In geometry lessons in high school We've all been told about the triangle. However, within school curriculum we receive only the most necessary knowledge and learn the most common and standard methods of calculation. Are there unusual ways finding this value?
As an introduction, let us remember which triangle is considered right-angled, and also define the concept of area.
A right triangle is a closed geometric figure, one of the angles of which is equal to 90 0. Integral concepts in the definition are legs and hypotenuse. Legs mean two sides that form a right angle at the point of connection. The hypotenuse is the side opposite the right angle. A right triangle can be isosceles (its two sides will be the same size), but will never be equilateral (all sides will be the same length). We will not discuss the definitions of height, median, vectors and other mathematical terms in detail. They are easy to find in reference books.
Square right triangle. Unlike rectangles, the rule about
the work of the parties in the determination does not apply. If we speak in dry terms, then the area of a triangle is understood as the property of this figure to occupy a part of the plane, expressed by a number. Quite difficult to understand, you will agree. Let's not try to delve deeply into the definition; that's not our goal. Let's move on to the main thing - how to find the area right triangle? We will not perform the calculations themselves, we will only indicate the formulas. To do this, let's define the notation: A, B, C - sides of the triangle, legs - AB, BC. Angle ACB is straight. S is the area of the triangle, h n n is the height of the triangle, where nn is the side on which it is lowered.
Method 1. How to find the area of a right triangle if the size of its legs is known
Method 2. Find the area of an isosceles right triangle
Method 3. Calculating area using a rectangle
We complete the right triangle to a square (if the triangle
isosceles) or rectangle. We get a simple quadrilateral made up of 2 identical right triangles. In this case, the area of one of them will be equal to half the area of the resulting figure. S of a rectangle is calculated by the product of the sides. Let's denote this value M. The desired area value will be equal to half M.
Method 4. “Pythagorean pants.” The famous Pythagorean theorem
We all remember its formulation: “the sum of the squares of the legs...”. But not everyone can
say, what does some “pants” have to do with it? The fact is that Pythagoras initially studied the relationship between the sides of a right triangle. Having identified patterns in the ratio of the sides of squares, he was able to derive a formula known to all of us. It can be used in cases where the size of one of the sides is unknown.
Method 5. How to find the area of a right triangle using Heron's formula
This is also a fairly simple method of calculation. The formula involves expressing the area of a triangle through the numerical values of its sides. For calculations, you need to know the sizes of all sides of the triangle.
S = (p-AC)*(p-BC), where p = (AB+BC+AC)*0.5
In addition to the above, there are many other ways to find the size of such a mysterious figure as a triangle. Among them: calculation by the inscribed or circumscribed circle method, calculation using the coordinates of vertices, the use of vectors, absolute value, sines, tangents.
A right triangle is found in reality on almost every corner. Knowledge of the properties of a given figure, as well as the ability to calculate its area, will undoubtedly be useful to you not only for solving geometry problems, but also in life situations.
In elementary geometry, a right triangle is a figure that consists of three connected segments that form three angles (two acute and one straight). A right triangle is an original figure characterized by a number important properties, which form the foundation of trigonometry. Unlike a regular triangle, the sides of a rectangular figure have their own names:
It is the ratio of the legs and hypotenuse that forms the basis of trigonometry: sines, tangents and secants are defined as the ratio of the sides of a right triangle.
This figure has become widespread in reality. Triangles are used in design and technology, so calculating the area of a figure has to be done by engineers, architects and designers. The bases of tetrahedrons or prisms - three-dimensional figures that are easy to meet in everyday life - have the shape of a triangle. Additionally, a square is the simplest representation of a "flat" right triangle in reality. A square is a metalworking, drawing, construction and carpentry tool that is used to construct angles by both schoolchildren and engineers.
Square geometric figure is a quantitative assessment of how much of the plane is bounded by the sides of the triangle. The area of an ordinary triangle can be found in five ways, using Heron's formula or using such variables as the base, side, angle and radius of the inscribed or circumscribed circle. The most simple formula area is expressed as:
where a is the side of the triangle, h is its height.
The formula for calculating the area of a right triangle is even simpler:
where a and b are legs.
Working with our online calculator, you can calculate the area of a triangle using three pairs of parameters:
In problems or everyday situations you will be given different combinations of variables, so this form of the calculator allows you to calculate the area of a triangle in several ways. Let's look at a couple of examples.
Let's say you want to tidy up the kitchen walls. ceramic tiles, which has the shape of a right triangle. In order to determine the consumption of tiles, you must find out the area of one cladding element and total area treated surface. Suppose you need to process 7 square meters. The length of the legs of one element is 19 cm, then the area of the tile will be equal to:
This means that the area of one element is 24.5 square centimeters or 0.01805 square meters. Knowing these parameters, you can calculate that to finish 7 square meters of wall you will need 7/0.01805 = 387 elements of facing tiles.
Let in school task in geometry, you need to find the area of a right triangle, knowing only that the side of one leg is 5 cm, and the opposite angle is 30 degrees. Our online calculator comes with an illustration showing the sides and angles of a right triangle. If side a = 5 cm, then its opposite angle is angle alpha, equal to 30 degrees. Enter this data into the calculator form and get the result:
Thus, the calculator not only calculates the area of a given triangle, but also determines the length adjacent leg and hypotenuse, as well as the value of the second angle.
Right triangles are found in our lives literally on every corner. Determining the area of such figures will be useful to you not only when solving school assignments in geometry, but also in everyday and professional activities.
Instructions
Task 1.
Find the lengths of all sides of a triangle if it is known that one leg exceeds the length of the other by 1 cm, and the length of the triangle is 28 cm.
Solution.
Write down the basic formula for area S = (a*b)/2 = 28. It is known that b = a + 1, substitute this value into the formula: 28 = (a*(a+1))/2.
Open the brackets and get quadratic equation with one unknown a^2 + a - 56 = 0.
Find this by calculating the discriminant D = 1 + 224 = 225. The equation has two solutions: a_1 = (-1 + √225)/2 = (-1 + 15)/2 = 7 and a_2 = (-1 - √ 225)/2 = (-1 - 15)/2 = -8.
The second one doesn't make sense because the length of a segment cannot be negative, so a = 7 (cm).
Find the length of the second leg b = a + 1 = 8 (cm).
The length of the third side remains. By the Pythagorean theorem for a right triangle c^2 = a^2 + b^2 = 49 + 64, hence c = √(49 + 64) = √113 ≈ 10.6 (cm).
Task 2.
Find the lengths of all sides of a right triangle if you know that its area is 14 cm and angle ACB is 30°.
Solution.
Write down the basic formula S = (a*b)/2 = 14.
Now express the lengths of the legs through the product of the hypotenuse and trigonometric functions using the property of a right triangle:
a = c*cos(ACB) = c*cos(30°) = c*(√3/2) ≈ 0.87*c.
b = c*sin(ACB) = c*sin(30°) = c*(1/2) = 0.5*c.
Substitute the resulting values into the area formula:
14 = (0.87*0.5*c^2)/2, from where:
28 ≈ 0.435*c^2 → c = √64.4 ≈ 8 (cm).
You have found the length of the hypotenuse, now find the lengths of the other two sides:
a = 0.87*c = 0.87*8 ≈ 7 (cm), b = 0.5*c = 0.5*8 = 4 (cm).
Video on the topic
First, let's agree on notation. A leg is the side of a right triangle that is adjacent to a right angle (i.e., makes an angle of 90 degrees with the other side). We agree to denote the lengths of the legs as a and b. We will call the values of the acute angles of a right triangle opposite the legs A and B, respectively. The hypotenuse is the side of a right triangle that is opposite the right angle (that is, it is opposite the right angle and forms acute angles with the other sides of the triangle). We denote the length of the hypotenuse by c. Let us denote the required area by S.
Instructions
Apply the formula S = (a^2)/(2*tg(A)) if you are given only one of the legs (a), but the angle (A) opposite to this leg is also known. The sign "^2" indicates squaring.
Use the formula S=(a^2)*tg(B)/2 d if you are given only one of the legs (a), but the angle (B) adjacent to this leg is also known.
Video on the topic
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The relationships between the sides and angles of a right triangle are discussed in the branch of mathematics called trigonometry. To find the sides of a right triangle, it is enough to know the Pythagorean theorem, the definitions of trigonometric functions, and have some means for finding the values of trigonometric functions, for example, a calculator or Bradis tables. Let us consider below the main cases of problems of finding the sides of a right triangle.
You will need
Instructions
If one of the acute angles, for example, A, and one of the legs, for example, a, is given, then the hypotenuse and the other leg are calculated from the relations: b=a*tg(A), c=a*sin(A).
If you do not know the value of the sine or cosine of one of the angles necessary for calculation, you can use the Bradis tables; they provide the values of trigonometric functions for a large number of angles. In addition, most modern calculators are capable of calculating sines and cosines of angles.
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In a figure such as a right triangle, there is necessarily a clear relationship between the sides relative to each other. Knowing two of them, you can always find the third. You will learn how this can be done from the instructions given below.
You will need
Instructions
Square both sides and add them together a2+b2. The result obtained is the hypotenuse ( basis) squared c2. Next, you just need to extract the root of the last one, and the hypotenuse is found. This method is simple and easy to use on . The main thing in the process of finding parties triangle Thus, do not forget to extract the root of the preliminary result in order to avoid the most common mistake. The formula was derived thanks to the most famous Pythagorean theorem in the world, which in all sources has the form: a2+b2 = c2.
Divide one of the legs a by the sine of the opposite angle sin α. If the sides and sines are known in the condition, this option for finding the hypotenuse will be acceptable. Formula in in this case will have a very simple form: c=a/sin α. Be careful in all calculations.
Multiply side a by two. The hypotenuse has been calculated. This is perhaps the most elementary way of finding our side. But, unfortunately, this method is used only in one case - if the side that lies opposite the angle in degree measure equal to the number thirty. If there is one, you can be sure that it will always be exactly half of the hypotenuse. Accordingly, all you have to do is double it and you’re ready.
Divide leg a by the cosine of the adjacent angle cos α. This method is only suitable if you know one of the legs and the cosine of the angle adjacent to it. This method is reminiscent of the one already presented to you earlier, in which the leg is also used, but instead of the cosine, the sine of the opposite angle is used. Only in this case it will have a slightly different modified appearance: с=a/ cos α. That's all.
Tre square, one of the angles of which is right (equal to 90°) is called rectangular. Its longest side always lies opposite the right angle and is called the hypotenuse, and the other two sides are called legs. If the lengths of these three sides are known, then find the values of all angles of three square and will not be difficult, since in fact you only need to calculate one of the angles. There are several ways to do this.
Instructions
Use to calculate the quantities (α, β, γ) the definitions of trigonometric functions through a rectangular triangle. Such, for example, for the sine of an acute angle as the ratio of the length of the opposite leg to the length of the hypotenuse. This means that if the lengths of the legs (A and B) and the hypotenuse (C), then, for example, you can find the sine of the angle α lying opposite leg A by dividing the length sides And for the length sides C (hypotenuse): sin(α)=A/C. Having found out the value of the sine of this angle, you can find its value in degrees using the inverse function of the sine - arcsine. That is, α=arcsin(sin(α))=arcsin(A/C). In the same way you can find the size of an acute angle in a triangle. square Yes, but this is not necessary. Since the sum of all angles is three square a is 180°, and in three square If one of the angles is 90°, then the value of the third angle can be calculated as the difference between 90° and the value of the found angle: β=180°-90°-α=90°-α.
Instead of defining the sine, you can use the definition of the cosine of an acute angle, which is formulated as the ratio of the length of the leg adjacent to the desired angle to the length of the hypotenuse: cos(α)=B/C. And here use the reverse trigonometric function(arccosine) to find the angle in degrees: α=arccos(cos(α))=arccos(B/C). After this, as in the previous step, all that remains is to find the value of the missing angle: β=90°-α.
You can use a similar tangent - it is expressed by the ratio of the length of the leg opposite the desired angle to the length of the adjacent leg: tan(α)=A/B. Again, determine the angle in degrees using the inverse trigonometric function -: α=arctg(tg(α))=arctg(A/B). The formula for the missing angle will remain unchanged: β=90°-α.
Video on the topic
A triangle is considered to be right-angled if one of its angles is right. Side triangle located opposite the right angle is called the hypotenuse, and the other two sides- legs. To find the lengths of the sides of a rectangular triangle, you can use several methods.
Instructions
1. The meanings of two sides are known
In this case, the area of a right triangle is calculated by the formula:
S= 0.5ab
2. One leg and hypotenuse are known
Under such conditions, it is most logical to use the Pythagorean theorem and the above formula:
S = 0.5∙sqrt(c^2-a^2) ∙a,
where sqrt – Square root, c^2-a^2 – radical expression denoting the difference between the square of the hypotenuse and the leg.
3. Given the values of all sides of the triangle
For such problems, you can use Heron's formula:
S = (p-a)(p-b),
where p is the semi-perimeter, which is found by the following expression: p = 0.5∙ (a+b+c)
4. One leg and angle are known
Here it is worth turning to trigonometric functions. For example, tg(1) = 1/сtg(1) = b/a. That is, thanks to this relationship, it is possible to determine the value of the unknown leg. Next, the task comes down to the first point.
5. Known hypotenuse and angle
In this case, the trigonometric functions of sine and cosine are also used: сos(2)=1/sin(2) = b/c. Then the solution to the problem comes down to the second point of the article.
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c2=a2+b2, where c is the hypotenuse, a and b are the legs. That is, the hypotenuse will be equal to the square root of the sum of the squares of the legs. To find any of the legs, it is enough to subtract the square of the other leg from the square of the hypotenuse and take the square root from the resulting difference.
Thus, knowing the angle and one of the sides, you can use these formulas to calculate the other side. Both sides are also connected by trigonometric relations. The ratio of the opposite to the adjacent is called tangent, and the ratio of adjacent to the opposite is called cotangent. These relationships can be expressed by the formulas tgA=a/b or ctgA=b/a.
As you may remember from your school geometry curriculum, a triangle is a figure formed from three segments connected by three points that do not lie on the same straight line. A triangle forms three angles, hence the name of the figure. The definition may be different. A triangle can also be called a polygon with three angles, the answer will also be correct. Triangles are divided according to the number of equal sides and the size of the angles in the figures. Thus, triangles are distinguished as isosceles, equilateral and scalene, as well as rectangular, acute and obtuse, respectively.
There are a lot of formulas for calculating the area of a triangle. Choose how to find the area of a triangle, i.e. Which formula to use is up to you. But it is worth noting only some of the notations that are used in many formulas for calculating the area of a triangle. So, remember:
S is the area of the triangle,
a, b, c are the sides of the triangle,
h is the height of the triangle,
R is the radius of the circumscribed circle,
p is the semi-perimeter.
Here are the basic notations that may be useful to you if you have completely forgotten your geometry course. Below are the most understandable and not complex options calculating the unknown and mysterious area of a triangle. It is not difficult and will be useful both for your household needs and for helping your children. Let's remember how to calculate the area of a triangle as easily as possible:
In our case, the area of the triangle is: S = ½ * 2.2 cm * 2.5 cm = 2.75 sq. cm. Remember that area is measured in square centimeters (sqcm).
A right triangle is a triangle in which one angle is equal to 90 degrees (hence called right). A right angle is formed by two perpendicular lines (in the case of a triangle, two perpendicular segments). In a right triangle there can be only one right angle, because... the sum of all angles of any one triangle is equal to 180 degrees. It turns out that the other 2 angles should share the remaining 90 degrees, for example 70 and 20, 45 and 45, etc. So, you remember the main thing, all that remains is to find out how to find the area of a right triangle. Let's imagine that we have such a right triangle in front of us, and we need to find its area S.
1. The simplest way to determine the area of a right triangle is calculated using the following formula:
In our case, the area of the right triangle is: S = 2.5 cm * 3 cm / 2 = 3.75 sq. cm.
In principle, there is no longer any need to verify the area of the triangle in other ways, because Only this one will be useful and will help in everyday life. But there are also options for measuring the area of a triangle through acute angles.
2. For other calculation methods, you must have a table of cosines, sines and tangents. Judge for yourself, here are some options for calculating the area of a right triangle that can still be used:
We decided to use the first formula and with some minor blots (we drew it in a notebook and used an old ruler and protractor), but we got the correct calculation:
S = (2.5*2.5)/(2*0.9)=(3*3)/(2*1.2). We got the following results: 3.6=3.7, but taking into account the shift of cells, we can forgive this nuance.
If you are faced with the task of calculating the formula for an isosceles triangle, then the easiest way is to use the main and what is considered to be the classical formula for the area of a triangle.
But first, before finding the area of an isosceles triangle, let’s find out what kind of figure this is. An isosceles triangle is a triangle in which two sides have the same length. These two sides are called lateral, the third side is called the base. Do not confuse an isosceles triangle with an equilateral triangle, i.e. a regular triangle with all three sides equal. In such a triangle there are no special tendencies to the angles, or rather to their size. However, the angles at the base in an isosceles triangle are equal, but different from the angle between equal sides. So, you already know the first and main formula; it remains to find out what other formulas for determining the area of an isosceles triangle are known:
