Mathematics
Introduction
In our day-to-day life, we all come across various objects of contrasting shapes and sizes, such as a circle, triangle, rectangle, circle, oval, and many others. You can find the area and perimeter of any shape or size, no matter whether they are regular or irregular. Area and perimeter accord us beneficial details which aid us to comprehend these different forms of objects. In mathematics, area and perimeter are considered crucial properties of two-dimensional figures. The perimeter for rectangles is always less than their area.
Let us first understand what a rectangle is? A rectangle is a two-dimensional plane figure, having four sides. It is a quadrilateral that has all four right angles. It is a parallelogram having equal angles.
There are some universal properties of a rectangle such as:
- The opposite sides of a rectangle are parallel and equal to each other.
- It has a 90 degree interior angle each.
- The sum total of all the interior angles is 360 degree.
- The diagonals bisect each other.
- The length of both the diagonals is the same.
- The perimeter of a rectangle, with side lengths a and b, will be 2a + 2b units.
- The area of a rectangle, with side lengths a and b, will be ab sin 90 = ab square units.
- A rectangle’s diagonal is the diameter of its circumcircle.
- The length of each diagonal, in case a and b are the sides of a rectangle, is d = √a2 + b2, and the diagonals bisect each other at different angles. Therefore, one is acute, and another is an obtuse angle.
- If you rotate a rectangle along the line joining the midpoint of the shorter parallel sides, then you can obtain a cylinder. The cylinder’s height will be equal to the length of a rectangle, and the diameter will be equivalent to the width of a rectangle.
- If you rotate a rectangle along the line joining the midpoint of the longer parallel sides, then you can obtain a cylinder. The cylinder’s height will be equal to the width of a rectangle, and the diameter will be equivalent to the length of a rectangle.
What are Area and Perimeter?
During our maths classes, we all have calculated the area and perimeter of a rectangle, square, triangle, and circle. Before getting to the toughest part first, try to understand the meaning of perimeter and area in a mathematical sense. Before solving and finding the area and perimeter of a rectangle, it is paramount to have a conceptual clarity of area and perimeter.
Area: To determine the boundaries of a two-dimensional object, the area plays a significant role. The enclosed space within the frame of an object is called the area. The dimensions and properties determine the area of all the shapes. Distinct shapes have distinct areas. The kite’s area will be different from the area of a square.
In case two objects have identical shapes, then it is not requisite that the area covered by them will be equal, until the dimension of both the shapes is also the same.
For example
If there are two rectangle boxes with length L3 and L4, and breath B3 and B4, the areas of the rectangles say, C1 and C2, will be equal only if L3 = L4 and B3 = B4.
Perimeter: Based on the dimension of the shapes, the perimeter of various shapes can be equal to each other. On the basis of their dimensions, the perimeter of various shapes can match in length with each other.
For example
From a metal wire having a length L, a circle can be made, and with the same wire, a square can be made too, whose sides are equal in length.
Perimeter and Area Formula for Rectangle
The perimeter of a Rectangle, P = 2(a+b) (Length + Width)
Area of Rectangle = a x b (Length x Width)
Here, a and b are the length and width of a rectangle.
How can we find the Area of a Rectangle?
The area of a rectangle depends on its sides. The formula for the area of a rectangle is equal to the product of the length and breadth of the rectangle. There are various ways of measuring the size of the objects. In order to calculate the area of a rectangle, it is necessary that length and breadth should be known. To find the area of a rectangle, following steps need to be followed:-
Step 1. First, note the data given in the question.
Step 2. Then, when length and width values are multiplied, the square-units dimension is obtained.
Step 3. Always write the answer in square units.
The above-mentioned steps need to be followed to calculate the area of a rectangle.
What will be the area of a rectangle whose length is 10cm and the width is 5cm?
Answer:
Length = 10 cm
Width = 5 cm
Area of a rectangle = Length x Width
10 x 5 = 50
So, the area of rectangle = 50 cm2
Why is the area of rectangle LB?
The area of the rectangle is LB, as we calculate the square units of the rectangle. When total squares in the length are multiplied with total squares in the width of the rectangle, it gives the total squares in the rectangle.
Why is the area of the rectangle is length x breadth?
The rectangle is divided into two equal right-angled triangles by its diagonals. So the area of the rectangle will be equal to the sum of these two equal right-angled triangles.
Suppose, the diagonal AB divides the rectangle into two equal right-angled triangles, i.e., ∆ACD and ∆ ABD; Area of rectangle ABCD = Area of ∆ACD + Area of ABD
Since ACD and ABD both are right-angled equal triangle; therefore,
Area of ∆ACD = Area Of ABD = ½ x base x height = ½ x b x h
Area (ABCD) = 2(½ x b x h)
Area (ABCD) = b x h
The breadth of the rectangle is equal to the base of triangles, and the height is the breath of the rectangle; therefore,
Area (ABCD) = L X B (Length x Breadth)
Some Special formula related to the rectangle area
- How to find the area with the diagonal of rectangle and width?
Formula : Area = Width* √(Diagonal2 – Width2)
- How to find a missing side length when the area is known?
You have to apply this formula: Length = Area/Width
How can we find the Perimeter of a Rectangle?
The perimeter of the rectangle is a sum of all the sides of a rectangle. Since the opposite sides of a rectangle are equal, the perimeter will be twice the width and length of the rectangle, which is indicated by “P.”
So, to find the perimeter of a rectangle, you have to add all the sides of it. As you know, there are four sides to a rectangle and adding them will give its perimeter.
Here is an example to show the formula’s application:-
What will be the perimeter of a rectangle whose length is 10cm and width is 15cm?
Answer:
Length = 10 cm
Width = 15 cm
As we know,
The perimeter of a rectangle = 2(Length + Width)
Now substitute the value of the length and width
Perimeter – P = 2 (10 + 15)cm
P = 2 x 25 cm
So the perimeter of a rectangle = 50 cm
So here first, we added length with the width and then multiplied it with 2, as per the formula.
When the perimeter and one side is given
Suppose,
The perimeter of a rectangle is 15 cm, and the length of one side is 4cm. What will be the other side?
Let the missing side be b
As we know the perimeter of a rectangle = 2(a+b)
15 = 2 x (4 +b)
b = 7.5 – 4 = 3.5cm
Derivation of Formula for Perimeter of a Rectangle
As we know, the perimeter is equal to the sum of all the sides of the polygon. Therefore, for a rectangle, the perimeter (P) is;
P = sum of all four sides of a rectangle
P = a + b + a + b
P = 2(a + b)
Hence, we derived
Perimeter of a rectangle = 2(Length + Width)
To find out the area and perimeter of a rectangle is a part of basic geometry and requires practice. It has various real life uses such as in architecture and mapping. Understanding the formula of area and perimeter of a rectangle thoroughly is very important to solve the mathematical problems and to excel in your exams.
Now that you have a fair idea of the key points to focus on, you will be more confident in calculating the area and perimeter of a rectangle. Comprehend this concept with the expert teachers at Cuemath. Know more about them with a free trial class. They have a personalized learning plan for every student, and focus on building a strong foundation on the concepts in such a way that students feel at ease with math and stop fearing it. Enjoy learning!
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