In elementary geometry, a right triangle is a figure consisting of three segments connected at points, with angles two of which are acute and one straight (that is, equal to 90°). Right triangle is characterized by a number of important properties, many of which form the basis of trigonometry (for example, the relationship between its sides and angles). Since school, we all know how to calculate area of a right triangle, and in everyday life we encounter this geometric figure quite often, sometimes without even noticing it. It finds quite wide application in technology and therefore engineers, designers and architects often have to solve such a problem.
Architects need to determine this value when they design buildings with pediments, which are the completion of the facades and have triangular shape bounded by a cornice and on the sides by roof slopes. Often the angle between the slopes is straight, and in such cases the pediment has the shape of a right triangle. It is necessary to determine its area for the simple reason that it is necessary to know exactly the amount of building material required for its arrangement. It should be noted that gables are mandatory elements of low-rise buildings (country houses, cottages, dachas).
Finding the area of a right triangle
Formula for calculating the area of a right triangle
| S | ab |
a- leg
b- leg
S- area of a right triangle
Form right triangle have many of the details from which modern furniture is made. As you know, in order to make the most efficient use of room space, all elements of the furnishings must be placed in it in an optimal way. You can make good use of areas such as corners using triangular-shaped tables, the tops of which in most cases are right-angled triangles with legs adjacent to the walls. When designing and calculating these elements, furniture production designers use the formula according to which finding the area of a right triangle is carried out based on the length of its sides. In addition, they often have to develop designs for tables attached directly to the walls, which include supporting elements, which also represent right triangles.
Builders engaged in facing work often in their professional activities have to use ceramic tiles in the shape of a right triangle with legs of the same or different lengths. They also have to determine the area of these elements in order to find out the required number.
Form right triangle It also has such an important and necessary measuring tool as a square. It is used to construct and control right angles, and it is used very widely and by many: from ordinary schoolchildren in geometry lessons to designers of ultra-modern technology.
A triangle is a flat geometric figure with one angle equal to 90°. Moreover, in geometry it is often necessary to calculate the area of such a figure. We will tell you how to do this further.
The simplest formula for determining the area of a right triangle
Initial data, where: a and b are the sides of the triangle extending from the right angle.
That is, the area is equal to half the product of the two sides that extend from the right angle. Of course, there is Heron's formula used to calculate the area of a regular triangle, but to determine the value you need to know the length of the three sides. Accordingly, you will have to calculate the hypotenuse, and this is extra time.
Find the area of a right triangle using Heron's formula
This is a well-known and original formula, but for this you will have to calculate the hypotenuse on two legs using the Pythagorean Theorem.
In this formula: a, b, c are the sides of the triangle, and p is the semi-perimeter.
Find the area of a right triangle using the hypotenuse and angle
If none of the legs are known in your problem, then you will not be able to use the simplest method. To determine the value, you need to calculate the length of the legs. This can be done simply by using the hypotenuse and the cosine of the adjacent angle.
b=c×cos(α)
Once you know the length of one of the legs, using the Pythagorean theorem you can calculate the second side coming out of the right angle.
b 2 =c 2 -a 2
In this formula, c and a are the hypotenuse and leg, respectively. Now you can calculate the area using the first formula. In the same way, you can calculate one of the legs, given the second and the angle. In this case, one of the required sides will be equal to the product of the leg and the tangent of the angle. There are other ways to calculate area, but knowing the basic theorems and rules, you can easily find the desired value.
If you do not have any of the sides of the triangle, but only have the median and one of the angles, then you can calculate the length of the sides. To do this, use the properties of the median to divide a right triangle into two. Accordingly, it can act as a hypotenuse if it comes out of an acute angle. Use the Pythagorean theorem and determine the length of the sides of the triangle that extend from the right angle.
As you can see, knowing the basic formulas and the Pythagorean Theorem, you can calculate the area of a right triangle, having only one of the angles and the length of one of the sides.
The area of a right triangle can be found in several ways. A right angle in any figure adds properties to it and this can be used to correctly and quickly solve problems.
Right triangle
First, let's discuss the right triangle itself, its features and properties. A right triangle is a triangle that contains an angle.
A right triangle cannot be obtuse, because then the sum of the angles of the triangle would exceed 180 degrees, and this is impossible.
In a right triangle, two of the three altitudes coincide with the sides - the legs. For the same reason, the point of intersection of the altitudes of a right triangle coincides with the vertex at a right angle.
Rice. 1. All heights of a right triangle.
The same point will be the center of the circumscribed circle.
Area of a triangle
The area of a triangle is usually found using the standard formula, as half the product of the base and the height drawn to this base.
$$S=(1\over2)*a*h$$
You can find the area as half the product of the sides and the sine of the angle between them:
$$S=(1\over2)*a*b*sin(g)$$
There are complicated formulas for finding area, but they are used extremely rarely.
Area of a right triangle
The area of a right triangle is found using the same formulas, but in some cases these formulas can be simplified.
For example, you can take advantage of the fact that the altitudes in a right triangle coincide with the legs. Then the standard formula becomes:
$S=(1\over2)*a*b$, where a and b are the legs of a right triangle.
This is one of the simplest formulas for the area of a right triangle. Let's try to transform the second formula.
$$S=(1\over2)*a*b*sin(g)$$
If we remember that the sine of an angle is the ratio of the opposite side to the hypotenuse. In our case, we denote the opposite leg as the letter f, because a is an adjacent leg, and an acute angle can only be made between the leg and the hypotenuse. So b is the hypotenuse.
$S=(1\over2)*a*b*sin(g)= (1\over2)*a*b*(f\over(b))=(1\over2)a*f$ - everything turns out the same same formula.
Rice. 2. Drawing to conclusion.
This means that we carried out the first conclusion correctly, and a right triangle has only one special formula for finding the area. If it does not work, you can use general formulas. These are two possible ways to calculate the area.
For example, if the hypotenuse is known according to the conditions of the problem, then you can try to find the height falling on the hypotenuse and determine the area using the general formula. Using the same principle, you can find the area through the sine if the hypotenuse and leg are known.
Rice. 3. Altitude drawn to the hypotenuse.
The main thing to remember is that any problem always has 3 solutions and solve each in the most convenient way.
What have we learned?
We talked about right triangles and derived the formula for the area of a right triangle using the legs. We discussed the general formulas for the area of triangles and said that each of these formulas would work for solving a right triangle.
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A right triangle is a triangle in which one of the angles is 90°. Its area can be found if two sides are known. You can, of course, take the long route - find the hypotenuse and calculate the area using , but in most cases this will only take extra time. That is why the formula for the area of a right triangle looks like this:
The area of a right triangle is equal to half the product of the legs.
An example of calculating the area of a right triangle.
Given a right triangle with legs a= 8 cm, b= 6 cm.
We calculate the area: 
Area is: 24 cm 2
The Pythagorean theorem also applies to a right triangle. – the sum of the squares of the two legs is equal to the square of the hypotenuse.
The formula for the area of an isosceles right triangle is calculated in the same way as a regular right triangle.
An example of calculating the area of an isosceles right triangle:
Given a triangle with legs a= 4 cm, b= 4 cm. Calculate the area:
Calculate the area: = 8 cm 2
The formula for the area of a right triangle based on the hypotenuse can be used if one leg is given in the condition. From the Pythagorean theorem we find the length of the unknown leg. For example, given the hypotenuse c and leg a, leg b will be equal to:
Next, calculate the area using the usual formula. An example of calculating the formula for the area of a right triangle based on the hypotenuse is identical to that described above.
Let's consider an interesting problem that will help consolidate knowledge of formulas for solving a triangle.
Task: The area of a right triangle is 180 square meters. see, find the smaller leg of the triangle if it is 31 cm less than the second.
Solution: let's designate the legs a And b. Now let’s substitute the data into the area formula: we also know that one leg is smaller than the other a – b= 31 cm
From the first condition we obtain that
We substitute this condition into the second equation: 
Since we found the sides, we remove the minus sign.
It turns out that the leg a= 40 cm, a b= 9 cm.
