Find out the side of a triangle by its side and angle. Triangle parameters according to given parameters. Formula for the area of a triangle given one side and two angles
In mathematics, when considering a triangle, a lot of attention is paid to its sides. Because these elements form this geometric figure. The sides of a triangle are used to solve many geometry problems.
Definition of the concept
Segments connecting three points that do not lie on the same line are called sides of a triangle. The elements under consideration limit part of the plane, which is called the interior of this geometric figure.
Mathematicians in their calculations allow generalizations regarding the sides of geometric figures. Thus, in a degenerate triangle, three of its segments lie on one straight line.
Characteristics of the concept
Calculating the sides of a triangle involves determining all other parameters of the figure. Knowing the length of each of these segments, you can easily calculate the perimeter, area and even the angles of the triangle.
Rice. 1. Arbitrary triangle.
By summing the sides of a given figure, you can determine the perimeter.
P=a+b+c, where a, b, c are the sides of the triangle
And to find the area of a triangle, then you should use Heron's formula.
$$S=\sqrt(p(p-a)(p-b)(p-c))$$
Where p is the semi-perimeter.
The angles of a given geometric figure are calculated using the cosine theorem.
$$cos α=((b^2+c^2-a^2)\over(2bc))$$
Meaning
Some properties of this geometric figure are expressed through the ratio of the sides of a triangle:
- Opposite the smallest side of a triangle is its smallest angle.
- The external angle of the geometric figure in question is obtained by extending one of the sides.
- Opposite equal angles of a triangle are equal sides.
- In any triangle, one of the sides is always greater than the difference of the other two segments. And the sum of any two sides of this figure is greater than the third.
One of the signs that two triangles are equal is the ratio of the sum of all sides of the geometric figure. If these values are the same, then the triangles will be equal.
Some properties of a triangle depend on its type. Therefore, you should first take into account the size of the sides or angles of this figure.
Forming triangles
If the two sides of the geometric figure in question are the same, then this triangle is called isosceles.
Rice. 2. Isosceles triangle.
When all the segments in a triangle are equal, you get an equilateral triangle.
Rice. 3. Equilateral triangle.
It is more convenient to carry out any calculation in cases where an arbitrary triangle can be classified as a specific type. Because then finding the required parameter of this geometric figure will be significantly simplified.
Although a correctly chosen trigonometric equation allows you to solve many problems in which an arbitrary triangle is considered.
What have we learned?
Three segments that are connected by points and do not belong to the same straight line form a triangle. These sides form a geometric plane, which is used to determine the area. Using these segments, you can find many important characteristics of a figure, such as perimeter and angles. The aspect ratio of a triangle helps to find its type. Some properties of a given geometric figure can only be used if the dimensions of each of its sides are known.
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In geometry, an angle is a figure that is formed by two rays that emerge from one point (called the vertex of the angle). In most cases, the unit of measurement for angle is degree (°) - remember that a full angle, or one revolution, is 360°. You can find the angle value of a polygon by its type and the values of other angles, and if given a right triangle, the angle can be calculated from two sides. Moreover, the angle can be measured using a protractor or calculated using a graphing calculator.
Steps
How to find interior angles of a polygon
- For example, a triangle has 3 sides and 3 interior angles, and a square has 4 sides and 4 interior angles.
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Calculate the sum of all interior angles of the polygon. To do this, use the following formula: (n - 2) x 180. In this formula, n is the number of sides of the polygon. The following are the sums of the angles of commonly encountered polygons:
- The sum of the angles of a triangle (a polygon with 3 sides) is 180°.
- The sum of the angles of a quadrilateral (a polygon with 4 sides) is 360°.
- The sum of the angles of a pentagon (a polygon with 5 sides) is 540°.
- The sum of the angles of a hexagon (a polygon with 6 sides) is 720°.
- The sum of the angles of an octagon (a polygon with 8 sides) is 1080°.
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Divide the sum of all the angles of a regular polygon by the number of angles. A regular polygon is a polygon with equal sides and equal angles. For example, each angle of an equilateral triangle is calculated as follows: 180 ÷ 3 = 60°, and each angle of a square is calculated as follows: 360 ÷ 4 = 90°.
- An equilateral triangle and a square are regular polygons. And at the Pentagon building (Washington, USA) and road sign"Stop" shape of a regular octagon.
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Subtract the sum of all known angles from the total sum of the angles of the irregular polygon. If the sides of a polygon are not equal to each other, and its angles are also not equal to each other, first add up the known angles of the polygon. Now subtract the resulting value from the sum of all the angles of the polygon - this way you will find the unknown angle.
- For example, if given that the 4 angles of a pentagon are 80°, 100°, 120° and 140°, add up these numbers: 80 + 100 + 120 + 140 = 440. Now subtract this value from the sum of all the angles of the pentagon; this sum is equal to 540°: 540 - 440 = 100°. Thus, the unknown angle is 100°.
Advice: the unknown angle of some polygons can be calculated if you know the properties of the figure. For example, in an isosceles triangle, two sides are equal and two angles are equal; In a parallelogram (which is a quadrilateral), opposite sides are equal and opposite angles are equal.
Measure the length of the two sides of the triangle. Longest side right triangle called the hypotenuse. The adjacent side is the side that is near the unknown angle. The opposite side is the side that is opposite the unknown angle. Measure the two sides to calculate the unknown angles of the triangle.
Advice: use a graphing calculator to solve the equations, or find an online table with the values of sines, cosines, and tangents.
Calculate the sine of an angle if you know the opposite side and the hypotenuse. To do this, plug the values into the equation: sin(x) = opposite side ÷ hypotenuse. For example, the opposite side is 5 cm and the hypotenuse is 10 cm. Divide 5/10 = 0.5. Thus, sin(x) = 0.5, that is, x = sin -1 (0.5).
Count the number of sides of the polygon. To calculate the interior angles of a polygon, you first need to determine how many sides the polygon has. Note that the number of sides of a polygon is equal to the number of its angles.
A triangle is a geometric number consisting of three segments that connect three points that do not lie on the same line. The points that form a triangle are called its points, and the segments are side by side.
Depending on the type of triangle (rectangular, monochrome, etc.), you can calculate the side of the triangle in different ways, depending on the input data and the conditions of the problem.
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To calculate the sides of a right triangle, the Pythagorean theorem is used, according to which the square of the hypotenuse equal to the sum square feet.
If we label the legs as "a" and "b" and the hypotenuse as "c", then the pages can be found with the following formulas:
If the acute angles of a right triangle (a and b) are known, its sides can be found with the following formulas:
Cropped triangle
A triangle is called an equilateral triangle in which both sides are the same.
How to find the hypotenuse in two legs
If the letter "a" is identical to the same page, "b" is the base, "b" is the angle opposite the base, "a" is the adjacent angle to calculate pages can use following formulas:
Two corners and a side
If one page (c) and two angles (a and b) of any triangle are known, the sine formula is used to calculate the remaining pages:
You must find the third value y = 180 - (a + b) because
the sum of all angles of a triangle is 180°; 
Two sides and an angle
If two sides of a triangle (a and b) and the angle between them (y) are known, the cosine theorem can be used to calculate the third side.
How to determine the perimeter of a right triangle
A triangular triangle is a triangle, one of which is 90 degrees and the other two are acute. calculation perimeter such triangle depending on the amount of information known about it.
You'll need it
- Depending on the case, skills 2 three sides of the triangle, as well as one of its acute angles.
instructions
first Method 1. If all three pages are known triangle Then, regardless of whether perpendicular or non-triangular, the perimeter is calculated as: P = A + B + C, where possible, c is the hypotenuse; a and b are legs.
second Method 2.
If a rectangle has only two sides, then using the Pythagorean theorem, triangle can be calculated using the formula: P = v (a2 + b2) + a + b or P = v (c2 - b2) + b + c.
third Method 3. Let the hypotenuse be c and an acute angle? Given a right triangle, it will be possible to find the perimeter this way: P = (1 + sin?
fourth Method 4. They say that in the right triangle the length of one leg is equal to a and, on the contrary, has an acute angle. Then calculate perimeter This triangle will be carried out according to the formula: P = a * (1 / tg?
1/son? + 1)
fifths Method 5.
Online triangle calculation
Let our leg lead and be included in it, then the range will be calculated as: P = A * (1 / CTG + 1 / + 1 cos?)
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The Pythagorean theorem is the basis of all mathematics. Determines the relationship between the sides of a true triangle. There are now 367 proofs of this theorem.
instructions
first The classic school formulation of the Pythagorean theorem sounds like this: the square of the hypotenuse is equal to the sum of the squares of the legs.
To find the hypotenuse in a right triangle of two Catets, you must apply to construct a square of the lengths of the legs, assemble them and take Square root from the amount. In the original formulation of his statement, the market is based on the hypotenuse, which is equal to the sum of the squares of 2 squares produced by Catete. However, the modern algebraic formulation does not require the introduction of a domain representation.
second For example, a right triangle whose legs are 7 cm and 8 cm.
Then, according to the Pythagorean theorem, the square hypotenuse is equal to R + S = 49 + 64 = 113 cm. The hypotenuse is equal to the square root of the number 113.
Angles of a right triangle
The result was an unfounded number.
third If the triangles are legs 3 and 4, then the hypotenuse = 25 = 5. When you take the square root, you get a natural number. The numbers 3, 4, 5 form a Pygagorean triplet, since they satisfy the relation x? +Y? = Z, which is natural.
Other examples of a Pythagorean triplet are: 6, 8, 10; 5, 12, 13; 15, 20, 25; 9, 40, 41.
fourth In this case, if the legs are identical to each other, the Pythagorean theorem turns into a more primitive equation. For example, suppose such a hand is equal to the number A and the hypotenuse is defined for C, and then c? = Ap + Ap, C = 2A2, C = A? 2. In this case you don't need A.
fifths The Pythagorean theorem is a special case, greater than the general cosine theorem, which establishes the relationship between the three sides of a triangle for any angle between two of them.
Tip 2: How to determine the hypotenuse for legs and angles
The hypotenuse is the side in a right triangle that is opposite the 90 degree angle.
instructions
first In the case of known catheters, as well as the acute angle of a right triangle, the hypotenuse can have a size equal to the ratio of the leg to the cosine / sine of this angle, if the angle was opposite / e include: H = C1 (or C2) / sin, H = C1 (or C2?) / cos?. Example: Let ABC be given an irregular triangle with hypotenuse AB and right angle C.
Let B be 60 degrees and A 30 degrees. The length of the stem BC is 8 cm. The length of the hypotenuse AB should be found. To do this you can use one of the above methods: AB = BC / cos60 = 8 cm. AB = BC / sin30 = 8 cm.
The hypotenuse is the longest side of a rectangle triangle. It is located at a right angle. Method for finding the hypotenuse of a rectangle triangle depending on the source data.
instructions
first If your legs are perpendicular triangle, then the length of the hypotenuse of the rectangle triangle can be discovered by a Pythagorean analogue - the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs: c2 = a2 + b2, where a and b are the length of the legs of the right triangle .
second If one of the legs is known and at an acute angle, the formula for finding the hypotenuse will depend on the presence or absence of under certain angle in relation to the known leg - adjacent (the leg is located close), or vice versa (the opposite case is located nego.V of the specified angle is equal to the fraction of the hypotenuse of the leg in the cosine angle: a = a / cos; E, on the other hand, the hypotenuse is the same as the ratio sinusoidal angles: da = a / sin.
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Useful tips
An angular triangle whose sides are related as 3:4:5, called the Egyptian delta due to the fact that these figures were widely used by the architects of ancient Egypt.
This is also the simplest example of Jero's triangles, in which pages and area are represented by integers.
A triangle is called a rectangle whose angle is 90°. The side opposite the right corner is called the hypotenuse, the other is called the legs.
If you want to find how a right triangle is formed by some properties of regular triangles, namely the fact that the sum of the acute angles is 90°, which is used, and the fact that the length of the opposite leg is half the hypotenuse is 30°.
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Cropped triangle
One of the properties of an equal triangle is that its two angles are equal.
To calculate the angle of a right congruent triangle, you need to know that:
- This is no worse than 90°.
- The values of acute angles are determined by the formula: (180 ° -90 °) / 2 = 45 °, i.e.
Angles α and β are equal to 45°.
If the known value of one of the acute angles is known, the other can be found using the formula: β = 180º-90º-α or α = 180º-90º-β.
This ratio is most often used if one of the angles is 60° or 30°.
Key Concepts
The sum of the interior angles of a triangle is 180°.
Because it's one level, two remain sharp.
Calculate triangle online
If you want to find them, you need to know that:
other methods
The values of the acute angles of a right triangle can be calculated from the average - with a line from a point on the opposite side of the triangle, and the height - the line is a perpendicular drawn from the hypotenuse at a right angle.
Let the median extend from the right corner to the middle of the hypotenuse, and let h be the height. In this case it turns out that: 
- sin α = b / (2 * s); sin β = a / (2 * s).
- cos α = a / (2 * s); cos β = b / (2 * s).
- sin α = h/b; sin β = h/a.
Two pages
If the lengths of the hypotenuse and one of the legs are known in a right triangle or on both sides, then trigonometric identities are used to determine the values of the acute angles:
- α = arcsin (a/c), β = arcsin (b/c).
- α = arcos (b/c), β = arcos (a/c).
- α = arctan (a / b), β = arctan (b / a).
Length of a right triangle
Area and Area of a Triangle
perimeter
The circumference of any triangle is equal to the sum of the lengths of the three sides. General formula to find triangular triangle:
where P is the circumference of the triangle, a, b and c of its sides.
Perimeter of an equal triangle can be found by successively combining the lengths of its sides or multiplying the side length by 2 and adding the base length to the product.
The general formula for finding an equilibrium triangle will look like this:
where P is the perimeter of an equal triangle, but either b, b is the base.
Perimeter of an equilateral triangle can be found by sequentially combining the lengths of its sides or by multiplying the length of any page by 3.
The general formula for finding the rim of equilateral triangles will look like this:
where P is the perimeter of an equilateral triangle, a is any of its sides.
region
If you want to measure the area of a triangle, you can compare it to a parallelogram. Consider triangle ABC:
If we take the same triangle and fix it so that we get a parallelogram, we get a parallelogram with the same height and base as this triangle:
In this case, the common side of the triangles is folded together along the diagonal of the molded parallelogram.
From the properties of a parallelogram. It is known that the diagonals of a parallelogram are always divided into two equal triangles, then the surface of each triangle is equal to half the range of the parallelogram.
Since the area of a parallelogram is the same as the product of its base height, the area of the triangle will be equal to half of this product. Thus, for ΔABC the area will be the same
Now consider a right triangle:
Two identical right triangles can be bent into a rectangle if it leans against them, which is each other hypotenuse.
Since the surface of the rectangle coincides with the surface of the adjacent sides, the area of this triangle is the same:
From this we can conclude that the surface of any right triangle is equal to the product of the legs divided by 2.
From these examples it can be concluded that the surface of each triangle is the same as the product of the length, and the height is reduced to the substrate divided by 2.
The general formula for finding the area of a triangle would look like this:
where S is the area of the triangle, but its base, but the height falls to the bottom a.
A triangle is called a right triangle if one of its angles is 90º. The side opposite the right angle is called the hypotenuse, and the other two are called the legs.
To find the angle in a right triangle, some properties of right triangles are used, namely: the sum of the acute angles is 90º, and also the fact that opposite the leg, the length of which is half the length of the hypotenuse, lies an angle equal to 30º.
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Isosceles triangle
One of the properties of an isosceles triangle is that its two angles are equal. To calculate the angles of a right isosceles triangle you need to know that:
- A right angle is 90º.
- The values of acute angles are determined by the formula: (180º-90º)/2=45º, i.e. angles α and β are equal to 45º.
If the size of one of the acute angles is known, the second can be found using the formula: β=180º-90º-α, or α=180º-90º-β. Most often this ratio is used if one of the angles is 60º or 30º.
Key Concepts
The sum of the interior angles of a triangle is 180º. Since one angle is right, the remaining two will be acute. To find them you need to know that:
other methods
The values of the acute angles of a right triangle can be calculated by knowing the value of the median - a line drawn from the vertex to the opposite side of the triangle, and the height - a straight line, which is a perpendicular dropped from a right angle to the hypotenuse. Let s be the median drawn from the right angle to the middle of the hypotenuse, h be the height. In this case it turns out that: 
- sin α=b/(2*s); sin β =a/(2*s).
- cos α=a/(2*s); cos β=b/(2*s).
- sin α=h/b; sin β =h/a.
Two sides
If the lengths of the hypotenuse and one of the legs, or two sides, are known in a right triangle, trigonometric identities are used to find the values of the acute angles:
- α=arcsin(a/c), β=arcsin(b/c).
- α=arcos(b/c), β=arcos(a/c).
- α=arctg(a/b), β=arctg(b/a).
The first are the segments that are adjacent to the right angle, and the hypotenuse is the longest part of the figure and is located opposite the angle of 90 degrees. Pythagorean triangle is called the one whose sides are equal natural numbers; their lengths in this case are called “Pythagorean triple”.
Egyptian triangle
In order for the current generation to recognize geometry in the form in which it is taught in school now, it has developed over several centuries. The fundamental point is considered to be the Pythagorean theorem. The sides of a rectangular is known throughout the world) are 3, 4, 5.
Few people are not familiar with the phrase “Pythagorean pants are equal in all directions.” However, in reality the theorem sounds like this: c 2 (square of the hypotenuse) = a 2 + b 2 (sum of squares of the legs).
Among mathematicians, a triangle with sides 3, 4, 5 (cm, m, etc.) is called “Egyptian”. The interesting thing is that which is inscribed in the figure is equal to one. The name arose around the 5th century BC, when Greek philosophers traveled to Egypt.
When building the pyramids, architects and surveyors used the ratio 3:4:5. Such structures turned out to be proportional, pleasant to look at and spacious, and also rarely collapsed.
In order to build a right angle, the builders used a rope with 12 knots tied on it. In this case, the probability of constructing a right triangle increased to 95%.
Signs of equality of figures
- An acute angle in a right triangle and a long side, which are equal to the same elements in the second triangle, are an indisputable sign of equality of figures. Taking into account the sum of the angles, it is easy to prove that the second acute angles are also equal. Thus, the triangles are identical according to the second criterion.
- When superimposing two figures on top of each other, we rotate them so that, when combined, they become one isosceles triangle. According to its property, the sides, or rather the hypotenuses, are equal, as well as the angles at the base, which means that these figures are the same.
Based on the first sign, it is very easy to prove that the triangles are indeed equal, the main thing is that the two smaller sides (i.e., the legs) are equal to each other.
The triangles will be identical according to the second criterion, the essence of which is the equality of the leg and the acute angle.
Properties of a triangle with a right angle
The height that is lowered from the right angle splits the figure into two equal parts.
The sides of a right triangle and its median can be easily recognized by the rule: the median that falls on the hypotenuse is equal to half of it. can be found both by Heron's formula and by the statement that it is equal to half the product of the legs.
In a right triangle, the properties of angles of 30°, 45° and 60° apply.
- With an angle of 30°, it should be remembered that the opposite leg will be equal to 1/2 of the largest side.
- If the angle is 45°, then the second acute angle is also 45°. This suggests that the triangle is isosceles and its legs are the same.
- The property of an angle of 60° is that the third angle has a degree measure of 30°.
The area can be easily found out using one of three formulas:
- through the height and the side on which it descends;
- according to Heron's formula;
- on the sides and the angle between them.
The sides of a right triangle, or rather the legs, converge with two altitudes. In order to find the third, it is necessary to consider the resulting triangle, and then, using the Pythagorean theorem, calculate the required length. In addition to this formula, there is also a relationship between twice the area and the length of the hypotenuse. The most common expression among students is the first one, as it requires fewer calculations.
Theorems applying to right triangle
Right triangle geometry involves the use of theorems such as:
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