area, perimeter

In this lesson, you will learn:

• Perimeter of two-dimensional figures.
• Area of two-dimensional figures.
• How to find the perimeter and area of a non-standard polygon.

Perimeter and Area help to quantify the physical space of two-dimensional figures. Knowledge of area and perimeter is applied practically by people on a daily basis, such as architects, engineers, and graphic designers.

The perimeter of a shape is defined as the total distance around the shape. Basically, it is the length of any shape if it is expanded in a linear form. The perimeter of different shapes can match in length with each other depending upon their dimensions. For example, if a circle is made of a metal wire of length L, then the same wire we can use to construct a square, whose sides are equal in length. For figures with straight sides such as triangles, rectangles, squares, or polygons; the perimeter is the sum of lengths for all the sides. Few real-life examples of where we need perimeter:

• Length of the border of a quilt or blanket.
• Length of rope needed to fence a park or a farm.
• Length of the frame needed for a picture

Let us take the example of applying a fence around your garden to protect it from animals and thieves.

Measure the length of the boundary of your garden. Here it is 15 meter + 10 meter + 15 meter + 10 meter =50 meters. You need to buy a 50-meter wire to fence the garden. 50 meters is the perimeter of this garden.

You measure the perimeter in linear units, which are one-dimensional. Examples of units of measure for perimeter are inches, centimeters, meters, or feet. 

Example 1: Find the perimeter of the given figure. All measurements are in inches.

Answer: 21 + 15 + 3 + 7 = 46 inches

The Perimeter of a circle is called its circumference.

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Area

The area of a two-dimensional figure describes the amount of surface the shape covers. We measure the area in square units of a fixed size. For example, you can write more on two sheets of paper than on a single sheet because it has twice the area of a single sheet and therefore twice as much space to write on.  Examples of square units of measure are square inches, square centimeters, or square miles. 

A few real-life situations where we use area are:

• Floor covering like carpets or tiles requires area measurement.
• Wallpaper or painting the walls.
• Wrapping a present: you need to know the area of the box to understand how much wrapping paper you need.
• When you buy a house, you want to know the area of the ground it has been built on, as well as the usable surface inside.

How to find the area of a polygon? When finding the area of a polygon, you count how many squares of a certain size will cover the region inside the polygon. For example, below is a 5 × 5 = 25 squares. Each square has a side of 1 unit.  Hence this square has an area of 25 square units. 

This helps us to derive the formula of the area of a square as s × s = s²(here s represents a side of a square). The unit will similarly be inch², cm², m². 

In a similar way, we can derive the formula for the area of other two-dimensional figures. This formula helps you to determine the area faster than counting the number of square units inside the polygon. Let’s look at a rectangle.

 

You can count the squares individually. This rectangle contains 8 square units in 4 rows. So the total number of squares is 8 × 4 = 32. Therefore, the area is 32 square units. It is much easier to multiply 8 times 4 to derive the area of this rectangle, and, more generally, the area of any rectangle can be found by multiplying length times width.

Area of a rectangle = length × width
Let us look at the area formulas for other polygons.

Polygon

Parallelogram  Area of parallelogram = Base × Height Height is the line perpendicular to the base.

Triangle Area of a triangle = 1/2 × Base × Height

Trapezoid Area of a trapezium = \(\frac{(b_1 + b_2)}{2} \times h\)

Perimeter and Area of a Circle

To calculate the perimeter and area of a circle, we need to know its radius (the distance from the center to any point on the boundary). The circumference of the circle is the perimeter of the circle.

Perimeter of a circle = 2 × π  × radius
Area of a circle = π × radius²

Here, π (pi) is a mathematical constant approximately equal to \(\frac{22}{7}\) or 3.14159.

Example: A square metallic frame has a perimeter of 264 cm. It is bent in the shape of a circle. Find the area of the circle. 
The perimeter of the square = Perimeter of the circle = 264

\(2 \times \frac{22}{7} \times r = 264 \\ r = 264 \times \frac{7}{22} \times \frac{1}{2} \\ r = 42\)

Area of the circle = \(\frac{22}{7} \times {42}^2\)= 5544 cm²

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Finding the perimeter and area of a non-standard polygon

In real life, not every plane figure can be clearly classified as a rectangle, square, or triangle. To find the area of a composite figure which consists of more than one shape, we need to find the sum of the area of all the shapes forming the composite figure. To find the perimeter of non-standard shapes, find the distance around the shape by adding together the length of each side. To find the area of non-standard shapes you need to create regions within the shape for which you can find the area and add these areas together. Let us take an example and find the perimeter and area of the below figure.

Let's divide this figure into a rectangle and a triangle and calculate their area separately.

• Area of rectangle ABCD = 18 × 12 = 216 m²
• Area of Triangle BCE = ½ × 18 × 13 = 117m²

Total area of the figure = 216 + 117 = 333 m²

Refer to the lesson "Estimate area" to understand how without the formula you can estimate the area of a figure.