Area

🏆Practice area of the square

In this article, we will learn what area is, and understand how it is calculated for each shape, in the most practical and simple way there is.
Shall we start?

What is the area?

Area is the definition of the size of something. In mathematics, which is precisely what interests us now, it refers to the size of some figure.
In everyday life, you have surely heard about area in relation to the surface of an apartment, plot of land, etc.
In fact, when they ask what the surface area of your apartment is, they are asking about its size and, instead of answering with words like "big" or "small" we can calculate its area and express it with units of measure. In this way, we can compare different sizes.

Units of measurement of area

Large areas such as apartments are usually measured in meters, therefore, the unit of measurement will be $m^2$ square meter.
On the other hand, smaller figures are generally measured in centimeters, that is, the unit of measurement for the area will be $cm^2$ square centimeter.
Remember:
Units of measurement for the area in $cm => cm^2$
Units of measurement for the area $m=>m^2$

Test yourself on area of the square!

Complete the sentence:

To find the area of a right triangle, one must multiply ________________ by each other and divide by 2.

Area

Now we will learn to calculate the area of (almost) all the shapes we know! Are we ready?

Area of the Square

$a$ Side of the square

$a\times a=Area~ of ~the ~square$

$A=a^2$

We will multiply the side of the square by itself

Another way:

$\frac{diagonal \times diagonal}{2}=Area~ of ~the ~square$

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Area of the Rectangle

$a\times b=Area~of~the~rectangle$

We will multiply one side of the rectangle by the adjacent side (the side with which it forms a $90^o$ degree angle)

Area of the triangle

$\frac{height~\times corresponding~side}{2}=Area~ of ~the ~triangle$

We will multiply the height by the corresponding side - that is, the side with which it forms a $90^o$ degree angle and divide the product by $2$.

Do you know what the answer is?

Area of the Rhombus

$a$ –> Side of the rhombus
$h$ –>  Height

$a\times h= Area~ of ~the~ rhombus$

We will multiply the height by the corresponding side, that is, the side with which it forms a right angle of $90^o$ degrees.

Another way :

$\frac{diagonal\times diagonal}{2}=Area~ of~ the~ rhombus$

Area of the parallelogram

$H$ –> Height
$B$ –>  The side that forms a $90^o$ degree angle with the height $H$.

We will multiply the height by the side to which the height reaches and forms with it a $90^o$ degree angle.

$B\times H=Area~ of ~the~ parallelogram$

Area of the Circle

$r$   The radius of the circumference
$π$  PI
It will be calculated as the number $3.14$

$π\times r^2=Area~ of ~the ~circle$

We will multiply PI $3.14$ by the radius of the circumference squared, that is $r^2$
Or, more simply, the formula is:

$r\times r\times 3.14=Area~ of ~the ~circle$

Area of the trapezoid

We will add the bases and multiply the result by the height of the trapezoid.
We will divide the result by $2$.

$\frac{(a+b)\times h}{2}=Area~ of~ the~ trapezoid$

Do you think you will be able to solve it?

Area of the Kite

We will multiply the diagonals and divide by $2$.

$\frac{ac\times db}{2}=Area~ of~ the~ trapezoid$

Area of Composite Figures

You don't have to worry about this pair of terms - composite figures. They are not called composite because they are complicated or difficult, but rather, they are composite figures because they are really made up of several figures that you already know.
The great key to calculating the area of this type of figures is to separate them into several simple figures on which you know how to calculate their area.

Let's look at an example

At first glance, it might scare us a bit since the figure seems very strange. But, very quickly we will remember the suggestion that we have written here above and apply it.
We will realize that we can divide the composite figure into two that we know and know how to calculate their area, rectangle and square.
We will calculate the area of each figure separately and then add them together.
In this way, we will obtain the area of the entire figure.

What is the difference between surface area and volume?

To understand the difference, let's remember a daily term we use in another context: superficial.
Superficial implies something or someone without depth, so, in geometry, the surface indicates the size of something flat, without depth. For example, if we draw a ball and paint it, that painted part would be its surface.
On the other hand, volume refers to the actual size of the ball, the space that we could fill inside it.
Volume is not the surface on the sheet of paper, but, really the size we can see (in a three-dimensional way) - the space it occupies in space.
The calculation of volume differs from the calculation of the surface.

Examples and exercises with solutions for area calculation

Exercise #1

Calculate the area of the right triangle below:

Step-by-Step Solution

As we see that AB is perpendicular to BC and forms a 90-degree angle

It can be argued that AB is the height of the triangle.

Then we can calculate the area as follows:

$\frac{AB\times BC}{2}=\frac{8\times6}{2}=\frac{48}{2}=24$

24 cm²

Exercise #2

Given the circle whose diameter is 7 cm

Step-by-Step Solution

First, let's remember the formula for the area of a circle:

$\pi r^2$

In the question, we are given the diameter of the circle, but we need the radius.

It is known that the radius is actually half of the diameter, therefore:

$r=7:2=3.5$

We replace in the formula

$\pi3.5^2=12.25\pi$

$12.25\pi$ cm².

Exercise #3

Calculate the area of the triangle ABC using the data in the figure.

Step-by-Step Solution

First, let's remember the formula for the area of a triangle:

(the side * the height that descends to the side) /2

In the question, we have three pieces of data, but one of them is redundant!

We only have one height, the line that forms a 90-degree angle - AD,

The side to which the height descends is CB,

Therefore, we can use them in our calculation:

$\frac{CB\times AD}{2}$

$\frac{8\times9}{2}=\frac{72}{2}=36$

36 cm²

Exercise #4

The trapezoid ABCD is shown below.

AB = 2.5 cm

DC = 4 cm

Height (h) = 6 cm

Calculate the area of the trapezoid.

Step-by-Step Solution

First, let's remind ourselves of the formula for the area of a trapezoid:

$A=\frac{\left(Base\text{ }+\text{ Base}\right)\text{ h}}{2}$

We substitute the given values into the formula:

(2.5+4)*6 =
6.5*6=
39/2 =
19.5

$19\frac{1}{2}$

Exercise #5

Shown below is the deltoid ABCD.

The diagonal AC is 8 cm long.

The area of the deltoid is 32 cm².

Calculate the diagonal DB.

Step-by-Step Solution

First, we recall the formula for the area of a kite: multiply the lengths of the diagonals by each other and divide this product by 2.

We substitute the known data into the formula:

$\frac{8\cdot DB}{2}=32$

We will reduce the 8 and the 2:

$4DB=32$

Divide by 4

$DB=8$