Arpita Srivastava Content Writer
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A rectangle is a form of quadrilateral that resembles a four-sided polygon. It has internal angles measuring 90 degrees each. The internal angles are equal to one another.
- Rectangles are usually characterized by their dimensions, mainly their length and width.
- The longer side is known as length.
- The shorter side of a rectangle is equivalent to the width.
- The opposite sides of the rectangle are equal to each other.
- Sides run parallel to each other.
- A rectangle is defined by five degrees of freedom.
- One degree of freedom for shape and size and three for position.
- The diagonals bisect each other at the point of intersection.
- Two rectangles are said to be incomparable if they do not fit inside each other.
- The four corners are also known as vertices.
- Some real life examples include door, blackboard and almirah.
Read More: Geometry Formula
Key Terms: Rectangle, Length, Width, Area, Perimeter, Diagonals, Quadrilateral, Polygon, Square, Golden Rectangle, Right Angle
What is a Rectangle?
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A rectangle is a two-dimensional figure with internal angles measuring 90 degrees each. All its parallel sides are equal to each other, and hence, they are called equiangular quadrilateral.
- In a rectangle, opposite sides are equal; hence, it is also called a parallelogram.
- The sides of a rectangle are measured by its length (l) and width (w).
- The lengths of the sides are both parallel and equal to one another.
- The word rectangle is a combination of two Latin words named rectus and angulus.
- In this, the adjacent sides, especially the corners, meet to form a right angle.
- If all the sides of a rectangle were equal to one another, then it would become a square.
Rectangle
Read More: Pythagoras Theorem
Types of Rectangle
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Rectangle is divided into two categories which are as follows:
Square
A square is a two-dimensional figure which consists of four equal sides and four equal angles. The internal angle of the square is equal to 90 degrees. All diagonals are equal in length. A square with vertices ABCD is denoted by ABCD.
ABCD.Golden Rectangle
Golden Rectangle is a type of perfect rectangle where the ratio of length and width is equivalent to the golden ratio. The golden ratio is indicated by the ratio 1: (1+⎷5)/2. It can be constructed with the help of a straightedge and compass.
Rectangle
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Properties of a Rectangle
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The properties of a rectangle are as follows:
- A rectangle has vertices that form an angle of 90 degrees each.
- The diagonals are bound to intersect one another by meeting at the centre point.
- These diagonals connect to two equidistant diagonal points.
- The sum total of the interior angles can be denoted by 360 degrees.
- This is due to the measurement of each vertex commonly being 90 degrees.
- The length of either diagonal is calculated by using the Pythagorean theorem.
- Pythagoras theorem is indicated by the equation: c2 = a2 + b2.
- Mathematically, every form of rectangle comes under the category of parallelograms.
- All parallelograms are not rectangles.
Read More: Three Dimensional Geometry
Area of a Rectangle
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Area is the measurement of the extension of a typical two-dimensional shape or region. It is the total amount of space that a shape usually takes in its two-dimensional form.
- The area is commonly estimated in square units.
- We often encounter different examples of rectangles every day, much like laptops and blackboards.
- Thus, the area of a rectangle can be determined by the following formula,
Area of a Rectangle = Length (L) × Width (W) sq. units.
Solved Example of a Area of RectangleGiven below are the example of a area of rectangle are as follows: Example 1: Alisha has a rectangular photo frame that is 7 inches long and 9 inches wide. Can you help Alisha find its area? Ans: We know the formula to calculate the area of a rectangle. Area of a Rectangle = (Length × Width). Thus, the area of the rectangular frame = 7 × 9 = 63 square inches Therefore, the area of the photo frame = 63 square inches |
Rectangle
Read More: Quadrilateral Angle Sum Property
Perimeter of a Rectangle
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The perimeter can be defined as the total distance around a shape or form. It helps measure the typical boundary that surrounds a shape. A rectangle, which is a kind of quadrilateral, has vertices equal to 90 degrees.
- A rectangle is also often termed as an equiangular quadrilateral.
- In a nutshell, the perimeter of a rectangle is the total distance that is covered by it.
- The formula for perimeter of a rectangle is:
P = 2 (Length + Width); where P equals to Perimeter.
Solved Example of a Perimeter of RectangleGiven below are the example of a perimeter of rectangle are as follows: Example 1: Find the perimeter of the rectangular field whose sides are 8 m and 10 m. Ans. Given,
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Read More: Mensuration
Diagonals of a Rectangle
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A diagonal is a line segment that helps in connecting either of the two non-adjacent vertices of a rectangle. If you look closely at the figure, AC and BD are known to be the diagonals that can actively connect with one another, forming the shape of a “cross”.
- A diagonal cuts a rectangle into two equally divided triangles.
- However, several things mostly vary in rectangles.
- The length, width and the number of squares the diagonal is going to cut through.
- The diagonals usually cut through each other, meeting at the centre point.
Diagonal of a Rectangle
Read More: Perimeter of a parallelogram
Things to Remember
- A rectangle is a quadrilateral with internal angles equal to 90 degrees each.
- The total sum of the angles of a rectangle is 360 degrees.
- Area is the total amount of space a shape usually occupies in its two-dimensional form.
- The formula for the area of a rectangle is equivalent to Length (L) × Width (W) sq. units.
- Perimeter is often referred to as the total distance around a shape or form.
- P = 2 (Length + Width), where P equals to Perimeter.
- A diagonal is known to be a line segment that connects either of the two non-adjacent vertices.
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Sample Questions
Ques. George wants a rectangular wooden frame that measures 7 inches long and 4 inches wide. What is the required area George needs for his frame? (3 marks)
The length of the rectangular frame is = 7 inches
The width of the rectangular frame is = 4 inches
Now, since we know the necessary formula required for measuring the area of a rectangle, we can say = Area = (Length * Width) square units.
Thus, Area = (7 inches * 4 inches) square units = 28 square inches.
Hence, the required area for the wooden frame is 28 square inches.
Ques. The respective length and breadth of a rectangle measures 12 cm and 8 cm. Figure out the area and perimeter of the respective rectangle. What is the length of the diagonal? (4 marks)
Ans. As we know that the area of a rectangle can be evaluated by the following formula, A = Length * Width
⇒A = 12 × 8
⇒ Thus, A = 96 cm2
As we already know, the perimeter is usually given by, P = 2 (Length + Width)
⇒ P = 2 (12 + 8)
⇒ P = 40
Now, the Diagonal Length can be determined by, D=
⇒ D = √122 + 82
⇒ D = √144 + 64
⇒ D = √208
⇒ D = 4√13 (Thus, the answer is 4√13)
Ques. The data shows measurements of a rectangular pool. The perimeter of it is 56 meters. The length of the pool is already mentioned, and is 16 meters respectively. Determine the width of the rectangular pool? (4 marks)
Now, as per the equation, the perimeter of the rectangular pool has been mentioned as 56 meters, while the length of it has been mentioned as 16 meters.
By substituting the units with real values, we can get,
56 = 32 + 2w (simplifying the following values)
Hence, we have to subtract 32 from each side.
24 = 2w
Now, we have to divide each side by 2, which gives,
12 = w
Thus, 12 meters is the width of the rectangular pool.
Ques. The length and width of a typical rectangle measures 3 cm and 4 cm respectively. Determine the area, perimeter, and the length of the diagonals that belong to the respective rectangle? (4 marks)
The length measures = 3 cm
The width measures = 4 cm
Now, the Area of a rectangle can be determined by = L × W
Hence,
Area can be determined by ( A ) = 3 × 4
Thus, the area is ( A ) = 12 cm2
The perimeter ( P ) = 2 ( L + W )
Therefore,
Perimeter of rectangle, as given in the equation, ( P ) = 2 ( 3 + 4)
The perimeter = 2 ( 7 )
Thus, the perimeter = 14 cm.
\(D = \sqrt{L^2 + W^2} \\ D = \sqrt{3^2 + 4^2} \\ D =\sqrt{9 + 16} \\ D = \sqrt {25} \\ D = 5 cm\)
The length of the Diagonal,
Ques. Ana wants to fence her garden. The perimeter of the fence is 30 meters. The breadth of the fence is 10 meters. Find out the length of the rectangular fence? (3 marks)
Clearly, we already know what the perimeter of a typical rectangle is. Which is to say, that the perimeter of a rectangle is = 2 (Length + Width).
Now, the perimeter is = 30 feet, while the width is = 10 feet.
Therefore,
P = 2 (L+B)
30 = 2 (L + 10)
15 = L + 10
L = 5 meters
Hence, the length of the rectangular fence is 5 meters.
Ques. Gina wants her carpenter to build a cardboard shelf with measurements of 9 cm and 6 cm respectively. Help Gina find out the total area of the cardboard shelf? (3 marks)
Ans. Gina wants her carpenter to build a cardboard shelf with the following measurements,
The length of the cardboard shelf is = 9 cm
The breadth of the cardboard shelf is = 6 cm
Now, Gina wants to find out the total area of the cardboard shelf. Since we already know the formula required to determine the area of a rectangle = (Length * Breadth) Square units.
Thus, the required equation can be = (9 cm * 6 cm) = 54 cm2
Hence, the area of the cardboard shelf is 54 cm2.
Ques. Alice has a new 200 by 150 m farmhouse, to prevent cows from grazing on her pasture she needs to fence the area. Find out the length of fencing wire required to optimally cover the area? (3 marks)
Ans. Alice has the following measurements for her new farmhouse,
The Length of the farmhouse = 200 m
The Breadth of the farmhouse = 150 m
To find out the total length of wire required to fence the farmhouse area, Alice needs to calculate the perimeter. Now by using the formula for perimeter, 2(l+b)
Length of the fence required (Perimeter) = 2 (200+150)
= 700 m
Thus, the required answer is 700 m.
Ques. The respective length and breadth of a rectangle measures 19 cm and 9 cm. Figure out the area and perimeter of the respective rectangle. What is the length of the diagonal? (4 marks)
⇒A = 19 × 9
⇒ Thus, A = 171 cm2
As we already know, the perimeter is usually given by, P = 2 (Length + Width)
⇒ P = 2 (19 + 9)
⇒ P = 56
Now, the Diagonal Length can be determined by, D= \(\sqrt{L^2 + W^2}\)
⇒ D = √192 + 92
⇒ D = √361 + 81
⇒ D = √442
Ques. Find the perimeter of the rectangular field whose sides are 8 m and 8 m? (2 marks)
Ans. Given,
- Length of a rectangular field = 8 m
- Breadth of a rectangular field = 8 m
- Perimeter of a rectangular field = 2 x (l + b)
- 2 x ( 8 + 8 )
- 2 x 16
- 32
- Perimeter of a rectangular field is 32 m
Ques. Amit wants her carpenter to build a cardboard shelf with measurements of 19 cm and 16 cm respectively. Help Amit find out the total area of the cardboard shelf? (3 marks)
Ans. Amit wants her carpenter to build a cardboard shelf with the following measurements,
The length of the cardboard shelf is = 19 cm
The breadth of the cardboard shelf is = 16 cm
Now, Amit wants to find out the total area of the cardboard shelf. Since we already know the formula required to determine the area of a rectangle = (Length * Breadth) Square units.
Thus, the required equation can be = (19 cm * 16 cm) = 304 cm2
Hence, the area of the cardboard shelf is 304 cm2.
Ques. The data shows measurements of a rectangular pool. The perimeter of it is 36 meters. The length of the pool is already mentioned, and is 10 meters respectively. Determine the width of the rectangular pool? (4 marks)
Now, as per the equation, the perimeter of the rectangular pool has been mentioned as 36 meters, while the length of it has been mentioned as 10 meters.
By substituting the units with real values, we can get,
36 = 20 + 2w (simplifying the following values)
Hence, we have to subtract 20 from each side.
16 = 2w
Now, we have to divide each side by 2, which gives,
8 = w Thus, 8 meters is the width of the rectangular pool.
Ques. Era wants to fence her garden. The perimeter of the fence is 40 meters. The breadth of the fence is 10 meters. Find out the length of the rectangular fence? (3 marks)
Clearly, we already know what the perimeter of a typical rectangle is. Which is to say, that the perimeter of a rectangle is = 2 (Length + Width).
Now, the perimeter is = 40 feet, while the width is = 50 feet.
Therefore,
P = 2 (L+B)
40 = 2 (L + 10)
20 = L + 10
L = 10 meters
Hence, the length of the rectangular fence is 10 meters.
Ques. Tina has a new 300 by 100 m farmhouse, to prevent cows from grazing on her pasture she needs to fence the area. Find out the length of fencing wire required to optimally cover the area? (3 marks)
Ans. Tina has the following measurements for her new farmhouse,
The Length of the farmhouse = 300 m
The Breadth of the farmhouse = 100 m
To find out the total length of wire required to fence the farmhouse area, Alice needs to calculate the perimeter. Now by using the formula for perimeter, 2(l+b)
Length of the fence required (Perimeter) = 2 (300+100)
= 800 m
Thus, the required answer is 800 m.
Ques. Find the perimeter of the rectangular field whose sides are 20 m and 20 m? (2 marks)
Ans. Given,
- Length of a rectangular field = 20 m
- Breadth of a rectangular field = 20 m
- Perimeter of a rectangular field = 2 x (l + b)
- 2 x ( 20 + 20 )
- 2 x 40
- 80
- Perimeter of a rectangular field is 80 m
Ques. Phil has a new 100 by 110 m farmhouse, to prevent cows from grazing on her pasture she needs to fence the area. Find out the length of fencing wire required to optimally cover the area? (3 marks)
Ans. Phil has the following measurements for her new farmhouse,
The Length of the farmhouse = 100 m
The Breadth of the farmhouse = 110 m
To find out the total length of wire required to fence the farmhouse area, Alice needs to calculate the perimeter. Now by using the formula for perimeter, 2(l+b)
Length of the fence required (Perimeter) = 2 (100+110)
= 420 m
Thus, the required answer is 420 m.
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