The Area of a Triangle

🏆Practice triangle area

The Formula For Calculating The Area Of A Triangle

The formula for calculating the area of a triangle of any type:

height times base divided by $2$.

$Area=\frac{Base\times Height}{2}$

Test yourself on triangle area!

Complete the sentence:

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

How to Calculate the Area of an Isosceles Triangle

Information:

• The side $CB$ has a length of $14$ cm.
• The height has a length of $17$ cm.

If we apply the formula, we multiply the height ($17$ cm) by the length of the base ($14$ cm).

By multiplying $17$ by $14$, we obtain $238$, a result that we must divide by $2$.

$238$ divided by $2$ equals $119$.

The area of this triangle is therefore: $119$.

$Area=\frac{14\times17}{2}=119$

Why is it Important to Know How to Calculate the Area of a Triangle?

Whether you are preparing for an test or are about to take your university entrance exams, it is essential to know how to calculate the area of a triangle, whether it is right angled, isosceles, etc.

So, how do you calculate a triangular area? This guide will clear up all of your doubts concerning one of the most frequently asked questions in geometry examinations.

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Characteristics of the Triangle

A triangle is a geometric figure composed of three sides that form three angles and three vertices.

The vertices of the triangle are marked with the letters $A,B$ and $C$; together creating the sides: $AB, BC, CA$.

There are different types of triangles and some of them even share characteristics. We will deal with those later in this article!

Before addressing the different types of triangles that exist and how to calculate their areas,
we must first know the terms that are usually used when we talk about a triangular area.

Terms Useful When Calculating a Triangular Area and About Triangles in General

• Straight line: A line extending in the same direction and having an infinite number of points.
• Segment: A fragment of a line between two points.
• Height: The height of a triangle is the length of a perpendicular line segment starting on one side and intersecting the angle opposite. The height is denoted by the letter 'h'.
• Median: The median is the segment extending from a given vertex to the midpoint of the side opposite that vertex.
• Bisector: A ray extending from a given vertex, dividing it into two equal angles.
• Perpendicular bisector: A line that is perpendicular to a side of the triangle and that passes through the midpoint of that side.
• Median segment: A median segment of a triangle is a segment that connects the midpoints of two sides of a triangle and is arranged parallel to the third side, its length being half of the third side.
• Opposite side: An opposite side is one that faces a given vertex.

Do you know what the answer is?

How to Calculate the Area of a Triangle

One of the most useful tips when calculating the area of a triangle (and thus solving the problem) is to understand that a triangle is a half-square.

There are triangles that are easily distinguishable as "half-squares" because of their shape, such as the isosceles right triangle.

However, it is important to note that triangles that do not appear to be "half-squares" are also "half-squares", since this is one of the features that characterize them.

What is the next step?

Calculate the area of the triangle.
The formula used to calculate the area of a triangle is as follows:

height times base divided by $2$.

$Area=\frac{Base\times Height}{2}$

How to Calculate the Area of Different Types of Triangles

Calculate the Area of an Equilateral Triangle

Information:

• The side $CB$ has a length of $15$ cm.
• The height has a length of $13$ cm.

Solution:

If we apply the formula, we multiply the height ($13$ cm) by the length of the base ($15$ cm).
By multiplying $13$ by $15$, we get $195$, a result that we must divide by $2$.

$192$ divided by $2$ equals $97.5$.

The area of this triangle is therefore: $97.5$.

$Area=\frac{13\times15}{2}=97.5$

Calculating the Area of an Isosceles Triangle

Information:

• The side $CB$ has a length of $14$ cm.
• The height has a length of $17$ cm.

Solution:

Again, if we apply the formula, we multiply the height ($17$ cm) by the length of the base ($14$ cm).
By multiplying $17$ by $14$, we get $238$, a result that we must divide by $2$.

$238$ divided by $2$ equals $119$.

Therefore, the area of this triangle is: $119$.

$Area=\frac{14\times17}{2}=119$

Calculating the Area of a Scalene Triangle

Information:

• The side $CB$ has a length of $9$ cm.
• The height has a length of $10$ cm.

Solution:

If we apply the formula, we multiply the height ($10$ cm) by the length of the base ($9$ cm).
By multiplying $10$ by $9$ we get $90$, a result that we must divide by $2$.

$90$ by $2$ equals $45$.

Therefore, the area of this triangle is $45$ cm².

$Area=\frac{9\times10}{2}=45$

Do you think you will be able to solve it?

Calculating the Area of a Right Triangle

Information:

• The side $CB$ has a length of $12$ cm.
• The height has a length of $14$ cm.

Note: in a right triangle or right-angled triangle, the base and the height correspond to the legs of the triangle.

Solution:

If we apply the formula, we multiply the height ($14$ cm) by the length of the base ($12$ cm).
By multiplying $14$ by $12$ we get $168$, a result that we must divide by $2$.

$168$ by $2$ equals $84$.

Therefore, the area of this triangle is: $84$.

$Area=\frac{12\times14}{2}=84$

Calculating the Area of an Obtuse Triangle

Information:

• The side $CB$ has a length of $13$ cm.
• The height has a length of $16$ cm.

In an obtuse triangle, the height is outside the triangle.
This means that we must extend the line of the base from point
$C$ to the point $D$ to find the height.

In this way we create a right triangle $\triangle ABD$,where the height we are looking for is the side $AD$.

However, remember that since we are trying to calculate the area of the obtuse triangle, we only have to consider the side as the base. $CB$ is the base.

Solution:
In this case, if we apply the formula, we multiply the height ($16$ cm) by the length of the base of the triangle whose area we want to find.
By multiplying $16$ by $13$ we obtain $208$, a result that we must divide by $2$.

$208$ divided by $2$ equals $104$.

Therefore, the area of this triangle is: $104$.

$Area=\frac{13\times16}{2}=104$

Heron's Formula

What is Heron's formula and what is it for?

Heron's formula, the invention of which is attributed to the Greek mathematician Heron of Alexandria, allows us to obtain the area of a triangle knowing the lengths of its three sides $a, b$ and $c$.

$A\text{rea}=\sqrt{s(s-a)(s-b)(s-c)}$

Where 's' is the perimeter of the triangle divided by $2$:

$s=\frac{a+b+c}{2}$

Exercises to Calculate the Area of a Triangle

Exercise 1

In the grounds of a hotel, they want to build a special triangular-shaped pool.

The length of the pool is $10$ m.

The width of the pool is $8$ m.

The pool is to be covered with tiles $2$ m long and $2$ m wide.

Question:

How many tiles are needed to cover the pool area?

Solution:

To find out how many tiles are needed, we must calculate the triangular area of the pool and the area of each tile, then divide these numbers.

$\frac{\text{S.triangle}}{S.tile}$

The result is equal to the number of tiles needed.

In a triangle, its length is equal to its height and its width is equal to the base of the triangle.

$\text{S.triangle=}\frac{10\cdot8}{2}=40$

h = length = $10$ meters.

base = width = $8$ meters.

Since the length is $2$ meters, the width is also $2$ meters.

Tile area: $2\cdot2=4$

$\frac{40}{4}=10$

$10$ tiles

Do you know what the answer is?

Exercise 2

The triangle $\triangle ABC$ is a right-angled triangle.

The area of the triangle is equal to $6cm^2$.

Calculate $X$ and the length of the side $BC$.

Solution:

We use the formula to calculate the area of the right triangle:

$\frac{AC\cdot BC}{2}=\frac{leg\times leg}{2}$

Then compare the expression with the area of the triangle ($6$).

$\frac{4\cdot(X-1)}{2}=6$

Multiplying the equation by the common denominator means that we multiply by $2$.

$4(X-1)=12$

We open the parentheses before the distributive property:

$4X-4=12$ / $+4$

$4X=16$ / $:4$

$X=4$

Replace $X=4$ into the expression $BC$ and we find:

$BC=X-1=4-1=3$

$BC=3$

$X=4$

Exercise 3

Given the triangle $\triangle PRS$, calculate the height $PQ$.

The length of the side $SR$ is equal to $4\operatorname{cm}$.

The area of the triangle $PSR$ is equal to $30$ cm².

Solution:

We use the formula to calculate the area of the triangle.

Please note: in the obtuse triangle, the height is outside of the triangle!

$\frac{Side\cdot\text{Height}}{2}=Triangular Area$

Double the equation by a common denominator.

$\frac{4\cdot PQ}{2}=30$ / $\cdot2$

Divide the equation by the coefficient of $PQ$.

$4PQ=60$ / $:4$

$PQ=15$

The length of the height $PQ$ is equal to $15 cm$.

Exercise 4

Given the right triangle $\triangle ADB$, calculate the area of the triangle. $\triangle ABC$.

The perimeter of the triangle is equal to $30\operatorname{ cm}$.

Given: $AB=15, AC=13, DC=5, CB=4$.

Solution:

Given that the perimeter of the triangle $ΔADC$ is equal to $30 cm$, we can calculate $AD$

$AD+DC+AD=PerimeterΔADC$

$AD+5+13=30$

$AD+18=30$ /$-18$

$AD=12$

Now we can calculate the area of the triangle $ΔABC$.

Please note: we are talking about an obtuse triangle, so its height is $AD$.

We use the formula to calculate the area of the triangle:

$\frac{sideheight\times side}{2}=$

$\frac{AD\cdot BC}{2}=\frac{12\cdot4}{2}=\frac{48}{2}=24$

The area of the triangle $ΔABC$ is equal to $24~cm²$.

Exercise 5

The triangle $ΔABC$ is isosceles. Therefore, $AB=AC$.

$AD$ is the height from the side $BC$.

Since $DC=10$, the length of the height $AD$ is $20%$ percent longer than the length of the side $BC$.

Calculate the area of the triangle $ΔABC$.

Solution:

Since it is an isosceles triangle (and, therefore, median), $DC=10$ and $BC=20$.

The height $AD$ is $20%$ percent longer than the length of $BC$.

That is:

$AD=1.2\cdot BC$

$\frac{100}{100}+\frac{20}{100}=\frac{120}{100}=1.2$

$AD=1.2\cdot20=24$

Hence, the area of the triangle $ΔABC$:

$SΔ\text{ABC}=\frac{AD\cdot BC}{2}=\frac{24\cdot20}{2}=\frac{480}{2}=240$

The area of the triangle $ΔABC$ is equal to $240~cm²$.

Do you think you will be able to solve it?

When are the rest of the terms we learned used?

The rest of the terms, such as median, bisector, etc., are used when we are missing some data. These terms help us to find new data when we have to solve a problem in which we are missing information.

Mistakes you may also make when you study for the exam...

Many students experience a sense of failure when they don't do as well in their exams as they would have liked. However, success in this regard is subjective. Instead of comparing yourself with your peers, you must take into consideration your own achievements and leave aside the results of your classmates. Often, the problem is not that you did not know something or that you did not understand how to calculate the area of a triangle, but rather that you did not prepare well for the exam.

To give an example: imagine an excellent pastry chef who knows many recipes, knows the products and succeeds in creating truly delicious pastries. If he had not practiced and had not prepared well (buying the products and appliances he needs, finding good recipes and having patience with the right timing, etc.), he would not have achieved good results. This is also true for students—good preparation is key!

What other mistakes are you making that are holding you back?

• Studying too intensively. Some students study for a week before the exam, perhaps ten hours a day or so. It is obvious that they are motivated and have good intentions, but the problem here is that sometimes this causes them to run out of energy before they even get to the exam. As a result, they are tired and burned out without enough time to go over all the material they are going to be tested on.
• Too much self-confidence. Did you get a 10 on the mock exam on triangular areas? If so, this does not mean that you should study two days before the exam. An exam requires preparation, both mental and practical. To prepare for an exam, it is recommended to study for at least a week.
• Stress and nerves. If exams make you anxious, it is best to start working on it beforehand. Students who suffer from this anxiety are usually those who know the subject matter inside out, but their self-confidence affects them negatively. Prepare yourself mentally because, otherwise, you may draw a blank on the exam and this will be reflected in your results.

Studying for the geometry exam with a stopwatch - why is it worth it?

Your study skills are as important as learning the subject matter. For this reason, a stopwatch can be your great ally. We recommend that you adopt its use from now on. The one that often comes with a cell phone is more than enough!

Once you have studied a subject area, it is advisable to practice and solve problems with a stopwatch, even when you are not preparing for exams. Why?

• The stopwatch gives you an indication of how long it takes you to solve a problem.
• The stopwatch lets you know which subject areas are your weakest.
• Thanks to using a stopwatch, you can also measure how fast you're progressing—if you're taking less time, you've made progress!

Do you know what the answer is?

You were expecting a 9, but you got a 7. What do you do now?

Many students have a hard time digesting the fact that they got a low grade and, more importantly, the disappointment that often accompanies it. It is very important for you to know that the more you let this affect you, the worse the impact on your academic achievement will be.

The grade you get on an exam is a kind of feedback that tells you what you are doing well and what you could improve. How can this feedback change the way you study?

• Use private tutors to reinforce your knowledge.
• Include extra study days before each exam.

Is it better for me to take private lessons with a friend rather than on my own?

We do not recommend it and the reason is very simple: a private lesson is "private", that is to say, it is adapted to your needs. When two friends study in the same private class, one of them will have to adapt to the rhythm of the other.

Thus, the idea of tailoring a "private lesson" to the student's needs is diluted. That said, if both you and your friend find the same topic difficult (for example, how to calculate the area of a triangle), you can take a joint private lesson.

Other methods to calculate the area of a triangle

If the question asks about a right triangle, you can multiply the legs (the sides of the triangle that are not the base) and divide by 2. This method is often a great shortcut to reaching the solution. That is why it is important that you know the formula and this specific feature of this triangle.

Also, if you are asked about an isosceles triangle, you should know that both the bisector and the median are considered the height of the triangle. With this knowledge, you can quickly work out the area of the triangle.

If you are interested in learning more about other triangle topics, you can access one of the following articles:

If you are interested in learning how to calculate areas of other geometric shapes, you can enter one of the following articles:

In Tutorela you will find a variety of articles about mathematics!

Do you think you will be able to solve it?

Help from a math tutor: when is it needed?

Many times, the study of mathematics arouses some anxiety among students in high school and further education. A private math class is ideal for those who want to get good grades on their exams, but don't know how.

A private lesson focuses on a certain aspect and includes more than just doing exercises, like:

• Learning to read the statement and understand what is being asked of you.
• An emphasis on understanding what is being asked of us and how we should answer.
• Finding information that can help us solve the problem.
• Studying formulas and tricks that can help us when it comes to finding the solution to a problem or exercise.

In the past, private lessons in mathematics or any other subject took place at the student's or teacher's home. Nowadays, it is also possible to have private lessons online. This offers a great way to learn advanced subject from the comfort of your own home, at the hours that best suit both the student and the teacher.

Triangles and other geometric shapes are aspects that students are exposed to as early as their first years in high school. Their grades in mathematics are what set the pace of their learning and can affect whether or not a student chooses to continue studying the subject at later on. Often, what holds people back when studying mathematics is not based on intelligence or aptitude, but rather arises from erroneous learning methods that do not help the student understand the subject matter effectively. A private mathematics teacher works side by side with the student, ensuring that in the end the student has understood their lessons.

examples with solutions for triangle area

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

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 #3

Calculate the area of the following triangle:

Step-by-Step Solution

The formula for calculating the area of a triangle is:

(the side * the height from the side down to the base) /2

That is:

$\frac{BC\times AE}{2}$

Now we replace the existing data:

$\frac{4\times5}{2}=\frac{20}{2}=10$

10

Exercise #4

Calculate the area of the triangle below, if possible.

Step-by-Step Solution

The formula to calculate the area of a triangle is:

(side * height corresponding to the side) / 2

Note that in the triangle provided to us, we have the length of the side but not the height.

That is, we do not have enough data to perform the calculation.

Cannot be calculated

Exercise #5

What is the area of the given triangle?

Step-by-Step Solution

This question is a bit confusing. We need start by identifying which parts of the data are relevant to us.

Remember the formula for the area of a triangle:

The height is a straight line that comes out of an angle and forms a right angle with the opposite side.

In the drawing we have a height of 6.

It goes down to the opposite side whose length is 5.

And therefore, these are the data points that we will use.

We replace in the formula:

$\frac{6\times5}{2}=\frac{30}{2}=15$