# Area in Math – Definition with Examples

## Definition

Definition

Area is defined as the total space taken up by a flat (2-D) surface or shape of an object.

Take a pencil and draw a square on a piece of paper. It is a 2-D figure. The space the shape takes up on the paper is called its Area

Now, imagine your square is made up of smaller unit squares. The area of a figure is counted as the number of unit squares required to cover the overall surface area of that particular 2-D shape. Square cms, square feet, square inches, square meters, etc., are some of the common units of area measurement.To find out the area of the square figures drawn below, draw unit squares of 1-centimeter sides. Thus, the shape will be measured in cm², also known as square centimeters.

Here, the area of the shapes below will be measured in square meters (m²) and square inches (in²).

How to calculate the area if there are also half unit squares in the grid?

To understand that, let us take one more example:

Step 1: Count the full squares.

There are 18 full squares.

Step 2: Count the half squares.

On counting, we see that there are 6 half squares.

Step 3: 1 full square $= 1$ square unit

So, 18 full square $= 18$ square units

1 half square $= \frac{1}{2}$ square unit

6 half squares $= 3$ square units

Total area $= 18 + 3 = 21$ square units.

Origin of the Term: Area

The term ‘area’ originated from Latin, meaning ‘a plain piece of empty land’. It also means ‘a particular amount of space contained within a set of boundaries’.

Look at the carpet in your home. To buy a carpet that fits the floor, we need to know its area. Or the carpet will be bigger or smaller than the space! Some other instances when we need to know the area are while fitting tiles on the floor, painting the wall or sticking wallpaper to it, or finding out the total number of tiles needed to build a swimming pool.

## Formulas for Calculating Area

We are surrounded by so many 2-D shapes: circle, triangle, square, rectangle, parallelogram, and trapezium. You can draw all of these shapes on your paper. Every shape is different and unique, so its area is also calculated differently. To find the area, first, identify the shape. Then, use the appropriate formula from the list given below to find its area.

## Areas of Composite Figures

Every plane figure cannot be classified as a simple rectangle, square, triangle, or typical shape in real life. Some figures are made up of more than one simple 2-D shape. Let us join a rectangle and a semicircle.

These shapes formed by the combination of two or more simple shapes are called “composite figures” or “composite shapes”.

For finding the area of a composite figure, we must find the sum of the area of all the shapes in it. So, the area of the shape we just drew will be the area of the rectangle, l$\times$ b plus half the area of the circle, ½ x πr², where l and b are length and breadth of the rectangle and r is the radius of the semicircle.

If we draw a semi – circle below a triangle, we get the composite shape:

The area of such a composite figure will be calculated by adding the area of the triangle and the area of the semicircle.

Area of the a composite figure =($\frac{1}{2}\times b\times h) (\frac{1}{2}+𝜋r^2$)

where r is the radius of the semicircle and b and h are the base and height of the triangle respectively.

Real-life Applications

Here are a few ways in which you can apply the knowledge of the area of figures in your daily life.

• We can find the area of a gifting paper to check whether it will be able to cover a box or not.
• We can find the area of a square or circle to find the area of the signal board.

## Solved Examples

1. A circle has a diameter of 20 cm. Find out the area of this circle.

Ans: For the circle, d = 20 cm.

Radius, r = $\frac{d}{2}$ = 10 cm

Therefore, A = πr²

= 3.14$\times 10\times 10$ = 314 cm$^{2}$

Area of the given circle is 314 $cm^{2}$.

1. The height of a triangle is 10 cm and the base is 20 cm. What is the area of this triangle?

Ans: Area of the triangle = $\frac{1}{2}\times b\times h$

= $\frac{1}{2}\times 20\times 10 = 100 cm^{2}$

Therefore, the area of the given triangle is 100 $cm^{2}$.

1. The width of a rectangle is half of its length. The width is measured to be 10 cm. What is the area of the rectangle?

Ans: For the rectangle, w = 10 cm and l = (10 $\times$ 2) = 20 cm.

Area of the rectangle, i.e., A = l $\times$ w

A = 20 $\times$ 10 = 200 cm$^{2}$

Therefore, the area of the given rectangle is 200 cm$^{2}$.

Example 4: What is the area of the following figure?

Solution:  Full square $= 1$ square unit

So, 14 full square $= 14$ square units

1 half square $= \frac{1}{2}$ square units

5 half squares $= 2.5$ square units

Total area $= 14 + 2.5 = 16.5$ square units.

## Practice Problems

1

### The side of a square is 7 cm. What is the area of this square?

20 cm$^{2}$
30 cm$^{2}$
40 cm$^{2}$
49 cm$^{2}$
CorrectIncorrect
Correct answer is: 49 cm$^{2}$
Area of the square = side $\times$ side A = 7 $\times$ 7 = 49 cm$^{2}$
2

### The height of a triangle is 25 cm and the base is 50 cm. What is the area of the triangle?

600 cm$^{2}$
625 cm$^{2}$
650 cm$^{2}$
700 cm$^{2}$
CorrectIncorrect
Correct answer is: 625 cm$^{2}$
Area of the triangle = $\frac{1}{2}\times b\times h = \frac{1}{2}\times 50\times 25$ = 625 cm$^{2}$
3

### What is the area of a circle that has a radius of 4 cm?

40 cm2
16 cm$^{2}$
60.27 cm$^{2}$
50.27 cm$^{2}$
CorrectIncorrect
Correct answer is: 50.27 cm$^{2}$
Area of the circle = 𝜋r$^{2}$ = 3.14$\times 4\times 4$ = 50.27 cm$^{2}$
4

### The base of a parallelogram is 50 cm and the perpendicular height is 20 cm. What is the area of this parallelogram?

400 cm$^{2}$
500 cm$^{2}$
750 cm$^{2}$
1000 cm$^{2}$
CorrectIncorrect
Correct answer is: 1000 cm$^{2}$
Area of the parallelogram = b$\times$ h = 50$\times$ 20 = 1000 cm$^{2}$

Conclusion

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