac{1}{4}\)(Perimeter of the Square)
Perimeter of an Equilateral Triangle:
Consider an equilateral triangle ABC as shown in fig. Each side of an equilateral triangle is of the same length. Therefore, AB = BC = CA.
Now,
Perimeter of \(\bigtriangleup{ABC}\) = AB + BC + CA
= AB + AB + AB { Since, AB = BC = CA }
= 3 AB
= 3 x(Length of a side of \(\bigtriangleup{ABC}\))
Hence,
Perimeter of an equilateral triangle = 3 x Length of a side of the triangle.
Results:
Result 1) A regular pentagon is a polygon with 5 equal sides.
Therefore, the perimeter of a regular pentagon = 5 x Length of a side
=> Length of a side of a regular pentagon = \(\frac{1}{5}\)(Perimeter of the Pentagon).
Result 2) A regular hexagon is a polygon with 6 equal sides.
Therefore, the perimeter of a regular hexagon = 6 x Length of a side.
=> Length of a side of a regular hexagon = \(\frac{1}{6}\)(Perimeter of the hexagon).
Area:
Area is a quantity that expresses the extent of a two-dimensional surface or shape, in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat.
The magnitude of the measurement of the region is called its area.
We have defined area of a closed plane as the measurement of the region enclosed by it. For measuring areas of plane figures, we express their areas in terms of the area of a square whose side is of length 1 cm as shown in fig. We say that its area is 1 square centimetre and is written in short as 1 sq. cm or 1 \(cm^{2}\)
This is standard unit of area, just as 1 cm is a standard unit of length. If a plane region R is divided into n squares, each of side 1 cm, then we say that the area of the region R is n \(cm^{2}\).
Area of a Rectangle:
In a rectangle, the larger side is called its length while the smaller side is called its breadth.
In order to obtain a formula for the area of a rectangle, let us consider the following experiment:
Experiment: Draw a rectangle ABCD on a sheet of paper such that length = AB = 7 cm and breadth BC = 4 cm.
On sides AB and BC step off segments of 1 cm each so that the side AB is divided into 7 equal segments and the side BC is divided into 4 equal segments of 1 cm each.
Now, draw lines parallel to the side BC through each point of division of the side AB. Also, draw lines parallel to AB through each point of division of BC.
We observe that the rectangular region ABCD is divided into a number of square regions, a side of each square region being 1 cm. The area of each of these square regions is 1 \(cm^{2}\).
Clearly, there are 4 rows of squares and each row contains 7 squares.
Therefore, total number of squares = 7 X 4 = 28.
Therefore, Area of rectangle ABCD = 28 \(cm^{2} = (7 x 4) cm^{2}\).
Similarly, for different lengths and breadths we can find the area of a rectangle:
|
Length (l) in cm |
Breadth (b) in cm |
No. of 1 cm squares |
Area |
Area in terms of length & breadth |
|
6 |
2 |
12 |
\(12 cm^{2}\) |
6 x 2 \(cm^{2}\) |
|
7 |
5 |
35 |
\(35 cm^{2}\) |
7 x 5\(cm^{2}\) |
|
8 |
4 |
32 |
\(32 cm^{2}\) |
8 x 4\(cm^{2}\) |
|
10 |
5 |
50 |
\(50 cm^{2}\) |
10 x 5\(cm^{2}\) |
|
12 |
7 |
84 |
\(84 cm^{2}\) |
12 x 7\(cm^{2}\) |
Therefore, Area of a rectangle = (Length x Breadth) square units.
=> Area = l x b square units.
Length = \((\frac{Area}{Breadth})\) units.
Breadth = \((\frac{Area}{Length})\) units.
Area of a Square:
Square is a rectangle whose length and breadth are equal. Therefore,
Area of a Square = Length x Breadth
= (Length x Length)
= \((Length)^2 = (Side)^2\)
= \({\frac{1}{2} X (diagonal)^2}\) sq.units.