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.

Triangle geometry

In elementary geometry, a right triangle is a figure that consists of three connected segments that form three angles (two acute and one straight). The right triangle is an original figure characterized by a number of important properties that form the foundation of trigonometry. Unlike a regular triangle, the sides of a rectangular figure have their own names:

• The hypotenuse is the longest side of a triangle, opposite the right angle.
• Legs are segments that form a right angle. Depending on the angle under consideration, the leg can be adjacent to it (forming this angle with the hypotenuse) or opposite (lying opposite the angle). There are no legs for non-right triangles.

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.

Right triangle in reality

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.

Area of ​​a triangle

The area of ​​a geometric figure is a quantitative estimate 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 simplest formula for 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:

• two legs;
• leg and adjacent angle;
• leg and opposite angle.

In tasks or everyday situations you will be given different combinations 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.

Real life examples

Ceramic tile

Let's say you want to cover the kitchen walls with ceramic tiles, which have 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 the total area of ​​the surface being treated. Let's say 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.

School task

Let's say in a school geometry problem 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 of the adjacent leg and hypotenuse, as well as the value of the second angle.

Conclusion

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.

In geometry there are often problems related to the sides of triangles. For example, it is often necessary to find a side of a triangle if the other two are known.

Triangles are isosceles, equilateral and unequal. From all the variety, for the first example we will choose a rectangular one (in such a triangle, one of the angles is 90°, the sides adjacent to it are called legs, and the third is the hypotenuse).

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Length of the sides of a right triangle

The solution to the problem follows from the theorem of the great mathematician Pythagoras. It says that the sum of the squares of the legs of a right triangle is equal to the square of its hypotenuse: a²+b²=c²

• Find the square of the leg length a;
• Find the square of leg b;
• We put them together;
• From the obtained result we extract the second root.

Example: a=4, b=3, c=?

• a²=4²=16;
• b² =3²=9;
• 16+9=25;
• √25=5. That is, the length of the hypotenuse of this triangle is 5.

If the triangle does not have a right angle, then the lengths of the two sides are not enough. For this, a third parameter is needed: this can be an angle, the height of the triangle, the radius of the circle inscribed in it, etc.

If the perimeter is known

In this case, the task is even simpler. The perimeter (P) is the sum of all sides of the triangle: P=a+b+c. Thus, by solving a simple mathematical equation we get the result.

Example: P=18, a=7, b=6, c=?

1) We solve the equation by moving all known parameters to one side of the equal sign:

2) We substitute the values ​​​​instead and calculate the third side:

c=18-7-6=5, total: the third side of the triangle is 5.

If the angle is known

To calculate the third side of a triangle given an angle and two other sides, the solution comes down to calculating the trigonometric equation. Knowing the relationship between the sides of the triangle and the sine of the angle, it is easy to calculate the third side. To do this, you need to square both sides and add their results together. Then subtract from the resulting product the product of the sides multiplied by the cosine of the angle: C=√(a²+b²-a*b*cosα)

If the area is known

In this case, one formula will not do.

1) First, calculate sin γ, expressing it from the formula for the area of ​​a triangle:

sin γ= 2S/(a*b)

2) Using the following formula, we calculate the cosine of the same angle:

sin² α + cos² α=1

cos α=√(1 — sin² α)=√(1- (2S/(a*b))²)

3) And again we use the theorem of sines:

C=√((a²+b²)-a*b*cosα)

C=√((a²+b²)-a*b*√(1- (S/(a*b))²))

Substituting the values ​​of the variables into this equation, we obtain the answer to the problem.

The following unique calculator is used to convert exotic units of length to...

Instructions

Let us denote the angles of the triangle by the letters A, B and C, as shown in the figure. Angle BAC is equal to 90º, the other two angles are denoted by the letters α and β. We denote the legs of the triangle by the letters a and b, and the hypotenuse by the letter c.

Then sinα = b/c, and cosα = a/c.

Similarly for the second acute angle of the triangle: sinβ = a/c, and cosβ = b/c.

Depending on which sides we know, we calculate the sines or cosines of the angles and look at the values ​​of α and β using the Bradis table.

Having found one of the angles, you can remember that the sum of the interior angles of a triangle is 180º. This means that the sum of α and β is equal to 180º - 90º = 90º.

Then, having calculated the value for α from the tables, we can use the following formula to find β: β = 90º - α

If one of the sides of the triangle is unknown, then we apply the Pythagorean theorem: a²+b²=c². Let us derive from it the expression for the unknown side through the other two and substitute it into the formula to find the sine or cosine of one of the angles.

Tip 2: How to find the hypotenuse in a right triangle

The hypotenuse is the side in a right triangle that lies opposite the right angle. The hypotenuse is the longest side in a right triangle. The remaining sides in a right triangle are called legs.

You will need

• Basic knowledge of geometry.

Instructions

The square of the length of the hypotenuse is equal to the sum of the squares of the legs. That is, to find the square of the length of the hypotenuse, you need to square the length of the legs and add.

The length of the hypotenuse is equal to the square root of the square of its length. To find its length, take the square root of a number equal to the sum of the squares of the legs. The resulting number will be the length of the hypotenuse.

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note

The length of the hypotenuse is a positive value, so when extracting a root, the radical expression must be greater than zero.

Helpful advice

In an isosceles right triangle, the length of the hypotenuse can be calculated by multiplying the leg by the root of two.

Sources:

• how to calculate the hypotenuse in a right triangle

Tip 3: How to find an acute angle in a right triangle

Directly carbonic the triangle is probably one of the most famous, from a historical point of view, geometric shapes. Pythagorean “pants” can only compete with “Eureka!” Archimedes.

You will need

• - drawing of a triangle;
• - ruler;
• - protractor

Instructions

The sum of the angles of a triangle is 180 degrees. In a rectangular triangle one angle (straight) will always be 90 degrees, and the rest are acute, i.e. less than 90 degrees each. To determine what angle is in a rectangular triangle is straight, use a ruler to measure the sides of the triangle and determine the largest. It is called the hypotenuse (AB) and is located opposite the right angle (C). The remaining two sides form a right angle and are called legs (AC, BC).

Once you have determined which angle is acute, you can either measure the angle using a protractor or calculate it using mathematical formulas.

To determine the angle using a protractor, align its top (let’s denote it with the letter A) with a special mark on the ruler in the center of the protractor; leg AC should coincide with its upper edge. Mark on the semicircular part of the protractor the point through which the hypotenuse AB passes. The value at this point corresponds to the angle in degrees. If there are 2 values ​​indicated on the protractor, then for an acute angle you need to choose the smaller one, for an obtuse angle - the larger one.

Find the resulting value in the Bradis reference tables and determine which angle the resulting numerical value corresponds to. Our grandmothers used this method.

Nowadays, it is enough to take a calculator with the function of calculating trigonometric formulas. For example, the built-in Windows calculator. Launch the "Calculator" application, in the "View" menu item, select "Engineering". Calculate the sine of the desired angle, for example, sin (A) = BC/AB = 2/4 = 0.5

Switch the calculator to inverse functions mode by clicking on the INV button on the calculator display, then click on the button for calculating the arcsine function (indicated on the display as sin to the minus first power). The following message will appear in the calculation window: asind (0.5) = 30. I.e. the value of the desired angle is 30 degrees.

Sources:

• Bradis tables (sines, cosines)

The method for calculating the unknown side of a triangle depends not only on the conditions of the task, but also on why it is being done. This problem is faced not only by schoolchildren in geometry lessons, but also by engineers working in various industries, interior designers, cutters and representatives of many other professions. The accuracy of calculations for different purposes may be different, but their principle remains the same as in a school problem book.

Problems of solving triangles (that’s what such problems are called) are dealt with by a special branch of geometry - trigonometry.

Along the length of two sides of the triangle

The well-known ancient mathematician Pythagoras suggested finding the length of the third side of a right triangle. The basis is a right triangle, that is, one in which one of the angles is equal to 90 degrees. The adjacent sides to a given angle are always designated as legs; accordingly, the third, largest side is called the “hypotenuse”. The Pythagorean theorem is as follows: “the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.”

To solve this problem, we denote the length of one leg as X (x), and the other as Y (y), the length of the hypotenuse can be denoted as Z (z). Now let's write down the formula for calculating the length of the hypotenuse: Z squared = X squared + Y squared. Based on this formula, we ultimately obtain the value of the square of the length of the hypotenuse. This means that to obtain the length of the hypotenuse, you also need to take the square root of the resulting sum of the lengths of the legs.

Previously, we looked at the ideal option when you need to determine the length of the hypotenuse. If the length of one of the legs in the problem is unknown, then, based on the indicated theorem, a derivative formula can be derived. The square of the length of one of the legs is equal to the value obtained by subtracting the square of the length of the other leg from the square of the length of the hypotenuse: X squared = Z squared - Y squared. Well, the last step is to extract square root from the obtained value.

For example, let’s take simple values ​​for the length of the legs: 2 and 3 centimeters. By simple mathematical operations we get Z squared = 4 + 9 = 13. This means Z is approximately equal to 3.6 centimeters. If we exclude the squaring of values, then it turns out that Z = 2 + 3 = 5 centimeters, which is not true.

By the length of two sides and by the angle between them

You can find the length of the third side of a triangle using the cosine theorem. This geometric theorem is as follows: the square of one of the sides of a triangle is equal to the value obtained by subtracting twice the product of the length of the known sides and the cosine of the angle that is located between them from the sum of the squares of the length of the known sides.

In mathematical form, this formula looks like this: Z squared=X²+Y²-2*X*Y*cosC. Here X, Y, Z denote the length of all sides of the triangle, and C is the value in degrees of the angle that is located between the known sides.

For example, we use a triangle, known parties which are equal to 2 and 4 centimeters, and the angle between them is 60 degrees. We use the formula indicated earlier and get: Z squared =4+16-2*2*4*cos60=20-8=12. The length of the unknown side is 3.46 centimeters.