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 sides 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.
Solving a triangle is finding all its six elements (i.e., three sides and three angles) from any three given elements that define the triangle.
This mathematical program finds the side \(c\), angles \(\alpha \) and \(\beta \) from user-specified sides \(a, b\) and the angle between them \(\gamma \)
The program not only gives the answer to the problem, but also displays the process of finding a solution.
This online calculator may be useful for high school students secondary schools in preparation for tests and exams, when testing knowledge before the Unified State Exam, for parents to control the solution of many problems in mathematics and algebra. Or maybe it’s too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get it done as quickly as possible? homework in mathematics or algebra? In this case, you can also use our programs with detailed solutions.
In this way, you can conduct your own training and/or training of your younger brothers or sisters, while the level of education in the field of solving problems increases.
If you are not familiar with the rules for entering numbers, we recommend that you familiarize yourself with them.
Rules for entering numbers
Numbers can be specified not only as whole numbers, but also as fractions.
The integer and fractional parts in decimal fractions can be separated by either a period or a comma.
For example, you can enter decimals so 2.5 or so 2.5
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Theorem
The sides of a triangle are proportional to the sines of the opposite angles:
$$ \frac(a)(\sin A) = \frac(b)(\sin B) = \frac(c)(\sin C) $$
Theorem
Let AB = c, BC = a, CA = b in triangle ABC. Then
Square side of triangle equal to the sum squares of the other two sides minus twice the product of these sides multiplied by the cosine of the angle between them.
$$ a^2 = b^2+c^2-2ba \cos A $$
Solving a triangle is finding all its six elements (i.e. three sides and three angles) by any three given elements that define a triangle.
Let's look at three problems involving solving a triangle. In this case, we will use the following notation for the sides of triangle ABC: AB = c, BC = a, CA = b.
Solving a triangle using two sides and the angle between them
Given: \(a, b, \angle C\). Find \(c, \angle A, \angle B\)
Solution
1. Using the cosine theorem we find \(c\):
3. \(\angle B = 180^\circ -\angle A -\angle C\)
Solving a triangle by side and adjacent angles
Given: \(a, \angle B, \angle C\). Find \(\angle A, b, c\)
Solution
1. \(\angle A = 180^\circ -\angle B -\angle C\)
Solving a triangle using three sides
Given: \(a, b, c\). Find \(\angle A, \angle B, \angle C\)
Solution
1. Using the cosine theorem we obtain:
$$ \cos A = \frac(b^2+c^2-a^2)(2bc) $$
2. Similarly, we find angle B.
3. \(\angle C = 180^\circ -\angle A -\angle B\)
Solving a triangle using two sides and an angle opposite a known side
Given: \(a, b, \angle A\). Find \(c, \angle B, \angle C\)
Solution
1. Using the theorem of sines, we find \(\sin B\) we get:
$$ \frac(a)(\sin A) = \frac(b)(\sin B) \Rightarrow \sin B = \frac(b)(a) \cdot \sin A $$
Let's introduce the notation: \(D = \frac(b)(a) \cdot \sin A \). Depending on the number D, the following cases are possible:
If D > 1, such a triangle does not exist, because \(\sin B\) cannot be greater than 1
If D = 1, there is a unique \(\angle B: \quad \sin B = 1 \Rightarrow \angle B = 90^\circ \)
If D If D 2. \(\angle C = 180^\circ -\angle A -\angle B\)
3. Using the sine theorem, we calculate the side c:
$$ c = a \frac(\sin C)(\sin A) $$
In geometry, an angle is a figure formed by two rays emanating from one point (the vertex of the angle). Angles are most often measured in degrees, with a complete angle, or revolution, being 360 degrees. You can calculate the angle of a polygon if you know the type of polygon and the magnitude of its other angles or, in the case of a right triangle, the length of two of its sides.
Count the number of angles in the polygon.
Find the sum of all the angles of the polygon. Formula for finding the sum of all internal corners of a polygon looks like (n - 2) x 180, where n is the number of sides as well as angles of the polygon. Here are the angle sums of some commonly encountered polygons:
Determine whether the polygon is regular. A regular polygon is one in which all sides and all angles are equal. Examples of regular polygons include an equilateral triangle and a square, while the Pentagon in Washington is built in the shape of a regular pentagon, and a stop sign is shaped like a regular octagon.
Add up the known angles of a polygon, and then subtract this sum from the total sum of all its angles. In most geometric problems of this kind we're talking about about triangles or quadrilaterals, since they require less input data, so we will do the same.
Determine what data you know. A right triangle is so called because one of its angles is right. You can find the magnitude of one of the two remaining angles if you know one of the following:
Determine which trigonometric function to use. Trigonometric functions express the relationships between two of the three sides of a triangle. There are six trigonometric functions, but the most commonly used are:
The circle inscribed in it (r). To do this, increase it by six times and divide by Square root from the three: A = r*6/√3.
Knowing the radius (R), you can also calculate the length sides(A) correct triangle. This radius is twice that used in the previous formula, so triple it and also divide by the square root of three: A = R*3/√3.
By (P) equilateral triangle calculate its length sides(A) is even simpler, since the lengths of the sides in this figure are the same. Just divide the perimeter by three: A = P/3.
In an isosceles triangle, calculating the length sides along a known perimeter it is a little more complicated - you also need to know the length of at least one of the sides. If the length is known sides A, lying at the base of the figure, find the length of any of the side (B) in half the difference between the perimeter (P) and the size of the base: B = (P-A)/2. And if the side side is known, then determine the length of the base by subtracting twice the side length from the perimeter: A = P-2*B.
Knowing the area (S) occupied by a regular triangle on a plane is also sufficient to find its length sides(A). Take the square root of the ratio of the area and the root of three, and double the result: A = 2*√(S/√3).
In , in from any other, to calculate the length of one of the sides it is enough to know the lengths of the other two. If the required side is (C), to do this, find the square root of the lengths of the known sides (A and B), squared: C = √(A²+B²). And if you need to calculate the length of one of the legs, then the square root should be taken from the lengths of the hypotenuse and the other leg: A = √(C²-B²).
Sources:
In the general case, i.e. when there is no information about whether the triangle is equilateral, isosceles, or right, one has to use trigonometric functions to calculate the lengths of its sides. The rules for their application are determined by theorems, which are called the theorem of sines, cosines and tangents.
Instructions
One way to calculate the lengths of the sides of an arbitrary triangle assumes the sine theorems. According to it, the ratio of the lengths of the sides of the angles opposite them triangle are equal. This allows us to derive a formula for the length of a side for those cases where at least one side and two angles at the vertices of the figure are known from the conditions of the problem. If neither of these two angles (α and β) lies between known party A and calculated B, then multiply the length of the known side by the sine of the known angle β adjacent to it and divide by the sine of another known angle a: B = A*sin(β)/sin(α).
If one (γ) of two (α and γ) known angles is formed by , the length of one of which (A) is given in , and the second (B) needs to be calculated, then apply the same theorem. The solution can be reduced to the formula obtained in the previous step, if we also recall the theorem on the sum of angles in a triangle - this value is always 180°. The angle β is unknown in the formula, which can be calculated using this theorem by subtracting the values of two known angles from 180°. Substitute this value into the equation and you will get the formula B = A*sin(180°-α-γ)/sin(α).
