Instructions
1. For two legs S = a * b/2, a, b – legs,
The second option for calculating area uses sines of known angles instead of cotangents. In this version square is equal to the square of the length of the known side, multiplied by the sines of each of the angles and divided by the double sine of these angles: S = A*A*sin(α)*sin(β)/(2*sin(α + β)). For example, for the same triangle with a known side of 15 cm, and adjacent to it corners at 40° and 60°, the area calculation will look like this: (15*15*sin(40)*sin(60))/(2*sin(40+60)) = 225*0.74511316*(-0.304810621)/( 2*(-0.506365641)) = -51.1016411/-1.01273128 = 50.4592305 square centimeters.
The version of calculating the area of a triangle involves angles. The area will be equal to the square of the length of the known side, multiplied by the tangents of each of the angles and divided by double the sum of the tangents of these angles: S = A*A*tg(α)*tg(β)/2(tg(α)+tg(β) ). For example, for the triangle used in the previous steps with a side of 15 cm and adjacent corners at 40° and 60°, the area calculation will look like this: (15*15*tg(40)*tg(60))/(2*(tg(40)+tg(60)) = (225*(-1.11721493 )*0.320040389)/(2*(-1.11721493+0.320040389)) = -80.4496277/-1.59434908 = 50.4592305 square centimeters.
A triangle is the simplest polygon having three vertices and three sides. A triangle, one of whose angles is right, is called a right triangle. For right triangles, all formulas for general triangles are applicable. However, they can be modified, taking into account the properties right angle.
Instructions
Basic for finding area triangle through the base as follows: S = 1/2 * b * h, where b is the side triangle, and h – triangle. Height triangle is a perpendicular drawn from the vertex triangle to the line containing the opposite. For rectangular triangle height k b coincides with leg a. This way you will get the formula to calculate the area triangle with angle: S = 1/2 * a * b.
Consider. Let in a rectangular a = 3, b = 4. Then S = 1/2 * 3 * 4 = 6. Calculate square the same triangle, but now let only one side be known, b = 4. And the angle α, tan α = 3/4 is also known. Then, from the expression for the trigonometric function tangent α, express leg a: tg α = a/b => a = b * tan α. Substitute this value into the formula to calculate the area of a rectangular triangle and we get: S = 1/2 * a * b = 1/2 *b^2 * tan α = 1/2 * 16 * 3/4 = 6.
Consider as a special case the calculation of the area of an isosceles rectangular triangle. An isosceles triangle is a triangle in which two sides are equal to each other. In the case of a rectangular triangle it turns out a = b. Write down the Pythagorean theorem for this case: c^2 = a^2 + b^2 = 2 * a^2. Next, substitute this value into the formula for calculating the area as follows: S = 1/2 * a * b = 1/2 * a^2 = 1/2 * (c^2 / 2) = c^2 / 4.
If the radii of the inscribed circle r and the circumcircle R are known, then square rectangular triangle is calculated by the formula S = r^2 + 2 * r * R. Let the radius of the inscribed circle in the triangle be r = 1, the radius of the circumscribed circle triangle circle R = 5/2. Then S = 1 + 2 * 1 * 5 / 2 = 6.
Video on the topic
The radius of a circle circumscribed around a right triangle is equal to half the hypotenuse: R = c / 2. The radius of a circle inscribed in a right triangle is found by the formula r = (a + b – c) / 2.
This is one of the simplest geometric figures, in which three segments connecting three points in pairs limit a part of the plane. Knowledge of some of the parameters of a triangle (lengths of sides, angles, radii of an inscribed or circumscribed circle, height, etc.) in various combinations allows one to calculate the area of this limited section of the plane.
Instructions
If the lengths of two sides of a triangle (A and B) and the magnitude of their angle (γ) are known, then the area (S) of the triangle will be equal to half the product of the lengths of the sides and the sine of the known angle: S=A∗B∗sin(γ)/2.
If the lengths of all three sides (A, B and C) in an arbitrary triangle are known, then to calculate its area (S) it is more convenient to introduce an additional variable - the semi-perimeter (p). This variable is calculated in half the sum of the lengths of all sides: p=(A+B+C)/2. Using this variable can be defined as the square root of the product of the semi-perimeter on this variable and the length of the sides: S=√(p∗(p-A)∗(p-B)∗(p-C)).
If, in addition to the lengths of all sides (A, B and C), the length of the radius (R) of a circle circumscribed near an arbitrary triangle is also known, then you can do without a semi-perimeter - the area (S) will be equal to the ratio of the product of the lengths of all sides to the quadruple radius of the circle: S=A ∗B∗C/(4∗R).
If the values of all angles of a triangle (α, β and γ) and the length of one of its sides (A) are known, then the area (S) will be equal to the ratio of the product of the square of the length of the known side by the sines of two angles adjacent to it to the double sine of the opposite one angle: S=A²∗sin(β)∗sin(γ)/(2∗sin(α)).
If the values of all angles of an arbitrary triangle (α, β and γ) and the radius (R) of the circumscribed circle are known, then the area (S) will be equal to twice the square of the radius and the sines of all angles: S=2∗R²∗sin(α)∗ sin(β)∗sin(γ).
Video on the topic
Finding the volume of a triangle is truly a non-trivial task. The fact is that a triangle is a two-dimensional figure, i.e. it lies entirely in one plane, which means that it simply has no volume. Of course, you can't find something that doesn't exist. But let's not give up! We can accept the following assumption: the volume of a two-dimensional figure is its area. We will look for the area of the triangle.
You will need
- sheet of paper, pencil, ruler, calculator
Instructions
Draw on a piece of paper using a ruler and pencil. By carefully examining the triangle, you can make sure that it really does not have a triangle, since it is drawn on a plane. Label the sides of the triangle: let one side be side "a", the other side "b", and the third side "c". Label the vertices of the triangle with the letters "A", "B" and "C".
Measure any side of the triangle with a ruler and write down the result. After this, restore a perpendicular to the measured side from the vertex opposite to it, such a perpendicular will be the height of the triangle. In the case shown in the figure, the perpendicular "h" is restored to side "c" from vertex "A". Measure the resulting height with a ruler and write down the measurement result.
It may be difficult for you to restore the exact perpendicular. In this case, you should use a different formula. Measure all sides of the triangle with a ruler. After this, calculate the semi-perimeter of the triangle "p" by adding the resulting lengths of the sides and dividing their sum in half. Having the value of the semi-perimeter at your disposal, you can use Heron's formula. To do this, you need to take the square root of the following: p(p-a)(p-b)(p-c).
You have obtained the required area of the triangle. The problem of finding the volume of a triangle has not been solved, but as mentioned above, the volume is not. You can find a volume that is essentially a triangle in the three-dimensional world. If we imagine that our original triangle has become a three-dimensional pyramid, then the volume of such a pyramid will be the product of the length of its base by the area of the triangle we have obtained.
note
The more carefully you measure, the more accurate your calculations will be.
Sources:
- Calculator “Everything to everything” - a portal for reference values
- volume of triangle
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 different in 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.
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
$S=\frac(1)(2) βh=\frac(β)(2)\cdot \frac(2)(β) \sqrt(ρ(ρ-α)(ρ-β)(ρ-γ) )=\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$
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.