Area Of Shapes

What is Area?

The amount of space covered by a flat surface or piece of land or an object is called its area. The coloured part of each of the following figures shows the amount of space covered by each of them on a sheet of paper.

Units of measurement of Area

The units used to measure area are based on the units of length, i.e. mm, cm, m, km etc. to measure a region we use a square as a unit. Hence, the unit of area is a unit square. This is written as unit2. Let us see who the different units are used to represent the area of a surface.

Unit                       Area

mm        –              mm2      (1 mm2 is equal to 1 square millimetres)

cm          –              cm2             (1 cm2 is equal to 1 square centimetres)

m            –              m2          (1 m2 is equal to 1 square metres)

km          –              km2        (1 km2 is equal to 1 square kilometres)

mi           –              mi2         (1 mi2 is equal to 1 square miles)

How to choose the correct unit when representing the area of a shape?

The unit for measuring the area depends on the size of the area being used. For example, the area, of the following are usually measured as under –

A Stamp – in m2

A notebook  –   in cm2

The floor of a room  –  in m2     

A city  –  in km2 or mi2

Let us understand the area with an example. Find out the areas of these shapes by counting the number of squares they cover. Give your answers in square units. Here we have some half squares as well. Count two half squares as 1 unit2

Number of full squares in this shape = 28

Number of half squares in this shape = 0

Area of the figure = 28 + 0 = 28 unit2.

Number of full squares in this shape = 30

Number of half squares in this shape = 16

Area of the figure = 30 + 1/2 x 16 = 30 + 8 = 38 unit2.

Area of a Rectangle

A rectangle is a type of quadrilateral that equal opposite sides and four right angles.

Here we have a rectangle, covered with small squares. Each small square stands for 1 square centimetre, that is, each small square is cm on each side.

Count the number of squares that cover each rectangle. This will give you the area of a rectangle in square centimetres. Also, write the length and breadth of each rectangle. What do you observe? Let’s find out.

Rectangle 1

Number of small squares = 18

Area  = 18 cm2 or 18 sq. cm

Length  = 6 cm

Breadth = 3 cm

Rectangle 2

Number of small squares = 20

Area  = 20 cm2 or 20 sq. cm

Length  = 5 cm

Breadth = 4 cm

From the above examples, you can see that there is a quicker way to find the area in each case without having to count the number of squares. Since 6 x 3 = 18 and 5 x 4 = 50, we can say that the area of each of the two rectangles can be found out by multiplying the measures of length and breadth of the rectangle.

Area of a rectangle having length ‘l’ and breadth ‘b’ is given by l x b

For example, let us consider a rectangle having a length of 8 cm and a breadth of 7 cm as shown in the figure below.

The area of this rectangle is given by 8 7 7 = 56 cm2.

Hence:

Area of a rectangle = length x breadth

How to calculate the length and breadth of a rectangle using its area?

We now know how to calculate the area if we have the length and the breadth of a rectangle.  But, if only the length or breadth and area are given, is it possible to know the breadth of the rectangle? Let us find out.

Since, Area of a rectangle = Length x Breadth,

Therefore,

Length = Area / Breadth ,     Breadth = Area / Length

Let us understand it more through an example.

Example 1

Find the length of a rectangle, if its area = 90cm2 and breadth = 6 cm

 Solution

We are given that area = 90cm2 and breadth = 10 cm

Now, Length = Area / Breadth

Hence,  Length = 90/6 cm = 15 cm

Therefore, the length of the rectangle having area = 90cm2 and breadth = 6 cm is 15 cm

Example 2

A grassy plot is 80 m x60 m. Two cross paths each 4 m wide are constructed at right angles through the centre of the field, such that each path is parallel to one of the sides of the rectangles. Find the total area used path. Also, find the cost of gravelling them at £ 5 per square metre.

Solution

An important strategy for solving such questions is to first visualise them. Therefore, we will construct the rectangle and plot the details we have been given in the question.

Let the ABDC and PQSR be the rectangular paths that are to be constructed at right angles through the centre of the field, such that each path is parallel to one of the sides of the rectangles. We have,

We have to find the total area used path.

First, we will calculate the area of the path ABDC.

Since the grassy plot is a rectangle and we know that area of a rectangle having length ‘l’ and breadth ‘b’ is given by l x b, therefore,

Area of the path ABDC =(80 x 4) m2 = 320 m2

Again, let us now find out the area of the path PQSR which can be calculated in the same manner as done above.

We have,

Area of the path PQSR = (60 x 4) m2 = 240 m2

But, with the help of our figure, we can clearly see that the area of EFHG is common to both the areas that we have calculated just now. Hence we need to subtract this area once from our result to avoid counting it twice.

Area of the path EFHG = (4 x 4) m2 = 16 m2

Therefore,

Total area of the path = Area of the path ABDC + Area of the path PQSR – Area of the path EFHG

Hence,

Total area of the path = ( 320 + 240 – 16 ) m2 = 544 m2

Now, let us cost for gravelling the path.

We have been given that the cost of gravelling the path is £5 per square metre.

Since the total area to be gravelled is 544 m2, therefore,

The total cost of gravelling the path would be £ (544 x 5) = £2720.

Area of a Square

A square is a quadrilateral that has four equal sides and four right angles.

For a square whose side is of ‘s’ units:

Area of a Square = Side x Side = s2 sq. units

For example, if we have a square whose one side is 6 cm, its area would be calculated as

Area = Side x Side = 6 x 6 = 36 cm2

Area of a Triangle

A triangle is a polygon that is made of three edges and three vertices. The vertices join together to make three sides of a triangle. The area occupied between these three sides is called the area of a triangle.

The area of a triangle is defined by 1/2 x b x h

Where b = base of the triangle (or any one side of the triangle)

And

H = Height of the triangle from that base (or side)

The following figure illustrates the base and the height of a triangle

The above formula is applicable irrespective of the fact whether a triangle is a scalene triangle ( having different sides), an isosceles triangle ( having two sides equal), or an equilateral triangle ( having all sides equal).

Let us understand this more through an example. Suppose we have a triangle that has one side as 6 cm and an altitude (height) of 8 cm on that base as shown in the following figure:

The area of this triangle is given by

 1/2 x b x h

Where b = 6 cm and h = 8 cm

Therefore, Area = 1/2 x 6 x 8 = 24 cm2

Area of a Circle

The space occupied by a circle is called its area.

The area of a circle having a radius ‘r’ (The distance from the centre to a point on the boundary) is given by πr2 where π = 22/7 or 3.14 (approx.)

For example, suppose we have a circle that has a radius of 7 cm as shown in the figure below.

 Its area is given by:

Area = πr2 = 22/7 x 7 x 7 = 154 cm2

Suppose, instead of the radius we are given the diameter of a circle, how do we calculate the area?

We know that in a circle, the radius is half of the diameter. Mathematically,

r = d/2, where ‘d’ is the diameter and ‘r’ is the radius.

So, we half the given diameter and obtain the radius.

Example 1

Suppose we are required to find the area of a circle having a diameter of 4.2 cm.

Here diameter (d) = 4.2 cm

By the relation between radius and diameter, we have, r =d/2

Hence r = 4.2/2 = 2.1 cm

Now, area of this circle = = πr2 = 22/7 x 4.2 x 4.2 = 55.44 cm2

Example 2

Four equal circles are described about four corners of a square so that each touches two of the others, as shown in the figure. Find the area of the shaded region, if each side of the square measures 14 cm.

Solution

Let ABCD be the given square each of which is 14 cm long. Clearly, the radius of each circle is 7 cm.

We have,

Area of the square of side 14 cm long = (14 x 14) cm2 = 196 cm2

Area of the quadrant of a circle of radius 7 cm

 = 1/4 πr2

 = 1/4 x 22/7 x 72 cm2

 = 38.5 cm2

Therefore, area of four quadrants = 4 x 38.5 cm2 = 154 cm2

Hence,

Area of the shaded region

= Area of the square ABCD  – Area of 4 quadrilaterals

= (196 – 154) cm2

= 42 cm2

Area of a Parallelogram

A parallelogram is a quadrilateral, whose each pair of opposite sides is parallel. Suppose we have a parallelogram ABCD such that AB and DC are a pair of its opposite sides so that AB ‖ BC. Similarly, BC and AD are a pair of opposite sides such that BC ‖ AD.

If DL ⟂AB, then any line, then we find any line segment with its end-points on the two sides AB and DC perpendicular to them has the length DL.so, we call AB as the base and DL the corresponding altitude.

Similarly, if DM ⟂BC, then any line segment with its end-points on the two sides AB and DC perpendicular to them has the length DM. so we can call BC as the base and DM as the corresponding altitude.

We have,

Area of a parallelogram = Base x Altitude

Altitude is also known as height. So, we can say that,

Area of a parallelogram = Base x Height

For example, let us find the area of a parallelogram having the base = 5 cm and altitude = 4.2 cm.

We know that

Area of a parallelogram = Base x Altitude

Here, base = 5 cm and altitude = 4.2 cm

Therefore,

Area = (5 x 4.2) cm2 = 21 cm2

Area of a Rhombus

Rhombus is a parallelogram with all its sides equal. Some properties of a rhombus are:

  1. All sides of a rhombus are equal.
  2. Since all the sides of a rhombus are equal, it is also known as an equilateral quadrilateral.
  3. The diagonals of a rhombus bisect each other at right angles.
  4. Apart from using the sides, the area of a rhombus can also be calculated using its diagonals.

Area of a Rhombus using its Sides

Since rhombus is a parallelogram with all its sides equal, therefore, the formula for the area of a parallelogram also holds true for calculating the area of a rhombus.

Hence

Area of a Rhombus = Base x Height

Example,

Let us have a rhombus whose altitude is 7 cm and the rhombus has a perimeter of 180 cm.

Solution

We are given that

The altitude of the rhombus = 7 cm

The perimeter of the rhombus = 180 cm.

We need to find the area of the rhombus. In order to do so, we must first find the side of the rhombus.

Now, remember that the perimeter of a closed shape is the sum of all its sides. Since the rhombus is a quadrilateral with all its sides equal, we can say that if the rhombus has a side “a”, then

a +a +a +a =180 cm

4a = 180 cm

a = 45 cm

Hence the side (base) of the rhombus = 45 cm

Now, to calculate the area of a rhombus, we know that

Area of a Rhombus = Base x Height

Therefore,

Area = (45 x 7) cm2 = 315 cm2

Area of a Rhombus using its Diagonals

We can also calculate the area of a rhombus if we know its diagonals. The formula for calculating the area of the rhombus when we know its diagonals is given by –

Area of a Rhombus = 1/2 x (Product of the diagonals)

If d1 and d2 are two diagonals of a rhombus, then,

Area of a Rhombus = 1/2 x (d1 x d2)

Example

If the two diagonals of a rhombus measure 9 cm and 12 cm respectively, find the area of the rhombus.

Solution

We know that we are given the diagonals, the area of the rhombus = 1/2 x (d1 x d2)

Here, the two diagonals are 9 cm and 12cm.

So, let

D1 = 9 cm and d2 = 12 cm.

Area of the rhombus = 1/2 x (d1 x d2) = 1/2 x (9 x 12) cm2 = 54 cm2

Hence, area of the rhombus = 54 cm2

Area of a Hexagon

A hexagon is a polygon having six sides. A regular hexagon is a hexagon that has its all sides equal.

Some important properties of a hexagon include:

  1. A hexagon has 9 diagonals

2. Sum of the interior angles of a hexagon is equal to 720o

The area of a regular hexagon can be calculated if we know one side of the hexagon. Let one side of a regular hexagon be “s”. Then the formula for calculating the area of a regular hexagon would be:

Area of a Hexagon = $\frac{3\sqrt{3}}{2}s^{2}$

Example

Find the area of a regular hexagon, whose each side measures 6 cm.

Solution

To find the area of the hexagon, we need to know its side.

We are given that each side of the hexagon = 6 cm

Now, the area of a hexagon = $\frac{3\sqrt{3}}{2}s^{2}$

So, area = $\frac{3\sqrt{3}}{2}6^{2} = \frac{3\sqrt{3}}{2}\times 36cm^{2}=54\sqrt{3}cm^{2}$

Remember

  • The amount of space covered by a flat surface or piece of land or an object is called its area.
  • The unit of area is a unit square. This is written as unit2 such as cm2, m2 km2 etc.
  • Area of a rectangle = length x breadth
  • Area of a Square = Side x Side = s2 sq. units
  • Area of a triangle = 1/2 x b x h, where b = base, h = height
  • The area of a circle having a radius ‘r’ = πr2 where π = 22/7 or 3.14 (approx.)
  • Area of a parallelogram = Base x Altitude
  • Area of a Rhombus = Base x Height
  • If d1 and d2 are two diagonals of a rhombus, then, Area of a Rhombus = 1/2  x (d1 x d2)
  • Area of a Hexagon = $\frac{3\sqrt{3}}{2}s^{2}$