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Rectangle is a closed two-dimensional flat shape with four sides, four angles, and four vertices. In a rectangle, the opposite sides are equal and parallel to one another. The four interior angles of a rectangle are all right angles. A rectangle is called an equiangular quadrilateral since all of its angles are equal. A rectangle is also referred to as a right-angled parallelogram because its sides are parallel. A rectangle has two diagonals that intersect each other. The length of both diagonals of a rectangle is the same.
Rectangle Formula is the set of formulas for the area, perimeter, and diagonal of a rectangle. A rectangle has four sides, hence the perimeter of rectangle is equal to the sum of its four sides. The perimeter of a rectangle is 2 (Length + Breadth). The area of rectangle is the product of the rectangle's length and breadth is the formula for calculating its area. The area of rectangle is Length x Breadth. The length of the diagonals of a rectangle is calculated using the formula √(l2 + b2) units.
Read More: NCERT Solutions for Class 7 Maths Perimeter & Area
| Table of Content |
Key Terms: Rectangle, Area of Rectangle, Perimeter of Rectangle, Diagonals, Length, Width, Quadrilateral, Parallelogram, Area
Rectangle
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Rectangle is a member of the parallelogram family, which is a category of quadrilaterals. A rectangle's feature is that all of its internal angles are 90 degrees. The adjacent sides of a rectangle do not necessarily need to be equal, but its opposite sides are equal.
Rectangle
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| Related Topics | ||
|---|---|---|
| Quadrilateral Angle Sum Property | Area of Parallelogram | Perimeter of a Parallelogram |
| Area of Parallelogram and Triangles | Difference between Square and Rectangle | Geometry Formula |
Area of a Rectangle
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The area of rectangle is the space that it occupies between its four sides or boundaries. It is the entire amount of space that a shape typically occupies when it is represented in two dimensions.
The area of rectangle formula is given as follows:
Area of Rectangle = Length (L) × Width (W)
Solved Examples
Example 1: Find the area of a rectangle whose length is 16 cm and width is 8 cm.
Solution: Given,
- Length = 16 cm
- Width = 8 cm
Area of a Rectangle = Length × Width
Area= 16 × 8 = 128
So, the area of rectangle = 128 cm2
Example 2: The length of a rectangular board is 15 cm. Its area is 120 sq. cm. Find its width.
Solution: Given that,
- Area of the Board = 120 sq. cm.
- Length of the Board = 15 cm
Area of a rectangle = length x width
So, Width = Area/Length
Thus, the width of the board = 120/15 = 8 cm
Read More: Area Formula
Derivation of Area of Rectangle
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PQRS is a rectangle in unit squares, each measuring one square inch. The total number of unit squares in a rectangle PQRS determines its area.
Derivation of Area of Rectangle
We know that a rectangle's area is always the product of its two sides. We will derive the area of the rectangle by splitting the figure into two right triangles of equal size. For instance, a diagonal is drawn from the vertex P to R in the given rectangle PQRS.
The rectangle is split into two identical right-angle triangles by the diagonal PR.
Therefore, the area of PQRS will be
⇒ Area (PQRS) = Area (PQR) + Area (PSR)
⇒ Area (PQRS) = 2 × Area (PQR) [Area (PQR) = ½ × base × height]
⇒ Area (PQRS) = 2 × (½ × b × h)
⇒ Area (PQRS) = b × h
Perimeter of a Rectangle
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Perimeter of rectangle is the total distance of a rectangle's outer boundary. A rectangle has vertices that are each 90 degrees. Many people also frequently refer to a rectangle as an equiangular quadrilateral. The perimeter of a rectangle is calculated by adding the four sides.
The perimeter of rectangle formula is given as
P = 2 (Length + Width)
Since a rectangle's perimeter is a linear measurement, the unit will be in meters, centimeters, inches, feet, etc.
Solved Examples
Example 1: Find the perimeter of a rectangle whose length and width are 15 cm and 8 cm, respectively.
Solution: Given:
- Length = 15 cm
- Width = 8 cm
Perimeter of a rectangle = 2 (length + width)
Perimeter, P = 2(15 + 8) cm
P = 2 x 23 cm
Therefore, the perimeter of a rectangle = 46 cm
Example 2: A rectangular yard has a length equal to 20 cm and a perimeter equal to 120 cm. Find its width.
Solution: Given,
- Perimeter of the yard = 120 cm
- Length of the yard = 20 cm
Let W be the width of the rectangular yard.
Perimeter, P = 2(length + width)
Substituting the values, we get;
120 = 2(20 + width)
20 + W = 60
W = 60 – 20 = 40
Hence, the width of the yard is 40 cm.
Read More: Difference Between Area and Perimeter
Derivation of Perimeter of a Rectangle
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The definition of the perimeter is "boundary." There are four sides in the diagram drawn above. We can get the rectangle's perimeter by adding those four sides together.
Sum of each side = L+ L+ B + B
So, 2L+ 2B
Therefore, the perimeter of the rectangle = 2(l + b)
Diagonal of Rectangle Formula
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The diagonals of a rectangle are equal and the length of diagonals of a rectangle can be calculated using the formula
d = √( l2 + w2)
Where
- l denotes the length of the rectangle.
- w denotes the width of the rectangle.
Diagonals of Rectangle
Solved Example
Example: Calculate the length of the diagonal of a rectangle whose dimensions are 3 units and 4 units.
Solution: It is given that the dimensions of the rectangle are
- Length = 3 units
- Width = 4 units
Diagonal of a rectangle = (d) = √( l2 + w2)
Substituting the values,
d = √(3)2+(4)2 = √25 = 5 units
Thus, the length of the diagonal of the given rectangle is 5 units.
Read More: Area Perimeter Formula
Important Rectangle Formulas
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The formula for the rectangle's area and perimeter is provided below. Given that "l" stands for the rectangle's length and "b" for its width, then,
| Parameter | Element | Formula |
|---|---|---|
| Perimeter of Rectangle | Perimeter of Rectangle Formula | 2 (l + b) units |
| Length of Rectangle | P/2 - b units | |
| Breadth of Rectangle | P/2 - l units | |
| Area of Rectangle | Area of Rectangle Formula | l * b sq. units |
| Length of Rectangle | A/b units | |
| Breadth of Rectangle | A/l units | |
| Diagonal of Rectangle | √(l2 + b2) units | |
Applications of Rectangle Formula
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The applications of rectangle formula are as follows:
- The usage of geometric objects with lengths, widths, angles, and areas for construction and astronomy is depicted in early documentation of Babylonian culture.
- The ability to carve stone into regular shapes like squares, triangles, and rectangles, as well as a knowledge of perimeter and area concepts, enabled the ancient Egyptians to build enormous constructions like pyramids.
- These rectangle formula concepts are extremely useful in current object modeling, land surveying, and other fields.
Read more:
| Chapter-Related Topics | ||
|---|---|---|
| Types of Polygon | Area of Polygons | Lines of Symmetry in a Rectangle |
| Special Parallelograms | Area Perimeter Formula | Quadrilateral Formulas |
Examples of Rectangle
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Some examples of rectangular figures are
- Agricultural fields and parks.
- Painting Canvas.
- Tiles and walkways with rectangular tiles.
- Daily life objects such as serving trays, tables, glass, etc.
Things to Remember
- A rectangle is a four-sided polygon with internal angles that are equal to 90 degrees.
- A rectangle is divided into four triangles by its two diagonals.
- All parallelograms are not rectangles, but all rectangles are parallelograms.
- The area of rectangle is given through the formula: Length x Breadth
- The perimeter of rectangle is calculated using the formula: 2 (Length + Breadth).
- The formula for the length of the diagonals of a rectangle is d = √(l2 + b2).
Sample Questions
Ques. Calculate the area and the perimeter of the rectangle box that measures 15 cm in length and 8 cm in breadth. (3 Marks)
Ans. Given,
- Length = 15 cm
- Breadth = 8 cm
Now applying the area of rectangle formula i.e. = length × breadth
= (15 × 8) cm2
= 120 cm2
Using the formula for finding the perimeter of rectangle = 2 (length + breadth)
= 2 (15 + 8) cm
= 2 × 23 cm
= 46 cm
Ques. Evaluate the breadth of the serving tray which is rectangular in shape whose area is 320 m2 and whose length is 40 m. Find its perimeter. (3 Marks)
Ans. We know that the breadth of the rectangular serving tray = Area/Length
= 320m/40m
= 8 m
Thus, the perimeter of the rectangular serving tray = 2 (length + breadth)
= 2(40 + 8) m
= 2 × 48 m
= 96 m
Ques. A rectangular yard has a length equal to 12 cm and a perimeter equal to 60 cm. Find its width. (3 Marks)
Ans. It is given that,
- Perimeter = 60 cm
- Length = 10 cm
Let W be the width.
From the formula,
Perimeter, P = 2(length + width)
Substituting the values,
60 = 2(12 + width)
12 + W = 30
W = 30 – 12 = 18
Hence, the width is 20cm.
Ques. Find the perimeter of a rectangle whose length and width are 12 cm and 8 cm, respectively. (3 Marks)
Ans. Given,
- Length = 12cm
- Width = 8 cm
Perimeter of Rectangle = 2(Length + Width)
= 2(12 + 8) cm
= 2 × 20 cm
Therefore, the perimeter of a rectangle = 40 cm.
Ques. Find the area of a rectangle whose length is 11.5 cm and breadth is 6.5 cm. (3 Marks)
Ans. According to the question,
- Length of the rectangle (l) = 11.5 cm
- Breadth of the rectangle (b) = 6.5 cm
Area of a rectangle = length × breadth
Area of the rectangle = 11.5 × 6.5
= 74.75 cm2.
Ques. George needs an 8-inch-long by a 4-inch-wide wooden rectangle with a frame. How much area is necessary for George's frame? (3 Marks)
Ans. The frame George wants has the following dimensions:
- The rectangular frame's length is 8 inches.
- The rectangular frame's width is 4 inches.
Now we are aware of the formula needed to calculate the area of a rectangle, we can use = Area = (Length * Width) square units.
Thus, Area = (8 inches * 4 inches) square units = 32 square inches.
Hence, the required area for the wooden frame is 32 square inches.
Ques. The respective length and breadth of a rectangle measure 12 cm and 8 cm. Find out the area and perimeter of the respective rectangle. What is the length of the diagonal? (5 Marks)
Ans. As we know that the area of a rectangle can be calculated by the following formula,
A = Length * Breadth
⇒A = 12 × 8
⇒ Thus, A = 96 cm2
Perimeter of a rectangle is usually given by,
P = 2 (Length + Breadth)
⇒ P = 2 (12 + 8)
⇒ P = 40
Now, the Diagonal Length can be calculated by, D= √L2+W2
⇒ D = √122 + 82
⇒ D = √144 + 64
⇒ D = √208
⇒ D = 4√13
Thus, the length of the diagonal is 4√13.
Ques. Ana desires a garden fence. The fence has a perimeter of 30 meters. The fence is 5 meters in width. Identify the rectangular fence's length. (3 Marks)
Ans. To find the perimeter, we already know what the perimeter of a rectangle is
P = 2 (Length + Width)
Now, the perimeter is = 30 m, while the width is = 5 m.
Therefore,
P = 2 (L+B)
30 = 2 (L + 5)
15 = L + 5
L = 10 meters
Hence, the length of the rectangular fence is 10 meters.
Ques. Rama wants her carpenter to build a cardboard shelf with measurements of 9 cm and 6 cm respectively. Help Rama find out the total area of the cardboard shelf. (3 Marks)
Ans. Rama needs a cardboard shelf with the dimensions listed below built by her carpenter.
- The cardboard shelf measures 9 cm in length.
- The cardboard shelf's width is 6 cm.
Rama now wants to know how big the cardboard shelf is overall. Since we now know the formula needed to calculate a rectangle's area
= (Length * Breadth) Square units.
= (9 cm * 6 cm) = 54 cm2
Thus, the area of the cardboard shelf is 54 cm2.
Ques. In order to keep cows from grazing on her pasture, Alice needs to fence the space around her new 200 m by 150 m farmhouse. Find out how much fencing wire is needed to cover the area effectively. (3 Marks)
Ans. The dimensions Alice has for her new farmhouse are as follows:
- Length= 200 m
- Breadth= 150 m
Alice needs to calculate the perimeter in order to determine the total amount of wire needed to fence the farmhouse area.
Now by using the formula for perimeter, 2(l+b)
Length of the fence required that is perimeter = 2 (200+150)
= 700 m
Thus, the required answer is 700 m.
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