Area of Shapes - Definition, Formula with Examples

What is the Area of Shapes?

A shape is a figure that is enclosed by a boundary. In our surroundings, we encounter countless objects in various shapes such as squares, rectangles, circles, and many more.

So, what is the area of a shape?

The area of a shape refers to the amount of space enclosed within its boundary or perimeter. It represents the surface a shape covers on a flat plane. The area of different geometric shapes can be calculated using specific mathematical formulas, depending on the type and dimensions of the shape.

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What is Area?

Area is a measure of the space a shape occupies on a surface. It is always expressed in square units, such as square meters (m²), square centimeters (cm²), or square feet (ft²). Simply put, area tells us how much space lies within the boundaries of a shape.

For example, if you are tiling a floor, painting a wall, or laying down a carpet, you need to know the area to determine how much material is required. Understanding area helps in planning, measuring, and estimating accurately in both everyday situations and mathematical problems.

Read more: Difference Between 2D and 3D Shapes

What are 2D Shapes?

2D shapes or two-dimensional shapes are flat figures that exist in only two dimensions, length and width. They do not have depth or thickness, which means they cannot be physically held or picked up like solid objects. Instead, they are drawn or represented on flat surfaces such as paper, whiteboards, or screens.

Common examples of 2D shapes include circles, rectangles, triangles, squares, parallelograms, trapeziums, and ellipses. These shapes are defined by their sides, angles, and curves. For example, a rectangle has four sides and four right angles, while a circle has a curved boundary with no corners.

Area of 2D Shapes Formula

Each two-dimensional shape has its own unique formula for calculating area, depending on its structure and measurements. Below are the most commonly used area of 2D formulas:

Square: The area of a square is the space covered by a square-shaped surface. Since all sides of a square are equal, its area is found by multiplying the side by itself.

Area of a Square = a × a = a2

Rectangle: The area of a rectangle can be defined as the amount of space covered by a flat surface of a rectangle. It is calculated as the product of its length and width.

Area of a Rectangle = l × w

Triangle: The area of a triangle refers to the region enclosed within its three sides. It is calculated as half the product of its base and corresponding height.

Area of a Triangle = ½ ×b × h

Circle: The area of a circle is the space enclosed by its curved boundary. It is calculated using the square of the radius multiplied by pi.

Area of a Circle = π × r2

Parallelogram: The area of a parallelogram is the region between its four sides, with opposite sides being equal and parallel. It is calculated by multiplying the base by the vertical height.

Area of a Parallelogram = b × h

What are 3D Shapes?

Three-dimensional (3D) shapes, also known as solid shapes, are geometric figures that have three dimensions: length, width (or breadth), and height (or thickness). 

These shapes occupy space and have both volume and surface area, unlike two-dimensional (2D) shapes, which only have length and width.

Many 3D shapes can be created by extending or rotating 2D shapes. For instance, rotating a circle forms a sphere, while extending a rectangle creates a cuboid.

Key Properties of 3D Shapes

The two main properties used to describe and measure 3D shapes are:

1. Surface Area – The total area of all the outer surfaces of a shape.

2. Volume – The amount of space enclosed within the shape.

Surface area is expressed in square units (e.g., m², cm²), while volume is expressed in cubic units (e.g., m³, cm³).

Surface Area of 3D Shapes Formula

The surface area of a 3D shape is the total area that its outer surface occupies. Below are the surface area of 3D shapes formula commonly used:

 Note: π (Pi) is approximately 3.1416.

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Relationship Between 2D and 3D Shapes

3D shapes are often formed by extending, rotating, or transforming 2D shapes. Understanding how a 2D shape transforms into a 3D object helps in visualizing and calculating properties like surface area and volume.

Let’s consider a circle with radius r.

If you:

• Draw multiple concentric circles inside the original circle,

• Cut the circle along a radius (from the center to the boundary),

• Then flatten or stretch the curved part into a sector or triangle,

You will form a right circular cone.

When flattened, the curved surface of the cone resembles a triangle:

• Base of the triangle = Circumference of the original circle = 2πr

• Height of the triangle = Radius of the original circle = r

The area of this triangle (which is equivalent to the curved surface area of the cone) can be calculated as:

Area = ½ × base × height = ½ × 2πr × r = πr2

This shows how the area of a 2D shape (circle) relates to the surface area of a 3D shape (cone).

Area of Shapes Solved Examples

1: The length of a rectangle is 12 cm and its breadth is 5 cm. Find its area.

Solution:

We know that the formula for the area of a rectangle is:

Area = length × breadth

Here,
Length = 12 cm

Breadth = 5 cm

Now substitute the values:

Area = 12 × 5  = 60 cm2

So, the area of the rectangle is 60 square centimetres.

= 60 cm²

The area of the rectangle is 60 cm².

2: Find the surface area of a cube whose edge is 4 cm.

Solution: We know that the formula for the total surface area of a cube is:

Surface Area = 6a2

Here,

Edge a = 4 cm

Now substitute the value:

Surface Area = 6×42 = 96cm2

So, the surface area of the cube is 96 square centimetres.

3. Find the total surface area of a cylinder with radius 7 cm and height 10 cm. (Use π = 22/7)

Solution: We know that the formula for the total surface area of a cylinder is:

Surface Area = 2πr(r + h)

Here,

Radius (r) =7 cm

Height (h) =10 cm

Now substitute the values:

Surface Area = 2 × 22/7 × 7(7+10)

2 × 22 × 17

44 × 17

Surface Area =748cm2

So, the total surface area of the cylinder is 748 square centimetres.

Also Read: Perimeter of a Triangle

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