The school curriculum provides for teaching children geometry from an early age. One of the most basic knowledge in this field is finding the area of various shapes. In this article we will try to give all possible ways to obtain this value, from the simplest to the most complex.
Warp
The first formula that children learn at school involves finding the area of a triangle through the length of its height and base. Height is a segment drawn from the vertex of the triangle at right angles to the opposite side, which will be the base. How to find the area of a triangle using these quantities?
If V is the height and O is the base, then the area is S=V*O:2.
Another option for obtaining the desired value requires us to know the lengths of two sides, as well as the size of the angle between them. If we have L and M - the lengths of the sides, and Q - the angle between them, then you can get the area using the formula S=(L*M*sin(Q))/2.
Heron's formula
In addition to all the other answers to the question of how to calculate the area of a triangle, there is a formula that allows us to obtain the value we need, knowing only the lengths of the sides. That is, if we know the lengths of all sides, then we do not need to draw the height and calculate its length. We can use the so-called Heron's formula.
If M, N, L are the lengths of the sides, then we can find the area of the triangle as follows. P=(M+N+L)/2, then the value we need is S 2 =P*(P-M)*(P-L)*(P-N). In the end, all we have to do is calculate the root.
For a right triangle, Heron's formula is slightly simplified. If M, L are legs, then S=(P-M)*(P-L).
Circles
Another way to find the area of a triangle is to use incircles and circumcircles. To get the value we need using an inscribed circle, we need to know its radius. Let's denote it "r". Then the formula by which we will carry out calculations will take the following form: S=r*P, where P is half of the sum of the lengths of all sides.
In a right triangle, this formula is slightly modified. Of course, you can use the one above, but it is better to use a different expression for calculations. S=E*W, where E and W are the lengths of the segments into which the hypotenuse is divided by the tangency point of the circle.
Speaking of the circumscribed circle, finding the area of the triangle is also not difficult. By introducing the designation R as the radius of the circumscribed circle, you can obtain the following formula necessary to calculate the required value: S= (M*N*L):(4*R). Where the first three quantities are the sides of the triangle.
Speaking about an equilateral triangle, through a number of simple mathematical transformations you can obtain slightly modified formulas:
S=(3 1/2 *M 2)/4;
S=(3*3 1/2 *R 2)/4;
S=3*3 1/2 *r 2 .
In any case, any formula that allows you to find the area of a triangle can be changed in accordance with the data of the task. So all written expressions are not absolutes. When solving problems, reflect to find the most appropriate solution.
Coordinates
When studying coordinate axes, the tasks facing students become more complex. However, not so much as to panic. In order to find the area of a triangle from the coordinates of the vertices, you can use the same, but slightly modified Heron's formula. For coordinates it takes the following form:
S=((x 2 -x 1) 2 *(y 2 -y 1) 2 *(z 2 -z 1) 2) 1/2.
However, no one forbids, using coordinates, calculating the lengths of the sides of a triangle and then, using the formulas written above, calculating the area. To convert coordinates to length, use the following formula:
l=((x 2 -x 1) 2 +(y 2 -y 1) 2) 1/2.
Notes
The article used standard notations for quantities that are used in most problems. In this case, the power "1/2" means that you need to extract the root of the entire expression under the brackets.
Be careful when choosing a formula. Some of them lose their relevance depending on the initial conditions. For example, the circumcircle formula. It is able to calculate the result for you in any case, but there may be a situation when a triangle with the given parameters may not exist at all.
If you are sitting at home and doing homework, then you can use an online calculator. Many sites provide the ability to calculate various quantities using given parameters, and it doesn’t matter which ones. You can simply enter the initial data into the fields, and the computer (website) will calculate the result for you. This way you can avoid mistakes made due to carelessness.
We hope our article answered all your questions regarding calculating the area of a variety of triangles, and you will not have to look for additional information elsewhere. Good luck with your studies!
A triangle is a geometric figure that consists of three straight lines connecting at points that do not lie on the same straight line. The connection points of the lines are the vertices of the triangle, which are designated by Latin letters (for example, A, B, C). The connecting straight lines of a triangle are called segments, which are also usually denoted by Latin letters. The following types of triangles are distinguished:
- Rectangular.
- Obtuse.
- Acute angular.
- Versatile.
- Equilateral.
- Isosceles.
General formulas for calculating the area of a triangle
Formula for the area of a triangle based on length and height
S= a*h/2,
where a is the length of the side of the triangle whose area needs to be found, h is the length of the height drawn to the base.
Heron's formula
S=√р*(р-а)*(р-b)*(p-c),
where √ is the square root, p is the semi-perimeter of the triangle, a,b,c is the length of each side of the triangle. The semi-perimeter of a triangle can be calculated using the formula p=(a+b+c)/2.
Formula for the area of a triangle based on the angle and the length of the segment
S = (a*b*sin(α))/2,
where b,c is the length of the sides of the triangle, sin(α) is the sine of the angle between the two sides.
Formula for the area of a triangle given the radius of the inscribed circle and three sides
S=p*r,
where p is the semi-perimeter of the triangle whose area needs to be found, r is the radius of the circle inscribed in this triangle.
Formula for the area of a triangle based on three sides and the radius of the circle circumscribed around it
S= (a*b*c)/4*R,
where a,b,c is the length of each side of the triangle, R is the radius of the circle circumscribed around the triangle.
Formula for the area of a triangle using the Cartesian coordinates of points
Cartesian coordinates of points are coordinates in the xOy system, where x is the abscissa, y is the ordinate. The Cartesian coordinate system xOy on a plane is the mutually perpendicular numerical axes Ox and Oy with a common origin at point O. If the coordinates of points on this plane are given in the form A(x1, y1), B(x2, y2) and C(x3, y3 ), then you can calculate the area of the triangle using the following formula, which is obtained from the vector product of two vectors.
S = |(x1 – x3) (y2 – y3) – (x2 – x3) (y1 – y3)|/2,
where || stands for module.
How to find the area of a right triangle
A right triangle is a triangle with one angle measuring 90 degrees. A triangle can have only one such angle.
Formula for the area of a right triangle on two sides
S= a*b/2,
where a,b is the length of the legs. Legs are the sides adjacent to a right angle.
Formula for the area of a right triangle based on the hypotenuse and acute angle
S = a*b*sin(α)/ 2,
where a, b are the legs of the triangle, and sin(α) is the sine of the angle at which the lines a, b intersect.
Formula for the area of a right triangle based on the side and the opposite angle
S = a*b/2*tg(β),
where a, b are the legs of the triangle, tan(β) is the tangent of the angle at which the legs a, b are connected.
How to calculate the area of an isosceles triangle
An isosceles triangle is one that has two equal sides. These sides are called the sides, and the other side is the base. To calculate the area of an isosceles triangle, you can use one of the following formulas.
Basic formula for calculating the area of an isosceles triangle
S=h*c/2,
where c is the base of the triangle, h is the height of the triangle lowered to the base.
Formula of an isosceles triangle based on side and base
S=(c/2)* √(a*a – c*c/4),
where c is the base of the triangle, a is the size of one of the sides of the isosceles triangle.
How to find the area of an equilateral triangle
An equilateral triangle is a triangle in which all sides are equal. To calculate the area of an equilateral triangle, you can use the following formula:
S = (√3*a*a)/4,
where a is the length of the side of the equilateral triangle.
The above formulas will allow you to calculate the required area of the triangle. It is important to remember that to calculate the area of triangles, you need to consider the type of triangle and the available data that can be used for the calculation.
To determine the area of a triangle, you can use different formulas. Of all the methods, the easiest and most frequently used is to multiply the height by the length of the base and then divide the result by two. However, this method is far from the only one. Below you can read how to find the area of a triangle using different formulas.
Separately, we will look at ways to calculate the area of specific types of triangles - rectangular, isosceles and equilateral. We accompany each formula with a short explanation that will help you understand its essence.
Universal methods for finding the area of a triangle
The formulas below use special notation. We will decipher each of them:
- a, b, c – the lengths of the three sides of the figure we are considering;
- r is the radius of the circle that can be inscribed in our triangle;
- R is the radius of the circle that can be described around it;
- α is the magnitude of the angle formed by sides b and c;
- β is the magnitude of the angle between a and c;
- γ is the magnitude of the angle formed by sides a and b;
- h is the height of our triangle, lowered from angle α to side a;
- p – half the sum of sides a, b and c.
It is logically clear why you can find the area of a triangle in this way. The triangle can easily be completed into a parallelogram, in which one side of the triangle will act as a diagonal. The area of a parallelogram is found by multiplying the length of one of its sides by the value of the height drawn to it. The diagonal divides this conditional parallelogram into 2 identical triangles. Therefore, it is quite obvious that the area of our original triangle must be equal to half the area of this auxiliary parallelogram.
S=½ a b sin γ
According to this formula, the area of a triangle is found by multiplying the lengths of its two sides, that is, a and b, by the sine of the angle formed by them. This formula is logically derived from the previous one. If we lower the height from angle β to side b, then, according to the properties of a right triangle, when we multiply the length of side a by the sine of angle γ, we obtain the height of the triangle, that is, h.
The area of the figure in question is found by multiplying half the radius of the circle that can be inscribed in it by its perimeter. In other words, we find the product of the semi-perimeter and the radius of the mentioned circle.
S= a b c/4R
According to this formula, the value we need can be found by dividing the product of the sides of the figure by 4 radii of the circle described around it.
These formulas are universal, as they make it possible to determine the area of any triangle (scalene, isosceles, equilateral, rectangular). This can be done using more complex calculations, which we will not dwell on in detail.
Areas of triangles with specific properties
How to find the area of a right triangle? The peculiarity of this figure is that its two sides are simultaneously its heights. If a and b are legs, and c becomes the hypotenuse, then we find the area like this:
How to find the area of an isosceles triangle? It has two sides with length a and one side with length b. Consequently, its area can be determined by dividing by 2 the product of the square of side a by the sine of angle γ.
How to find the area of an equilateral triangle? In it, the length of all sides is equal to a, and the magnitude of all angles is α. Its height is equal to half the product of the length of side a and the square root of 3. To find the area of a regular triangle, you need to multiply the square of side a by the square root of 3 and divide by 4.
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 is 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
$S=\frac(1)(2) βh=\frac(β)(2)\cdot \frac(2)(β) \sqrt(ρ(ρ-α)(ρ-β)(ρ-γ) )=\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$
A triangle is one of the most common geometric shapes, which we become familiar with in elementary school. Every student faces the question of how to find the area of a triangle in geometry lessons. So, what features of finding the area of a given figure can be identified? In this article we will look at the basic formulas necessary to complete such a task, and also analyze the types of triangles.
Types of triangles
You can find the area of a triangle in completely different ways, because in geometry there is more than one type of figure containing three angles. These types include:
- Obtuse.
- Equilateral (correct).
- Right triangle.
- Isosceles.
Let's take a closer look at each of the existing types of triangles.
This geometric figure is considered the most common when solving geometric problems. When the need arises to draw an arbitrary triangle, this option comes to the rescue.
In an acute triangle, as the name suggests, all the angles are acute and add up to 180°.
This type of triangle is also very common, but is somewhat less common than an acute triangle. For example, when solving triangles (that is, several of its sides and angles are known and you need to find the remaining elements), sometimes you need to determine whether the angle is obtuse or not. Cosine is a negative number.
B, the value of one of the angles exceeds 90°, so the remaining two angles can take small values (for example, 15° or even 3°).
To find the area of a triangle of this type, you need to know some nuances, which we will talk about later.
Regular and isosceles triangles
A regular polygon is a figure that includes n angles and whose sides and angles are all equal. This is what a regular triangle is. Since the sum of all the angles of a triangle is 180°, then each of the three angles is 60°.
A regular triangle, due to its property, is also called an equilateral figure.
It is also worth noting that only one circle can be inscribed in a regular triangle, and only one circle can be described around it, and their centers are located at the same point.
In addition to the equilateral type, one can also distinguish an isosceles triangle, which is slightly different from it. In such a triangle, two sides and two angles are equal to each other, and the third side (to which equal angles are adjacent) is the base.
The figure shows an isosceles triangle DEF whose angles D and F are equal and DF is the base.
Right triangle
A right triangle is so named because one of its angles is right, that is, equal to 90°. The other two angles add up to 90°.
The largest side of such a triangle, lying opposite the 90° angle, is the hypotenuse, while the remaining two sides are the legs. For this type of triangle, the Pythagorean theorem applies:
The sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
The figure shows a right triangle BAC with hypotenuse AC and legs AB and BC.
To find the area of a triangle with a right angle, you need to know the numerical values of its legs.
Let's move on to the formulas for finding the area of a given figure.
Basic formulas for finding area
In geometry, there are two formulas that are suitable for finding the area of most types of triangles, namely for acute, obtuse, regular and isosceles triangles. Let's look at each of them.
By side and height
This formula is universal for finding the area of the figure we are considering. To do this, it is enough to know the length of the side and the length of the height drawn to it. The formula itself (half the product of the base and the height) is as follows:
where A is the side of a given triangle, and H is the height of the triangle.
For example, to find the area of an acute triangle ACB, you need to multiply its side AB by the height CD and divide the resulting value by two.
However, it is not always easy to find the area of a triangle this way. For example, to use this formula for an obtuse triangle, you need to extend one of its sides and only then draw an altitude to it.
In practice, this formula is used more often than others.
On both sides and corner
This formula, like the previous one, is suitable for most triangles and in its meaning is a consequence of the formula for finding the area by side and height of a triangle. That is, the formula in question can be easily derived from the previous one. Its formulation looks like this:
S = ½*sinO*A*B,
where A and B are the sides of the triangle, and O is the angle between sides A and B.
Let us recall that the sine of an angle can be viewed in a special table named after the outstanding Soviet mathematician V. M. Bradis.
Now let's move on to other formulas that are suitable only for exceptional types of triangles.
Area of a right triangle
In addition to the universal formula, which includes the need to find the altitude in a triangle, the area of a triangle containing a right angle can be found from its legs.
Thus, the area of a triangle containing a right angle is half the product of its legs, or:
where a and b are the legs of a right triangle.
Regular triangle
This type of geometric figure is distinguished by the fact that its area can be found with the indicated value of only one of its sides (since all sides of a regular triangle are equal). So, when faced with the task of “finding the area of a triangle when the sides are equal,” you need to use the following formula:
S = A 2 *√3 / 4,
where A is the side of the equilateral triangle.
Heron's formula
The last option for finding the area of a triangle is Heron's formula. In order to use it, you need to know the lengths of the three sides of the figure. Heron's formula looks like this:
S = √p·(p - a)·(p - b)·(p - c),
where a, b and c are the sides of a given triangle.
Sometimes the problem is given: “the area of a regular triangle is to find the length of its side.” In this case, we need to use the formula we already know for finding the area of a regular triangle and derive from it the value of the side (or its square):
A 2 = 4S / √3.
Examination tasks
There are many formulas in GIA problems in mathematics. In addition, quite often it is necessary to find the area of a triangle on checkered paper.
In this case, it is most convenient to draw the height to one of the sides of the figure, determine its length from the cells and use the universal formula for finding the area:
So, after studying the formulas presented in the article, you will not have any problems finding the area of a triangle of any kind.
