how to find the length and width of a rectangle when given the perimeter

Area of Rectangle: The total space or region enclosed inside a rectangle is known as area. In other words, the space occupied within the perimeter (boundary) of a rectangle is called the area of the rectangle. There are many interesting applications of the area of the rectangle such as determining the area of the rectangular floor, calculating the height of the building, etc.

Also when it comes to solving mensuration or geometry problems, the area of the rectangle plays a huge role. In this article, let's understand the concept of the rectangle area in detail. Read on to find more.

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Area of Rectangle: Table of Contents

Area and Perimeter of a Rectangle Formula
Area of Rectangle and Square
How to Calculate the Area of Rectangle?
Area of Rectangle by Square Method
Derivation of Area of Rectangle
Solved Problems on Area of Rectangle
FAQs

What is the Area of the Rectangle?

The rectangle area depends on its sides. For example, the area of a rectangle of a blackboard is dependent on the measure of length and breadth (or width).

Basically, the formula for the area of a rectangle is equal to the product of the length and breadth (or width) of the rectangle. Whereas, when we speak about the perimeter of a rectangle, it is equal to the sum of all its four sides.

Hence, the region enclosed by the perimeter of the rectangle is its area.

Area of rectangle \(= {\text{Length}} \times {\text{Breadth}}\)

What is a Rectangle?

A rectangle is a quadrilateral whose opposite sides are equal and parallel to each other. Since a rectangle has four sides, it has four angles. All angles of a rectangle are \(90\) degrees, thus all the angles of the rectangle are right angles.

The day-to-day examples of the rectangle are doors of a room, computer screen, books, scale, etc.

Unit of Area of Rectangle

The area of any polygon is the amount of space it occupies or encloses in the plane. The area is usually measured in square units like square meters, square feet, square inches, etc. Area of larger shapes such as fields or cities is measured in square kilometers, hectares, or acres.

Area and Perimeter of a Rectangle Formula

The perimeter of the rectangle is the length of the boundary of the rectangle, which is given as the sum of all the sides of the rectangle.

The rectangle area is calculated in units by multiplying the breadth (or width) by the length of a rectangle.

Thus, the perimeter and the rectangle area is given by:

The perimeter of Rectangle Formula	\(P = 2\left( {l + b} \right)\)
Area of Rectangle Formula	\(A = l \times b\)

Area of Rectangle and Square

The area of any rectangle is calculated when its length and breadth (or width) are known. By multiplying length and breadth, the rectangle's area will be obtained in a square-unit dimension. In the case of a square, the area will become \(sid{e^2}.\) The main difference between square and rectangle is that the length and breadth are equal for square, whereas the length and breadth are different in a rectangle.

Area of Rectangle by Square Method

The simplest way to find the rectangle area is to count the number of square units in it.

Area of Rectangle = Number of Square Units Forming the Rectangle

Example 1: Let us first calculate the area of the rectangle using the square method. Consider a rectangle \(ABCD\) of length \({\text{5 cm}}\) and width \({\text{4 cm}}\)

To find the area of this rectangle, we divide it into equal squares of \(1\) unit square each.

Now, count the number of squares in the above figure. There is a total of \(20\) squares.

Hence, the area of the rectangle \(ABCD\) is \({\text{20}}\,{\text{c}}{{\text{m}}^{\text{2}}}\)

How to Calculate the Area of a Rectangle?

Follow the steps below to find the area of a rectangle:

Note the dimensions of length and breadth (or width) from the given data.
Multiply length and breadth (or width) values.
Write the answer in square units.

Area of Rectangle Derivation

Firstly, the diagonals of the rectangle divide it into two equal right-angled triangles. Therefore, the area of the rectangle will be equal to twice the area of the right-angled triangle.

Suppose \(ABCD\) is a rectangle.

How to Calculate the Area of a Rectangle?

Now, let diagonal \(AC\) divide the rectangle into two congruent right-angled triangles, i.e., \(\Delta ABC\) and \(\Delta ADC.\) So, the area of both right-angled triangles will be equal.

Hence, area of rectangle \(ABCD\; = \) \(2 \times \) Area of \(\Delta ABC\)

We know that area of the triangle \(= \frac{1}{2} \times {\text{base}} \times {\text{height}}\)
Area of the rectangle \( = 2 \times \frac{1}{2} \times AB \times BC\)
\(= 2 \times \left( {\frac{1}{2} \times l \times b} \right) = l \times b\)
Thus, the area of the rectangle \(= {\text{Length}} \times {\text{Breadth}}\)

Area of Rectangle using Diagonals

We know that diagonal divides the rectangle into two congruent right-angled triangles. Thus, the diagonal becomes the hypotenuse of the corresponding right-angled triangle.

We know that the diagonal of a rectangle is calculated by using the Pythagoras theorem as follows:

\({\left( {{\text{Diagonal}}} \right)^{\text{2}}}{\text{=}}{\left( {{\text{Length}}}\right)^{\text{2}}}{\text{ + }}{\left( {{\text{Breadth}}} \right)^{\text{2}}}\)

From the above,

\({\text{Length=}}\sqrt {{{{\text{(Diagonal)}}}^{\text{2}}}{\text{-(Breadth}}{{\text{)}}^{\text{2}}}} \) and \({\text{Breadth=}}\sqrt {{{{\text{(Diagonal)}}}^{\text{2}}}{\text{-(Length}}{{\text{)}}^{\text{2}}}} \)

The area of the rectangle is given by

1. \({\text{Area}} = {\text{Length}} \times {\text{Breadth}} = {\text{Breadth}} \times \sqrt {{{\left( {{\text{Diagonal}}} \right)}^2} – {{\left( {{\text{Breadth}}} \right)}^2}} \)

2. \({\text{Area}} = {\text{Length}} \times {\text{Breadth}} = {\text{Length}} \times \sqrt {{{({\text{Diagonal}})}^2} – {{({\text{Length}})}^2}} \)

Area of Rectangle and Important Formulas

Let the length of the rectangle be \("l",\) breadth of the rectangle be \("b",\) the perimeter of rectangle be \("P"\) and area of rectangle be \("A",\) then

1. Area of the rectangle \(= l \times b\)

2. The perimeter of the rectangle \(= 2\left( {l + b} \right)\)

3. Length of the rectangle \( = \frac{A}{b}\)

4. Length of the rectangle \( = \frac{P}{2} – b\)

5. The breadth of the rectangle \( = \frac{A}{l}\)

6. The breadth of the rectangle \( = \frac{P}{2} – l\)

7. Diagonal of the rectangle \( = \sqrt {{l^2} + {b^2}} \)

Solved Problems On Area of Rectangle

1) Find the area, in a hectare, of the field whose length is \({\text{240}}\,{\text{m}}\) and width \({\text{110}}\,{\text{m}}\)

Solution:

Length \(\left( l \right) = 240\,{\text{m}}\) and width \((b) = 110~{\text{m}}\)
Area of the field (Rectangle) \( = l \times b\)
\( = 240 \times 110\)
\( = 26,400\,{{\text{m}}^2}\)
\( = \frac{{26,400}}{{10,000}} = 2.64\) hectare.

2) The length of a rectangular screen is \({\text{15}}\,{\text{cm}}\) Its area is \({\text{180}}\,{\text{sq}}{\text{.}}\,{\text{cm}}\) Find its width.,

Solution: Area of the screen \(= 180\,{\text{sq}}.{\text{cm}}\)
Length of the screen \({\text{15}}\,{\text{cm}}\)
Area of the rectangle = \(l \times b\)
So, \({\text{Width=}}\frac{{{\text{Area}}}}{{{\text{Length}}}}\)
\( = \frac{{180}}{{15}} = 12~{\text{cm}}\)

3) Find the perimeter and area of the rectangle of length \({\text{17}}\,{\text{cm}}\) and breadth \({\text{13}}\,{\text{cm}}\)

Solution:
Given: length \((l) = 17~{\text{cm}}\) and breadth \((b) = 13~{\text{cm}}\)
Perimeter of rectangle \( = 2\left( {l + b} \right)\)
\( = 2\left( {17 + 13} \right) = 2\left( {30} \right)\)
\(=60 \mathrm{~cm}\)
We know that area of rectangle \( = l \times b\)
\( = 17 \times 13\)
\({\text{=221}}\,{\text{sq}}{\text{.cm}}\)

4) Find the area of a rectangle of length \({\text{43}}\,{\text{m}}\) and width \({\text{13}}\,{\text{cm}}\)

Solution:

Given: length of the rectangle \((l) = 43\,{\text{m}}\).
Width of the rectangle \((b) = 13~{\text{m}}\)
We know that area of the rectangle \( = l \times b.\)
\( = 43 \times 13\)
\(= 559\,{{\text{m}}^2}\)
Area of the given rectangle is \(= 559\,{{\text{m}}^2}\).

5) The length and width of the rectangular farm are \(80\) yards and \(60\) yards. Find the area of the farm.

Solution:
Given: Length of the rectangle \((l) = 80\,{\text{yd}}\)
Width of the farm \((b) = 60\,{\text{yd}}\)
We know that area of the rectangle is length \( \times \) breadth.
So, area of the farm is \(= 80 \times 60 = 4800\,{\text{sq}}{\text{.yd}}\).

6) The width of the rectangle is \({\text{8}}\,{\text{cm}}\) and its diagonal is \({\text{17}}\,{\text{cm}}\). Find the area of the rectangle and its perimeter.

Solution:

Using Pythagoras theorem,
\(B{D^2} = C{D^2} + B{C^2}\)
⇒ \({17^2} = C{D^2} + {8^2}\)
⇒ \(C{D^2} = 289 – 64\)
⇒ \(C{D^2} = 225\)
⇒ \(CD = 15\)
Therefore, length of rectangle \({\text{=15}}\,{\text{cm}}\)
So, area of rectangle \( = l \times b\)
\(= 15 \times 8\,{\text{c}}{{\text{m}}^2}\)
\(= 120\,{\text{c}}{{\text{m}}^2}\)
Also, perimeter of rectangle \(= 2(15 + 8)\,{\text{cm}}\)
\(= 2 \times 23\,~{\text{cm}}\)
\(= 46\,~{\text{cm}}\).

7) How many envelopes can be made from a sheet of paper \({\text{100}}\,{\text{cm}}\) by \(75\,{\text{cm}}\) supposing \(1\) envelope requires \({\text{20}}\,{\text{cm}}\) by \({\text{5}}\,{\text{cm}}\) piece of paper?

Solution:

Area of the sheet \( = 100 \times 75\,{\text{c}}{{\text{m}}^2} = 7500\,{\text{c}}{{\text{m}}^2}\)
Area of envelope \( = 20 \times 5\,{\text{cm}} = 100\,{\text{c}}{{\text{m}}^2}\)
Number of envelopes that can be made \({\text{=}}\frac{{{\text{ Area of the sheet }}}}{{{\text{ area of the envelope }}}}\)
\( = \frac{{7500}}{{100}}\)
\( = 75\) envelopes

8) The length and width of a rectangular wall are \({\text{75}}\,{\text{m}}\) and \({\text{32}}\,{\text{m}}\) respectively. Find the cost of painting the wall if the rate of painting is \({\text{Rs}}{\text{.3}}\,{\text{per}}\,{\text{sq}}{\text{.}}\,{\text{m}}\)

Solution:
Given: Length of the wall \({\text{=75}}\,{\text{m}}\)
The breadth of the wall \({\text{=32}}\,{\text{m}}\)
Area of the wall \(= {\text{length}} \times {\text{breadth}}\)
\(= 75{\text{~m}} \times 32{\text{~m}}\)
\({\text{=2400}}\,{\text{sq}}{\text{.m}}\)
For \({\text{=1}}\,{\text{sq}}{\text{.m}}\) of painting costs \({\text{Rs}}.\,3\)
Thus, for \({\text{2400}}\,{\text{sq}}{\text{.m}}\) the cost of painting the wall will be \( = 3 \times 2400\)
\( = Rs.7200\)

9) The length and breadth of a rectangular wall are\( {\text{70}}\,{\text{m}}\) and \({\text{30}}\ {\text{m}}\) respectively. Find the cost of painting the wall if the rate of painting is \({\text{Rs 3 per sq}}{\text{. m}}\)

Answer:
Length of the wall \({\text{=70}}\,{\text{m}}\)
The breadth of the wall \({\text{=30}}\,{\text{m}}\)
Area of the wall \( = {\text{length}} \times {\text{breadth}} = 70~{\text{m}} \times 30~{\text{m}} = 2100\,{\text{sq}}.{\text{m}}\)
For \({\text{1}}\,{\text{sq}}{\text{.}}\,{\text{m}}\) of painting costs \({\text{Rs}}{\text{. 3}}\)
Thus, for \({\text{2100}}\,{\text{sq}}{\text{.}}\,{\text{m}}\) the cost of painting the wall will be \( = 3 \times 2100 = {\text{Rs}}\,6300\).

10) A floor whose length and width is \({\text{50}}\,{\text{m}}\) and \({\text{40}}\,{\text{m}}\) respectively needs to be covered by rectangular tiles. The dimension of each tile is \({\text{1}}\,{\text{m}} \times 2\,{\text{m}}\). Find the total number of tiles that would be required to fully cover the floor.

Answer:
Length of the floor \(= 50\,{\text{m}}\)
Breadth of the floor \(= 40\,{\text{m}}\)
Area of the floor \(={\text{length}} \times {\text{breadth}} = 50\,{\text{m}} \times 40\,{\text{m}} = 2000\,{\text{sq}}{\text{.}}\,{\text{m}}\)
Length of one tile \(= 2\,{\text{m}}\)
Breadth of one tile \(= 1\,{\text{m}}\)
Area of one tile \(={\text{length}} \times {\text{breadth}} = 2\,{\text{m}} \times 1\,{\text{m}} = 2\,{\text{sq}}{\text{.}}\,{\text{m}}\)
No. of tiles required \( = \) area of floor/area of a tile \( = \frac{{2000}}{2}\;= \;1000\) tiles

Frequently Asked Questions

Q.1. What is the area of rectangle and square?
Ans: The area of a rectangle and square is different. For square, the area is calculated using the formula: \({\left({ {\text{side}}}\right)^2}\), whereas for rectangle, it is length times breadth.

Q.2. What is the formula for the area of rectangle?
Ans: The formula for the area of rectangle is:
\({\text{Area}} = {\text{Length}} \times {\text{Breadth}}\)

Q.3. What is the perimeter of rectangle?
Ans: The perimeter of the rectangle is the sum of all its four sides.,
Hence, \({\text{Perimeter}}\left({{\text{rectangle}}} \right) = 2\left({{\text{Length}} + {\text{Breadth}}} \right)\)

Q.4. What is area of rectangle?
Ans: The area of the rectangle is the region occupied by the sides of the rectangle.

Q.5. What is the unit of area of rectangle?
Ans: The unit of area of any shape is a square unit. Hence, the unit of area of rectangle is \({\text{c}}{{\text{m}}^{\text{2}}}{\text{,}}\,{{\text{m}}^{\text{2}}}{\text{,}}\,{\text{k}}{{\text{m}}^{\text{2}}}\) etc.

Q.6. How to find a length when the area and width of the rectangle is known?
Ans: When the area of rectangle and width of the rectangle is known, then length can be calculated by using:
\({\text{Length=}}\frac{{{\text{Area}}}}{{{\text{Width}}}}\)

Q.7. How to find width when the area and length are known?
Answer: When the area of the rectangle and length of the rectangle is known, then width can be calculated by using:
\({\text{Width=}}\frac{{{\text{Area}}}}{{{\text{Length}}}}\)

Q.8. Why do we calculate the area of rectangle?
Ans: We calculate the area of a rectangle to find the area occupied by the rectangle within its perimeter.

Q.9. Is the area of rectangle the same as the area of square?
Ans: No, the area of the square is not necessarily the same as the area of the rectangle because every square is a rectangle with length and breadth equal, but all rectangles are not square. The formula to calculate the area of a rectangle is length×breadth and that of the square is \({\text{sid}}{{\text{e}}^2}\).

Q.10. What is the area of rectangle \(ABCD\)?

Ans: The area of the rectangle \(ABCD\) is given by the product of length and breadth.
Area of rectangle \(ABCD = AB \times BC\)

Check Properties of Rectangle Here

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