About Area and Perimeter in Math
What is the perimeter?
The perimeter of any geometric figure is the total length of the figure’s sides or edges. If we want to calculate the perimeter, the geometrical figures should be closed. The perimeter of a polygon is equal to the total length of all of its sides or edges.
Perimeter is also referred to as the length of the boundary of a shape.
What are rectangles and squares?
A rectangle is a quadrilateral whose opposite sides are parallel and equal in length. It is a quadrilateral with all four angles being 90 degrees, or right angles. The diagonals are equal in length and bisect each other. A rectangle with side lengths of l and w has a perimeter of 2 (l + w) units.
A square is defined as a rectangle with four equal sides. A square is a regular quadrilateral, meaning it has four equal sides and angles. The interior angles are all 90 degrees. The diagonals of the square are equal and bisect at 90 degrees. The diagonal divides the square into two congruent isosceles right triangles.
How can we find the area of a rectangle?
The area occupied by a rectangle within its boundary is calculated using the area of a rectangle formula. The area of a rectangle is calculated by multiplying its length and width. Thus, the product “l \(\times\) w” is the formula for the area ‘A‘ of a rectangle whose length and width are ‘l‘ and ‘w‘, respectively.
To find the area of a rectangle, follow the steps below:
Step 1: From the given data, take note of the length and breadth dimensions.
Step 2: Multiply the length and breadth.
Step 3: The product written in square units is the required answer.
Solved Examples on Area and Perimeter
Example 1: Find the perimeter and area of the rectangle. Make a different rectangle with the same perimeter as the first. Which rectangle has the larger area?
Solution:
Perimeter of the rectangle = length + width + length + width
= 5 + 2 + 5 + 2
= 14\(m\)
Area of the rectangle = length (l) \(\times\) width (w)
= 5 \(\times\) 2
= 10\(m^2\)
The diagram below has the same perimeter, 14, but the area will be different:
Perimeter of the rectangle = length + width + length + width
= 4 + 3 + 4 + 3
= 8 + 6
= 14\(m\)
Area of the rectangle = length (l) \(\times\) width (w)
= 4 \(\times\) 3
= 12\(m^2\)
Hence, the second rectangle has a larger area than the first one.
Example 2: Alex wants to create a rectangular photo frame that has a boundary length of 10 feet. How long and wide should the photo frame be to give it the most area?
Solution:
The perimeter is given as 10 feet.
P = 2 \(\times\)(l + w) = 10 feet
Now we need to get the maximum area for a given perimeter. Try playing around with different length and width dimensions such that the perimeter is the same but you get different areas. You can see that the maximum area is obtained when the length and width are equal to each other.
So l = w,
P = 2 \(\times\)(l + l) = 2 \(\times\)2l = 4 \(\times\) l
4 \(\times\) l = 10
l =\(\frac{10}{4}\) = 2.5 feet
Area of the rectangle = length (l) \(\times\) width (w)
= 2.5 \(\times\) 2.5 = 6.25 square feet
Hence, Alex should make the photo frame 2.5 feet long and 2.5 feet wide.
Example 3: Find the perimeter and area of the given rectangle. Make a new rectangle with the same area as the first. Which rectangle has the greater perimeter?
Solution:
Area of the rectangle = length (l) \(\times\) width (w)
= 5 \(\times\) 3
= 15\(m^2\)
Perimeter of the rectangle = length + width + length + width
= 5 + 3 + 5 + 3
= 16\(m\)
The rectangle below has the same area, 15 m2, but the perimeter will be different:
Area of the rectangle = length (l) \(\times\) width (w)
= 6 \(\times\) 2.5
= 15\(m^2\)
Perimeter of the rectangle = length + width + length + width
= 6 + 6 + 2.5 + 2.5
= 17\(m\)
Hence, the second rectangle has a bigger perimeter than the first one.
Example 4: James has 400 square foam tiles that are one-foot-long and one-foot-wide. He wants to use all of the tiles to create a rectangular exercise space. How long and wide should he make the exercise area in order to have the smallest perimeter possible?
Solution:
First, let us find the area of the space. There are 400 one-foot long and one-foot-wide tiles, so no matter how they are arranged, the area is going to be 400 \(\times\) 1 \(\times\) 1 = 400 square feet.
Now we need to get the least perimeter for a given area. Try playing around with different length and width dimensions such that the area is the same but you get different perimeters. You can see that the smallest perimeter is obtained when both the length and width of the rectangle are equal to each other.
So, Area = l \(\times\) w = 400
When l = w,
A = l \(\times\) l = 400
We know that 20 \(\times\) 20 = 400. So l = 20 feet
That is, the length and width of the rectangular exercise space are 20 feet.
Perimeter of the rectangle = length + width + length + width
= 20 + 20 + 20 + 20
= 80\(m\)
Hence, James should make the rectangular exercise space 20 meters long and 20 meters wide.
In our daily lives, we use area and perimeter for a variety of purposes. For example, we must know the floor area of a house before purchasing it, and we must know the perimeter of the garden before purchasing a wire to fence it.
To begin with, the area of a shape is its surface area or the flat space it covers, whereas the perimeter of a shape is the distance around its boundary. Secondly, the perimeter is measured in linear units, while the area is measured in square units.
Yes, a square is considered a rectangle because it has all of the properties of a rectangle, including all four interior angles being 90 degrees, opposite sides being parallel and equal, and two equal diagonals bisecting each other. So, all squares are rectangles, but all rectangles are not squares. When the adjacent sides of the rectangle are equal, it is considered a square.