Area of a shape

Table of Contents 


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Introduction to Area

The word "area" means vacant surface in Latin.

Area can be defined as the amount of space covered by a flat surface of a particular shape.

It is measured in terms of the "number of" square units (square centimeters, square inches, square feet, etc.)

Most objects or shapes have edges and corners.

The length and breadth of these edges are considered while calculating the area of a specific shape.


How to Calculate the Area

We use a grid to calculate the area of a shape.

The grid is made up of many squares of sides 1 unit by 1 unit.

The area of each of these squares is 1 square unit.

Hence, each square is known as a unit square.

Look at the figure shown below.

Let us find the area of the shape drawn in the grid. 

A grid with unit squares.

The area of this shape is the number of shaded unit squares.

Let's assume that each side of the unit square is 1 unit.

A grid with unit squares. There are numbers in the shaded unit squares.

Thus, Area = 9 sq. units.

When the shape does not occupy the complete unit square, we can approximate and find its value.

If it occupies about \(\dfrac{1}{2}\) of the unit square, we can combine two such halves to form an area of 1 sq. unit. 

area of a shape in the grid. Some unit squares are completely shaded and some are shaded half.

Therefore, the area occupied by the shape = 4 full squares and 8 half squares

Together this forms an area of 8 sq. units.

If the shaded region is less than \(\dfrac{1}{2}\), we can omit those parts.

For regular shapes, we have certain formulas to calculate their area.

Note that this is only an approximate value.

The simulation below generates some shapes.

Can you count the squares and find its area?

You can verify your answer in the end.


Area of a Square

The area of a square is the space occupied by a square.

Look at the orange square shown in the grid below.

It occupies 25 square units.

Area of a square - There is an orange square within another square. The orange square is further divided into 25 square units with numbers written inside each of the individual units.

From the figure, we can observe that the length of each side of the orange square is 5 units.

Therefore, the area of the square is the product of its sides.

Area of square = side \(\times\) side = 25 square units

\(\begin{align} \text {Area of a Square} = s \!\times\! s \text{ units}^2 \end{align}\)

Area of a Rectangle

The area of a rectangle is the space occupied by a rectangle.

Consider the yellow rectangle in the grid.

It has occupied 6 units.

A rectangle shown in a grid. It occupies 6 units. The length is 2 units and breadth is 3 units.

In the above example, the area of the rectangle whose length is 2 units and breadth is 3 units is \(6\text{ unit}^2\).

We have, \(2 \times 3 = 6\)

The area of a rectangle is obtained by multiplying its length and breadth which is the same as counting the unit squares.

Thus, the formula for the area, \(A\), of a rectangle whose length and breadth are \(l\) and \(b\) respectively is the product \({l \times b}\).

\begin{align}\text{Area of a Rectangle} = l \!\times\! b \text{ units}^2 \end{align}

Area of a Circle

A circle is a curved shape.

The area of a circle is the amount of space enclosed within the boundary of a circle.

Learn more about \(\pi\) and radius before we get to the formula for the area of a circle.

radius of a circle with formula

The area of circle is given by the formula:

\( \pi \text{r}^2\)

where \(r\) is the radius of the circle and \( \text{Pi} - \pi\)  is the mathematical constant whose value is approximated to \(3.14\)  or \( \frac{22}{7} \)


Area of Shapes - Formula

Here is the list of formulas for the area of shapes.

Shape Area of Shapes - Formula
Square
Areas of shapes: area of a square formula 
\(x^2\)
Rectangle
Areas of shapes: area of a rectangle formula
\(l \times b\)

Circle
Area of shapes: Area of a circle formula with radius r 

\( \pi \text{r}^2\)

Triangle

Area of triangle formula is shown using a triangle with height h and base b.

\(\frac{1}{2} \times \text{b} \times \text{h}\)

Parallelogram

  Areas of shapes: area of a parallelogram formula

\(b \times h\)

Isosceles Trapezoid

Areas of shapes: area of a trapezoid formula

\( \frac{1}{2}(a+b)h \)

Rhombus

Areas of shapes: area of a rhombus formula

\( \frac{1}{2} \times d_1 \times d_2 \)

Kite

Areas of shapes: area of a kite formula

\( \frac{1}{2} \times d_1 \times d_2 \)

The simulation below shows how the area is calculated in a triangle, square, rectangle and circle.

Try to apply the formula and calculate the area and then verify your answer.


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Solved Examples

Example 1

 

 

Find the area of a square of side 7 cm.

Solution:

A square with sides measuring 7 cm is shown.

Area of a square of side 7 cm is:

\[\begin{align} 7 \text{ cm} \times 7 \text{ cm}= 49 \;\text{cm}^2 \end{align}\]

\( \therefore\) Area of the square \(= 49 \text { cm}^2\)
Example 2

 

 

The dimensions of a rectangle are 15 cm and 8 cm.

Find its area.

Solution:

Solved example: Find the area of a rectangle of sides 15 cm and 8 cm.

Area of a rectangle is the product of its length and breadth.

Area = \(15 \times 8= 120\text{ cm}^2\)

\(\therefore\) The area of the rectangle is \(= 120 \text { cm}^2\)
Example 3

 

 

Find the area of a circle with radius 14 cm.

Solution:

A circle with radius 14 cm is given. Find its area.

Given: Radius of the circle = 14 cm

Area of a circle is calculated by the following formula

\[\begin{align} &= \pi r^2 \\ &= \left(\frac{22}{7}\right) \times \left({14}\right) \times \left({14}\right) \\ &= \text{616} \text{ cm}^2 \end{align} \]

\(\therefore\) Area of the circle \(= 616 \text { cm}^2\)

Example 4

 

 

Calculate the area of the given shape by counting the squares.

count the squares in the grid and find the area of the shape

Solution:

Let's count all the full squares and half squares.

We can combine two half squares into 1 sq. unit.

 count the squares in the grid and find the area of the shape

There are 24 unit squares and 5 half squares which add up to \(26\dfrac{1}{2}\) sq. units.

\(\therefore\) Area of the given shape \(= 26\dfrac{1}{2}  \text {sq. units}  \)


Area - Measuring Units

Area is calculated in terms of the shape's length and breadth.

Length is unidimensional and measured in units such as centimetre (cm), metre (m), kilometre (km), feet (ft), yards (yd), inches (in), etc.

However, the area of a shape is a two-dimensional quantity. 

Hence, it is measured in square units like sq. cm or (cm2), sq. in or (in2), sq. ft or (ft 2), sq. m or (m2), etc.

 
Challenge your math skills
Challenging Questions
  1. Here is an activity for you. 
    A rectangle of side 5 X 6 units is given.
    Can you make a compound shape such that its area is 15 units2


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Practice Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

 

 
tips and tricks
Tips and Tricks
  1. We often memorise the formulas for calculating the area of shapes.
    An easier method would be to use grid lines to understand how the formula has been derived.
  2. We often get confused between area and perimeter of a shape.
    A thorough understanding can be built by tracing the surface of any shape and observing that the area is essentially the space or the region covered by the shape.

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


1. What is the formula for finding the area?

  • The formula for finding the area of a square is \(s^2\) where s is the length of the side.
  • The area of a rectangle is the product of its length (\(l\)) and breadth(\(b\)) \(\text{Area} = l \times b\)
  • The area of a triangle is half times the product of its base (\(b\)) and its height (\(h\)); \(\text{Area} = \frac{1}{2} \times h \times b\)
  • The area of a circle is given by the formula \( \pi r^2\) where \(r\) is the radius of the circle and \(\pi\) is the constant 3.14.

2. What is the formula for area of all shapes?

The formula for the commonly used shapes is listed in the section Area of Shapes -Formula.

The simulation also explains how to apply the formula to calculate the area of different shapes.

3. How do you find the area in math problems?

To find the area in math problems, we can either count the number of unit squares occupied by the shape in the grid or apply the formula listed in the table under the section Area of Shapes -Formula.

  
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