What is Area?

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.

What is a Polygon?

A polygon is a two-dimensional shape created by connecting straight lines. Triangles, hexagons, and pentagons are representations of polygons. The number of sides in a shape is indicated by its name. A triangle, for example, has three sides, while a quadrilateral has four. As a result, a triangle is any shape that can be drawn by connecting three straight lines, and a quadrilateral is any shape that can be drawn by linking four lines.

The surface area of the shape is defined as the "space enclosed within the perimeter or boundary" of that surface shape. We use math to compute the area for various situations.

What are Two-Dimensional Shapes?

Flat forms, also known as two-dimensional shapes (2D shapes), are shapes of just two dimensions: length and width. Two-dimensional shapes have no depth. Surface area and perimeter are the two measurements used to measure flat shapes. Shapes that can be drawn on a sheet of paper are known as two-dimensional shapes.

What are Three-Dimensional Shapes?

Strong forms, also known as three-dimensional shapes (3D shapes), are shapes of three dimensions, such as length, width, and depth. Volume and surface area are the two distinct metrics used to describe three-dimensional shapes. In most cases, three-dimensional shapes are generated by rotating two-dimensional shapes.

Deriving formulas

Any object's area can be defined as follows:

The area is the measure of the surface stretch of the two-dimensional or three-dimensional shape. You can also think of area as the amount of material, such as clothing, paper, or tiles, required to cover a two-dimensional plane's surface. The surface area of a three-dimensional object, such as a sphere, ball, cube, or cuboid, is referred to as the area of all of its faces.

The formula we use is determined by the shape for which you are attempting to calculate the total area. 

Rectangle

You just need the width (w) and length (l) to calculate the area. The area of a rectangle is calculated by multiplying the length by the width:

Area of a Rectangle = Length x Width

The size is always squared. This is often expressed in square units, which are obtained from linear units. 

Square

Let's start with a basic understanding of a square's shape and structure. On a plane, a square is a four-sided rectangular closed figure. A square's edges are all of the same lengths. The length and breadth of an object must be measured before it can be defined in two-dimensional geometry. The length and width of a square are equal in this case.

The area of a square is calculated by multiplying the side by side:

Area of a square = side x side

Area of a square= s²

Triangle

A triangle is a three-sided polygon. The area of a triangle is calculated just like a rectangle, i.e. multiplying the base and height of the triangle, and then the result is divided by 2 because a diagonal divides the rectangle into two equal triangles. Each triangle is one-half that of the rectangle.

Area of a triangle = (1/2) x base x height

Circle

The area of a circle in a two-dimensional plane is the space occupied by the circle. This can be easily computed using the equation A = πr², where r is the circle's radius. Square units, such as m², cm², etc., are used to denote the area.

Area of a circle = π r²

Rhombus

The sides of a rhombus are all the same length, making it an equilateral quadrilateral. The opposite sides of a rhombus are parallel, the opposite angles are identical, and the adjacent angles are supplementary. Diagonals in a rhombus bisect each other at right angles. The region of a rhombus could be measured in a number of ways, including using the base and height, diagonals, and trigonometry.

Area of a rhombus=½ × d1×d2

Where d1= diagonal 1 and d2= diagonal 2

Parallelogram

In a two-dimensional plane, a parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. The opposite sides of a parallelogram are equal, the opposite angles are identical, and the adjacent angles are supplementary. Each diagonal in a parallelogram divides the parallelogram into two congruent triangles.

Area of a parallelogram = (base x height)

Common Mistakes 

Misunderstanding the question is a common problem while calculating the area. To overcome this, drawing a correct figure with accurate labels and measurements written on it helps you understand the issue more quickly.

Formula

• Area of a rectangle = Length X Width
• Area of a square= Side² 
• Area of a triangle=(½) x Base x Height
• Area of a Circle  = π r², where r is the radius of the circle
• Area of a rhombus =½ × d1×d2, where d1 and d2 are the diagonals of the rhombus
• Area of a parallelogram = (base x height)

Context and Applications

The area formula has applications in architecture, surveying, and map design. The real-valued edition of the area for such a given location can be used to create knowledge tools like globes as well as geophysical maps. The very first step in interpreting the volume of a three-dimensional object like a cone, cylinder, ball, or cube is to measure the area of a two-dimensional shape.

Practice Problem

1. Calculate the area of a circular path with a radius of 7 meters.

Solution:

The radius of a circular path, r = 7m, is given.

We can deduce the following from the formula for the area of a circle:

Area of a Circle = π r² 

                         = 22/7 x 7 x 7

                         = 154 sq.m.

2. A square plot's side length is 5 meters. Determine the area of the square plot.

Solution:

Given a = 5m side length.

Area of a Square = a²

                            = 5 x 5

                            = 25 sq.m.

3. Calculate the area of a triangle with a height 12 and base 20.

Solution:

Height = h = 12

Base = b = 20

Area of a triangle= ½ × b × h

                           =  ½ × 20 × 12 = 120 square units