Area expresses the spread of a two-dimensional region, shape, or planar lamina, in a 2 dimensional plane. The two-dimensional surface of a three-dimensional object is known as its surface area. In practical terms, area can be understood as the quantity of material with a fixed thickness required to fashion a model of the shape. Or it can be imagined as the amount of paint necessary to cover the surface with a single coat. Area is the two-dimensional analog of the length of a curve in one-dimension or the volume of a solid in three-dimensional. There are many related concepts and formulas derived from the formula of area for various geometric shapes. Read on to find out more!

Introduction to Area

In the pair of figures given below, the first figure cover less space while the second figure covers more space. So, the figure covering more space is bigger in area.

Know About Area of Triangle here

Similarly, take a look at the images given below:

We can see from the figures above that a large or small figure refers to the amount of space that the figure takes upon the same surface. The area that an object or figure occupies on a plane surface must be measured.

How do we do that?

One of the methods is as follows:

We can try to imagine and figure out how many small or specific unit measurements of that specific object would make the larger object or figure. We can see how many such unit shapes are covered by that space if we draw the outlines of objects like leaves or petals on graph paper.

There are some real-life examples, where designers and architects use different shapes such as circles, triangles, quadrilaterals, parallelograms, rectangles, squares, polygons, etc. to calculate area.

Here are some random shapes which show the area covered in a basketball court and a swimming pool.

The area for any shape can be stated as:

1. The amount of material (such as wood, paper, cloth, marble) required to cover the surface in a (2)-dimensional plane.
2. For (3)-dimensional objects such as cube, sphere, cylinder, or cuboid, we calculate the surface area.

Thus, the space that an object or figure covers on any surface is known as its area.

What is Area?

Definition: The area is used to define the amount of space covered by a two-dimensional shape or surface. We measure the area in square units, \({\rm{c}}{{\rm{m}}^{\rm{2}}}\) or \({{\rm{m}}^{\rm{2}}}\)

In the below figure, if one square block is of \({\rm{1}}\,{\rm{cm}}\) length and \({\rm{1}}\,{\rm{cm}}\) width, then we can find the area of the yellow region just by counting the number of squares in the region i.e. \(25\) squares, so the area is \(5\;{\rm{cm}} \times 5\;{\rm{cm}} = 25\;{\rm{c}}{{\rm{m}}^2}\)

What is Area Formula?

The simplest way to find the area of any geometric shape is by using unit squares. A unit square is a square whose length of each side is 1 unit. Using this as a reference, we can calculate the area of a polygon which is the number of unit squares covered by the shape.

Space that is inside the boundary of a closed shape is termed as area.

Standard Unit of Area

The area is measured in square units. The SI unit of area is the square meter (also written as \({{\rm{m}}^{\rm{2}}}\), which is the same as the area of a square whose length of each side is \({\rm{1}}\,{\rm{m}}\).

How to Find Areas of Various Shapes in Geometry?

(i) Area of a Triangle

Area of a triangle \({\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{ \times base \times height}}\)

The above formula is used when the length of any side and the corresponding height is known or given.

For the above figure, the area of the triangle \( = \frac{1}{2} \times BC \times AD\).

(ii) Area of a Rhombus

Area of rhombus has different formulas in different cases, and the most used are given below:

Area of rhombus using diagonals	Area \( = \frac{1}{2} \times {d_1} \times {d_2}\)
Area of rhombus using base and height	Area \( = b \times h\)

Where, \({d_1} = \) Length of diagonal \(1\)
\({d_2} = \) Length of diagonal \(2\)
\(b = \)  Length of any side
\(h= \)  Height of rhombus

(iii) Area of a Trapezium

Area of a Trapezium \({\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{ \times sum}}\,{\rm{of}}\,{\rm{parallel}}\,{\rm{sides \times Distance}}\,{\rm{between}}\,{\rm{the}}\,{\rm{parallel}}\,{\rm{sides}}\)

Where \(a\) and \(b\) represent the length of parallel sides, \(h\) represents the distance between the parallel sides.

(iv) Area of a Rectangle

The area of a rectangle is calculated in units by multiplying the breadth (or width) by the length of a rectangle.

Area of rectangle \({\rm{ = length \times width}}\)

(v) Area of a Square

A square is a special kind of rectangle whose sides are equal, that is the length and breadth of a square are equal.

Thus,

Area of a square \({\rm{ = side \times side}}\)

or  Area of a square \( = {({\mathop{\rm side}\nolimits} )^2}\)

(vi) Area of a Circle

The area of a circle is obtained as \(\pi {r^2}\), where r is the radius of the circle. This is the basic formula for the area of a circle.

(vii) Area of a Parallelogram

Let us transform the parallelogram \(ABCD\) into a rectangle with the same base and height.

Now, since the bases and heights of the parallelogram and the rectangle are the same,

Area of a parallelogram \(=\) Area of a rectangle \(AEFD\)

\( = EF \times AE = BC \times AE\)

Therefore, Area of a parallelogram \({\rm{ = base \times height}}\)

Finding Area of Irregular Shapes Using Graph Paper

Let us consider the above figure placed on a squared paper or graph paper where every square measure \(1\;{\rm{cm}} \times 1\;{\rm{cm}}.\).

See the squares covered in the figure. Some of them are fully covered, some partially, some less than half, and some more than half.

The fully covered squares are shown in the shaded figure.

We know that the area is the number of squares that are required to cover the shape. But we are stuck on a small problem. The squares do not always fit exactly into the area we measure. Some fitfully, some are partially covered, and some are largely covered in the shape.

To get over this difficulty, we should use this convention:

1. The areas which are less than half a square, just ignore it.
2. If the area is more than half of a square, then count it as one square.
3. If the area is exactly half the square, then take it as (\frac{1}{2}) square unit.
4. If the area is one full square, then take it as (1) square unit.

Now count the squares in the given figure and fill the table given below.

	Covered Area	No. of squares	Estimated Area (sq. units)
(i)	Fully-filled squares	\(18\)	\(18 \times 1 = 18\)
(ii)	Half-filled squares	\(4\)	\(4 \times \frac{1}{2} = 2\)
(iii)	More than Half-filled squares	\(10\)	\(10 \times 1 = 10\)
(iv)	Less than Half-filled squares	\(6\)	\(6 \times 0 = 0\)

Total area \({\rm{ = 18 + 2 + 10 + 0 = 30 sq}}{\rm{. units}}\)

Solved Problems – Area

Q.1. If a square is inscribed in a circle, what will be the ratio of the area of the square to that of the circle?
Ans: Let the radius of the circle be \(r\).
Hence, area of the circle \( = \pi {r^2}\)
The length of the diagonal of square \(=2r\)
So, the length of the side of the square \( = \sqrt 2 r\)
Hence, area of the square \( = 2{r^2}\) Hence, the required ratio is \(2:\pi \).

Q.2. Find the base of a parallelogram if its area is \({\rm{60\;c}}{{\rm{m}}^{\rm{2}}}\) and its altitude is \({\rm{15}}\,{\rm{cm}}\).
Ans: Given, area of a parallelogram \( = {\rm{60}}\,{\rm{c}}{{\rm{m}}^{\rm{2}}} \Rightarrow b \times h = 60\)
Here, altitude (or) height \(\left( h \right) = {\rm{15}}\,{\rm{cm}}\)
\(b \times 15 = 60\)
\(b = 4\,{\rm{cm}}\)
So, the base of the parallelogram is \(4\,{\rm{cm}}\).

Q.3. The length of two parallel sides of a trapezium is \(6\;{\rm{cm}}\) and \(12\;{\rm{cm}}\). If its area is \(63\;{\rm{c}}{{\rm{m}}^2}\), then find the distance between the parallel sides.
Ans: Given:  \(a = 6\;{\rm{cm}},b = 12\;{\rm{cm}},A = 63\;{\rm{c}}{{\rm{m}}^2},h = ?\)
Area of a trapezium \(A = \frac{1}{2} \times \left({a + b} \right) \times h\)
\( \Rightarrow 63 = \frac{1}{2} \times \left({6 + 12} \right) \times h\)
\( \Rightarrow 126 = 18 \times h\)
\( \Rightarrow h = \frac{{126}}{{18}} = 7\;{\rm{cm}}\)
Therefore, \(7\;{\rm{cm}}\) is the distance between the parallel sides.

Q.4. Find the area of the triangle if the base is \(5\;{\rm{cm}}\) and the height along the base is  \(8\;{\rm{cm}}\).
Ans: Area of the triangle \({\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{ \times base \times height = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{ \times 5 \times 8 = 20\;c}}{{\rm{m}}^{\rm{2}}}\)

Q.5. A rectangle of length \(17\;{\rm{cm}}\) and breadth \(13\;{\rm{cm}}\) is given. Find the area of the rectangle.

Ans: Given the length \((l) = 17\;{\rm{cm}}\) and breadth \((b) = 13\;{\rm{cm}}\)
The perimeter of rectangle \( = 2\left(\ {l + b} \right)\)
\({\text{=2(17 + 13)=2(30)}}\)
\( = 60\;{\rm{cm}}\)
We know that area of rectangle \( = l \times b\)
 \( = 17 \times 13 = 221\;{\rm{c}}{{\rm{m}}^2}\)

Summary

In this article, we have learnt that an area is a space occupied by any two-dimensional object on the plane. The formulae for finding the area of different well-known shapes like circle, triangle, rectangle, square, parallelogram, trapezium, rhombus, etc are also covered in this article. We also discussed the unit of area and how to calculate the area of any irregular shape by drawing it on graph paper.

This graphical approach of calculating the area of any regular or irregular shape is very helpful and convenient to use.

FAQs

Q.1. What is area and volume?
Ans: The area is space covered by a 2-dimensional flat surface in a plane. The standard unit of measurement is square units. Volume is the space covered by a 3-dimensional surface. The standard unit of measurement is cubic units.

Q.2. Is area always squared?
Ans: Yes, area is always expressed in square units.

Q.3. How do you find an area of a shape?
Ans: The area of a rectangle is the product of its height by breadth. To find the area of a square we just multiply the side of the square by itself (all the sides of a square are equal). Similarly, the area of a triangle is half of the area of a parallelogram.

Q.4. How do you find the area?
Ans: Area is the space enclosed by a figure. For different figures, we have different formulae to calculate area. If any irregular shape is given, then the area is calculated by drawing it on graph paper and then counting the number of squares enclosed by the figure.

Q.5. What is the formula for the area of common geometrical shapes?
Ans: Here are the most used formulae to calculate the area of geometric shapes-

Area of a Triangle	\(\frac{{\rm{1}}}{{\rm{2}}}{\rm{ \times base \times height}}\)
Area of a Rhombus	\(\frac{1}{2} \times {d_1} \times {d_2}\), where \({d_1}\) and \({d_2}\) are the length of the diagonals
Area of a Trapezium	\(\frac{1}{2}(a + b) \times h\), where \(a\) and \(b\) represents the length of parallel sides, \(h\) represents the distance between the parallel sides.
Area of a Rectangle	\({\rm{length \times width}}\)
Area of a Square	\({{\rm{(side)}}^{\rm{2}}}\)
Area of a Circle	\(\pi {r^2}\), where \(r\) is the radius of the circle
Area of a Parallelogram	\({\rm{base \times height}}\)