The area of a triangle, knowing the lengths of the sides. How to calculate the area of a triangle
You can find over 10 formulas for calculating the area of a triangle on the Internet. Many of them are used in problems with known sides and angles of the triangle. However, there are a number of complex examples where, according to the conditions of the assignment, only one side and angles of a triangle are known, or the radius of a circumscribed or inscribed circle and one more characteristic. In such cases, a simple formula cannot be applied.
The formulas given below will allow you to solve 95 percent of problems in which you need to find the area of a triangle.
Let's move on to consider common area formulas.
Consider the triangle shown in the figure below
In the figure and below in the formulas, the classical designations of all its characteristics are introduced.
a,b,c – sides of the triangle,
R – radius of the circumscribed circle,
r – radius of the inscribed circle,
h[b],h[a],h[c] – heights drawn in accordance with sides a,b,c.
alpha, beta, hamma – angles near the vertices.
Basic formulas for the area of a triangle
1. The area is equal to half the product of the side of the triangle and the height lowered to this side. In the language of formulas, this definition can be written as follows
Thus, if the side and height are known, then every student will find the area.
By the way, from this formula one can derive one useful relationship between heights
2. If we take into account that the height of a triangle through the adjacent side is expressed by the dependence
Then the first area formula is followed by the second ones of the same type
Look carefully at the formulas - they are easy to remember, since the work involves two sides and the angle between them. If we correctly designate the sides and angles of the triangle (as in the figure above), we will get two sides a, b and the angle is connected to the third With (hamma).
3. For the angles of a triangle, the relation is true
The dependence allows you to use the following formulas for the area of a triangle in calculations:
Examples of this dependence are extremely rare, but you must remember that there is such a formula.
4. If the side and two adjacent angles are known, then the area is found by the formula
5. The formula for area in terms of side and cotangent of adjacent angles is as follows
By rearranging the indexes you can get dependencies for other parties.
6. The area formula below is used in problems when the vertices of a triangle are specified on the plane by coordinates. In this case, the area is equal to half the determinant taken modulo.
7. Heron's formula used in examples with known sides of a triangle.
First find the semi-perimeter of the triangle
And then determine the area using the formula
or
It is quite often used in the code of calculator programs.
8. If all the heights of the triangle are known, then the area is determined by the formula
It is difficult to calculate on a calculator, but in the MathCad, Mathematica, Maple packages the area is “time two”.
9. The following formulas use the known radii of inscribed and circumscribed circles. 
In particular, if the radius and sides of the triangle, or its perimeter, are known, then the area is calculated according to the formula
10. In examples where the sides and the radius or diameter of the circumscribed circle are given, the area is found using the formula
11. The following formula determines the area of a triangle in terms of the side and angles of the triangle.
And finally - special cases:
Area of a right triangle with legs a and b equal to half their product
Formula for the area of an equilateral (regular) triangle=
= one-fourth of the product of the square of the side and the root of three.
Concept of area
The concept of the area of any geometric figure, in particular a triangle, will be associated with a figure such as a square. For the unit area of any geometric figure we will take the area of a square whose side is equal to one. For completeness, let us recall two basic properties for the concept of areas of geometric figures.
Property 1: If geometric figures are equal, then their areas are also equal.
Property 2: Any figure can be divided into several figures. Moreover, the area of the original figure is equal to the sum of the areas of all its constituent figures.
Let's look at an example.
Example 1
Obviously, one of the sides of the triangle is a diagonal of a rectangle, one side of which has a length of $5$ (since there are $5$ cells), and the other is $6$ (since there are $6$ cells). Therefore, the area of this triangle will be equal to half of such a rectangle. The area of the rectangle is
Then the area of the triangle is equal to
Answer: $15$.
Next, we will consider several methods for finding the areas of triangles, namely using the height and base, using Heron’s formula and the area of an equilateral triangle.
How to find the area of a triangle using its height and base
Theorem 1
The area of a triangle can be found as half the product of the length of a side and the height to that side.
Mathematically it looks like this
$S=\frac(1)(2)αh$
where $a$ is the length of the side, $h$ is the height drawn to it.
Proof.
Consider a triangle $ABC$ in which $AC=α$. The height $BH$ is drawn to this side, which is equal to $h$. Let's build it up to the square $AXYC$ as in Figure 2.
The area of rectangle $AXBH$ is $h\cdot AH$, and the area of rectangle $HBYC$ is $h\cdot HC$. Then
$S_ABH=\frac(1)(2)h\cdot AH$, $S_CBH=\frac(1)(2)h\cdot HC$
Therefore, the required area of the triangle, by property 2, is equal to
$S=S_ABH+S_CBH=\frac(1)(2)h\cdot AH+\frac(1)(2)h\cdot HC=\frac(1)(2)h\cdot (AH+HC)=\ frac(1)(2)αh$
The theorem has been proven.
Example 2
Find the area of the triangle in the figure below if the cell has an area equal to one
The base of this triangle is equal to $9$ (since $9$ is $9$ squares). The height is also $9$. Then, by Theorem 1, we get
$S=\frac(1)(2)\cdot 9\cdot 9=40.5$
Answer: $40.5$.
Heron's formula
Theorem 2
If we are given three sides of a triangle $α$, $β$ and $γ$, then its area can be found as follows
$S=\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$
here $ρ$ means the semi-perimeter of this triangle.
Proof.
Consider the following figure:
By the Pythagorean theorem, from the triangle $ABH$ we obtain
From the triangle $CBH$, according to the Pythagorean theorem, we have
$h^2=α^2-(β-x)^2$
$h^2=α^2-β^2+2βx-x^2$
From these two relations we obtain the equality
$γ^2-x^2=α^2-β^2+2βx-x^2$
$x=\frac(γ^2-α^2+β^2)(2β)$
$h^2=γ^2-(\frac(γ^2-α^2+β^2)(2β))^2$
$h^2=\frac((α^2-(γ-β)^2)((γ+β)^2-α^2))(4β^2)$
$h^2=\frac((α-γ+β)(α+γ-β)(γ+β-α)(γ+β+α))(4β^2)$
Since $ρ=\frac(α+β+γ)(2)$, then $α+β+γ=2ρ$, which means
$h^2=\frac(2ρ(2ρ-2γ)(2ρ-2β)(2ρ-2α))(4β^2)$
$h^2=\frac(4ρ(ρ-α)(ρ-β)(ρ-γ))(β^2 )$
$h=\sqrt(\frac(4ρ(ρ-α)(ρ-β)(ρ-γ))(β^2))$
$h=\frac(2)(β)\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$
By Theorem 1, we get
$S=\frac(1)(2) βh=\frac(β)(2)\cdot \frac(2)(β) \sqrt(ρ(ρ-α)(ρ-β)(ρ-γ) )=\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$
Area of a triangle - formulas and examples of problem solving
Below are formulas for finding the area of an arbitrary triangle which are suitable for finding the area of any triangle, regardless of its properties, angles or sizes. The formulas are presented in the form of a picture, with explanations for their application or justification for their correctness. Also, a separate figure shows the correspondence between the letter symbols in the formulas and the graphic symbols in the drawing.
Note . If the triangle has special properties (isosceles, rectangular, equilateral), you can use the formulas given below, as well as additional special formulas that are valid only for triangles with these properties:
- "Formula for the area of an equilateral triangle"
Triangle area formulas
Explanations for formulas:
a, b, c- the lengths of the sides of the triangle whose area we want to find
r- radius of the circle inscribed in the triangle
R- radius of the circle circumscribed around the triangle
h- height of the triangle lowered to the side
p- semi-perimeter of a triangle, 1/2 the sum of its sides (perimeter)
α
- angle opposite to side a of the triangle
β
- angle opposite to side b of the triangle
γ
- angle opposite to side c of the triangle
h a, h b , h c- height of the triangle lowered to sides a, b, c
Please note that the given notations correspond to the figure above, so that when solving a real geometry problem, it will be visually easier for you to substitute the correct values in the right places in the formula.
- The area of the triangle is half the product of the height of the triangle and the length of the side by which this height is lowered(Formula 1). The correctness of this formula can be understood logically. The height lowered to the base will split an arbitrary triangle into two rectangular ones. If you build each of them into a rectangle with dimensions b and h, then obviously the area of these triangles will be equal to exactly half the area of the rectangle (Spr = bh)
- The area of the triangle is half the product of its two sides and the sine of the angle between them(Formula 2) (see an example of solving a problem using this formula below). Even though it seems different from the previous one, it can easily be transformed into it. If we lower the height from angle B to side b, it turns out that the product of side a and the sine of angle γ, according to the properties of the sine in a right triangle, is equal to the height of the triangle we drew, which gives us the previous formula
- The area of an arbitrary triangle can be found through work half the radius of the circle inscribed in it by the sum of the lengths of all its sides(Formula 3), simply put, you need to multiply the semi-perimeter of the triangle by the radius of the inscribed circle (this is easier to remember)
- The area of an arbitrary triangle can be found by dividing the product of all its sides by 4 radii of the circle circumscribed around it (Formula 4)
- Formula 5 is finding the area of a triangle through the lengths of its sides and its semi-perimeter (half the sum of all its sides)
- Heron's formula(6) is a representation of the same formula without using the concept of semi-perimeter, only through the lengths of the sides
- The area of an arbitrary triangle is equal to the product of the square of the side of the triangle and the sines of the angles adjacent to this side divided by the double sine of the angle opposite to this side (Formula 7)
- The area of an arbitrary triangle can be found as the product of two squares of the circle circumscribed around it by the sines of each of its angles. (Formula 8)
- If the length of one side and the values of two adjacent angles are known, then the area of the triangle can be found as the square of this side divided by the double sum of the cotangents of these angles (Formula 9)
- If only the length of each of the heights of the triangle is known (Formula 10), then the area of such a triangle is inversely proportional to the lengths of these heights, as according to Heron’s Formula
- Formula 11 allows you to calculate area of a triangle based on the coordinates of its vertices, which are specified as (x;y) values for each of the vertices. Please note that the resulting value must be taken modulo, since the coordinates of individual (or even all) vertices may be in the region of negative values
Note. The following are examples of solving geometry problems to find the area of a triangle. If you need to solve a geometry problem that is not similar here, write about it in the forum. In solutions, instead of the "square root" symbol, the sqrt() function can be used, in which sqrt is the square root symbol, and the radical expression is indicated in parentheses.Sometimes for simple radical expressions the symbol can be used √
Task. Find the area given two sides and the angle between them
The sides of the triangle are 5 and 6 cm. The angle between them is 60 degrees. Find the area of the triangle.
Solution.
To solve this problem, we use formula number two from the theoretical part of the lesson.
The area of a triangle can be found through the lengths of two sides and the sine of the angle between them and will be equal to
S=1/2 ab sin γ
Since we have all the necessary data for the solution (according to the formula), we can only substitute the values from the problem conditions into the formula:
S = 1/2 * 5 * 6 * sin 60
In the table of values of trigonometric functions, we will find and substitute the value of sine 60 degrees into the expression. It will be equal to the root of three times two.
S = 15 √3 / 2
Answer: 7.5 √3 (depending on the teacher’s requirements, you can probably leave 15 √3/2)
Task. Find the area of an equilateral triangle
Find the area of an equilateral triangle with side 3 cm.
Solution .
The area of a triangle can be found using Heron's formula:
S = 1/4 sqrt((a + b + c)(b + c - a)(a + c - b)(a + b -c))
Since a = b = c, the formula for the area of an equilateral triangle takes the form:
S = √3 / 4 * a 2
S = √3 / 4 * 3 2
Answer: 9 √3 / 4.
Task. Change in area when changing the length of the sides
How many times will the area of the triangle increase if the sides are increased by 4 times?
Solution.
Since the dimensions of the sides of the triangle are unknown to us, to solve the problem we will assume that the lengths of the sides are respectively equal to arbitrary numbers a, b, c. Then, in order to answer the question of the problem, we will find the area of the given triangle, and then we will find the area of the triangle whose sides are four times larger. The ratio of the areas of these triangles will give us the answer to the problem.
Below we provide a textual explanation of the solution to the problem step by step. However, at the very end, this same solution is presented in a more convenient graphical form. Those interested can immediately go down the solutions.
To solve, we use Heron’s formula (see above in the theoretical part of the lesson). It looks like this:
S = 1/4 sqrt((a + b + c)(b + c - a)(a + c - b)(a + b -c))
(see first line of picture below)
The lengths of the sides of an arbitrary triangle are specified by the variables a, b, c.
If the sides are increased by 4 times, then the area of the new triangle c will be:
S 2 = 1/4 sqrt((4a + 4b + 4c)(4b + 4c - 4a)(4a + 4c - 4b)(4a + 4b -4c))
(see second line in the picture below)
As you can see, 4 is a common factor that can be taken out of brackets from all four expressions according to the general rules of mathematics.
Then
S 2 = 1/4 sqrt(4 * 4 * 4 * 4 (a + b + c)(b + c - a)(a + c - b)(a + b -c)) - on the third line of the picture
S 2 = 1/4 sqrt(256 (a + b + c)(b + c - a)(a + c - b)(a + b -c)) - fourth line
The square root of the number 256 is perfectly extracted, so let’s take it out from under the root
S 2 = 16 * 1/4 sqrt((a + b + c)(b + c - a)(a + c - b)(a + b -c))
S 2 = 4 sqrt((a + b + c)(b + c - a)(a + c - b)(a + b -c))
(see fifth line of the picture below)
To answer the question asked in the problem, we just need to divide the area of the resulting triangle by the area of the original one.
Let us determine the area ratios by dividing the expressions by each other and reducing the resulting fraction.
A triangle is the simplest geometric figure, which consists of three sides and three vertices. Due to its simplicity, the triangle has been used since ancient times to take various measurements, and today the figure can be useful for solving practical and everyday problems.
Features of a triangle
The figure has been used for calculations since ancient times, for example, land surveyors and astronomers operate with the properties of triangles to calculate areas and distances. It is easy to express the area of any n-gon through the area of this figure, and this property was used by ancient scientists to derive formulas for the areas of polygons. Constant work with triangles, especially the right triangle, became the basis for an entire branch of mathematics - trigonometry.
Triangle geometry
The properties of the geometric figure have been studied since ancient times: the earliest information about the triangle was found in Egyptian papyri from 4,000 years ago. Then the figure was studied in Ancient Greece and the greatest contributions to the geometry of the triangle were made by Euclid, Pythagoras and Heron. The study of the triangle never ceased, and in the 18th century, Leonhard Euler introduced the concept of the orthocenter of a figure and the Euler circle. At the turn of the 19th and 20th centuries, when it seemed that absolutely everything was known about the triangle, Frank Morley formulated the theorem on angle trisectors, and Waclaw Sierpinski proposed the fractal triangle.
There are several types of flat triangles that are familiar to us from school geometry courses:
- acute - all the corners of the figure are acute;
- obtuse - the figure has one obtuse angle (more than 90 degrees);
- rectangular - the figure contains one right angle equal to 90 degrees;
- isosceles - a triangle with two equal sides;
- equilateral - a triangle with all equal sides.
- There are all kinds of triangles in real life, and in some cases we may need to calculate the area of a geometric figure.
Area of a triangle
Area is an estimate of how much of the plane a figure encloses. The area of a triangle can be found in six ways, using the sides, height, angles, radius of the inscribed or circumscribed circle, as well as using Heron's formula or calculating the double integral along the lines bounding the plane. The simplest formula for calculating the area of a triangle is:
where a is the side of the triangle, h is its height.
However, in practice it is not always convenient for us to find the height of a geometric figure. The algorithm of our calculator allows you to calculate the area knowing:
- three sides;
- two sides and the angle between them;
- one side and two corners.
To determine the area through three sides, we use Heron's formula:
S = sqrt (p × (p-a) × (p-b) × (p-c)),
where p is the semi-perimeter of the triangle.
The area on two sides and an angle is calculated using the classic formula:
S = a × b × sin(alfa),
where alfa is the angle between sides a and b.
To determine the area in terms of one side and two angles, we use the relation that:
a / sin(alfa) = b / sin(beta) = c / sin(gamma)
Using a simple proportion, we determine the length of the second side, after which we calculate the area using the formula S = a × b × sin(alfa). This algorithm is fully automated and you only need to enter the specified variables and get the result. Let's look at a couple of examples.
Examples from life
Paving slabs
Let's say you want to pave the floor with triangular tiles, and to determine the amount of material needed, you need to know the area of \u200b\u200bone tile and the area of the floor. Suppose you need to process 6 square meters of surface using a tile whose dimensions are a = 20 cm, b = 21 cm, c = 29 cm. Obviously, to calculate the area of a triangle, the calculator uses Heron’s formula and gives the result:
Thus, the area of one tile element will be 0.021 square meters, and you will need 6/0.021 = 285 triangles for the floor improvement. The numbers 20, 21 and 29 form a Pythagorean triple - numbers that satisfy . And that's right, our calculator also calculated all the angles of the triangle, and the gamma angle is exactly 90 degrees.
School task
In a school problem, you need to find the area of a triangle, knowing that side a = 5 cm, and angles alpha and beta are 30 and 50 degrees, respectively. To solve this problem manually, we would first find the value of side b using the proportion of the aspect ratio and the sines of the opposite angles, and then determine the area using the simple formula S = a × b × sin(alfa). Let's save time, enter the data into the calculator form and get an instant answer
When using the calculator, it is important to indicate the angles and sides correctly, otherwise the result will be incorrect.
Conclusion
The triangle is a unique figure that is found both in real life and in abstract calculations. Use our online calculator to determine the area of triangles of any kind.
Sometimes in life there are situations when you have to delve into your memory in search of long-forgotten school knowledge. For example, you need to determine the area of a triangular-shaped plot of land, or the time has come for another renovation in an apartment or private house, and you need to calculate how much material will be needed for a surface with a triangular shape. There was a time when you could solve such a problem in a couple of minutes, but now you are desperately trying to remember how to determine the area of a triangle?
Don't worry about it! After all, it is quite normal when a person’s brain decides to transfer long-unused knowledge somewhere to a remote corner, from which sometimes it is not so easy to extract it. So that you don’t have to struggle with searching for forgotten school knowledge to solve such a problem, this article contains various methods that make it easy to find the required area of a triangle.
It is well known that a triangle is a type of polygon that is limited to the minimum possible number of sides. In principle, any polygon can be divided into several triangles by connecting its vertices with segments that do not intersect its sides. Therefore, knowing the triangle, you can calculate the area of almost any figure.
Among all the possible triangles that occur in life, the following particular types can be distinguished: and rectangular.
The easiest way to calculate the area of a triangle is when one of its angles is right, that is, in the case of a right triangle. It is easy to see that it is half a rectangle. Therefore, its area is equal to half the product of the sides that form a right angle with each other.
If we know the height of a triangle, lowered from one of its vertices to the opposite side, and the length of this side, which is called the base, then the area is calculated as half the product of the height and the base. This is written using the following formula:
S = 1/2*b*h, in which
S is the required area of the triangle;
b, h - respectively, the height and base of the triangle.
It is so easy to calculate the area of an isosceles triangle because the height will bisect the opposite side and can be easily measured. If the area is determined, then it is convenient to take the length of one of the sides forming a right angle as the height.
All this is of course good, but how to determine whether one of the angles of a triangle is right or not? If the size of our figure is small, then we can use a construction angle, a drawing triangle, a postcard or another object with a rectangular shape.
But what if we have a triangular plot of land? In this case, proceed as follows: count from the top of the supposed right angle on one side a distance multiple of 3 (30 cm, 90 cm, 3 m), and on the other side measure a distance multiple of 4 in the same proportion (40 cm, 160 cm, 4 m). Now you need to measure the distance between the end points of these two segments. If the result is a multiple of 5 (50 cm, 250 cm, 5 m), then we can say that the angle is right.
If the length of each of the three sides of our figure is known, then the area of the triangle can be determined using Heron's formula. In order for it to have a simpler form, a new value is used, which is called semi-perimeter. This is the sum of all the sides of our triangle, divided in half. After the semi-perimeter has been calculated, you can begin to determine the area using the formula:
S = sqrt(p(p-a)(p-b)(p-c)), where
sqrt - square root;
p - semi-perimeter value (p = (a+b+c)/2);
a, b, c - edges (sides) of the triangle.
But what if the triangle has an irregular shape? There are two possible ways here. The first of them is to try to divide such a figure into two right triangles, the sum of the areas of which is calculated separately, and then added. Or, if the angle between two sides and the size of these sides are known, then apply the formula:
S = 0.5 * ab * sinC, where
a,b - sides of the triangle;
c is the size of the angle between these sides.
The latter case is rare in practice, but nevertheless, everything is possible in life, so the above formula will not be superfluous. Good luck with your calculations!
