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.
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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 m2 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 cm2 cm^2 square centimeter.
Remember:
Units of measurement for the area in cm=>cm2 cm => cm^2
Units of measurement for the area m=>m2 m=>m^2

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Test yourself on area of the square!

einstein

Complete the sentence:

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

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Area

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

Area of the Square

A1 - A represents the area of the square

aa Side of the square

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

A=a2 A=a^2

We will multiply the side of the square by itself

Another way:

diagonal×diagonal2=Area of the square\frac{diagonal \times diagonal}{2}=Area~ of ~the ~square

For more information, enter the link of Area of a square


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Area of the Rectangle

A2 - Rectangle area formula

a×b=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 90o 90^o degree angle)

For more information, enter the link of Rectangle area


Area of the triangle

A3 - Triangle Area Formula

height ×corresponding side2=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 90o90^o degree angle and divide the product by 22.

For more information, enter the link to Triangle Area


Do you know what the answer is?

Area of the Rhombus

A4 - Rhombus area formula

aa –> Side of the rhombus
hh –>  Height

a×h=Area of the rhombusa\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 90o 90^o degrees.

Another way :

diagonal×diagonal2=Area of the rhombus\frac{diagonal\times diagonal}{2}=Area~ of~ the~ rhombus

For more information, enter the link of Rhombus area


Area of the parallelogram

A5 - Parallelogram area formula

HH –> Height
BB –>  The side that forms a 90o 90^o degree angle with the height HH.

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

B×H=Area of the parallelogramB\times H=Area~ of ~the~ parallelogram

For more information, enter the link of Parallelogram area


Check your understanding

Area of the Circle

R - Circle area formula

rr   The radius of the circumference
ππ  PI
It will be calculated as the number 3.14 3.14 

π×r2=Area of the circleπ\times r^2=Area~ of ~the ~circle

We will multiply PI 3.143.14 by the radius of the circumference squared, that is r2 r^2 
Or, more simply, the formula is:

r×r×3.14=Area of the circler\times r\times 3.14=Area~ of ~the ~circle

For more information, enter the link of Circle area


Area of the trapezoid

A7 - Trapezoid area formula

We will add the bases and multiply the result by the height of the trapezoid.
We will divide the result by 22.

(a+b)×h2=Area of the trapezoid\frac{(a+b)\times h}{2}=Area~ of~ the~ trapezoid

For more information, enter the link of Trapezoid area


Do you think you will be able to solve it?

Area of the Kite

A8 - Area formula of the kite

We will multiply the diagonals and divide by 22.

ac×db2=Area of the trapezoid\frac{ac\times db}{2}=Area~ of~ the~ trapezoid

For more information, enter the link of Area of the kite


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

A9 - Area of composite figures

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.


Test your knowledge

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.

A10 - Volume of the cylinder and Volume of the cube


Examples and exercises with solutions for area calculation

Exercise #1

Calculate the area of the right triangle below:

101010666888AAACCCBBB

Video Solution

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:

AB×BC2=8×62=482=24 \frac{AB\times BC}{2}=\frac{8\times6}{2}=\frac{48}{2}=24

Answer

24 cm²

Exercise #2

Given the circle whose diameter is 7 cm

What is your area?

777

Video Solution

Step-by-Step Solution

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

 πr2 \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 r=7:2=3.5

We replace in the formula

π3.52=12.25π \pi3.5^2=12.25\pi

Answer

12.25π 12.25\pi cm².

Exercise #3

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

121212888999AAABBBCCCDDD

Video Solution

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:

CB×AD2 \frac{CB\times AD}{2}

8×92=722=36 \frac{8\times9}{2}=\frac{72}{2}=36

Answer

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.

2.52.52.5444h=6h=6h=6AAABBBCCCDDD

Video Solution

Step-by-Step Solution

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

A=(Base + Base) h2 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

Answer

1912 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.

S=32S=32S=32888AAABBBCCCDDD

Video Solution

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:

 8DB2=32 \frac{8\cdot DB}{2}=32

We will reduce the 8 and the 2:

4DB=32 4DB=32

Divide by 4

DB=8 DB=8

Answer

8 cm

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