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 right triangle, the length of two of its sides.

Steps

Calculating Polygon Angles

Count the number of angles in the polygon.

Find the sum of all the angles of the polygon. The formula for finding the sum of all interior angles of a polygon is (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:

• The sum of the angles of a triangle (three-sided polygon) is 180 degrees.
• The sum of the angles of a quadrilateral (four-sided polygon) is 360 degrees.
• The sum of the angles of a pentagon (five-sided polygon) is 540 degrees.
• The sum of the angles of a hexagon (six-sided polygon) is 720 degrees.
• The sum of the angles of an octagon (eight-sided polygon) is 1080 degrees.

1. 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 the known angles of the polygon and then subtract that sum from total amount all its corners. 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.

   • If two angles of a triangle are equal to 60 degrees and 80 degrees, respectively, add these numbers. The result will be 140 degrees. Then subtract this amount from the total sum of all angles of the triangle, that is, from 180 degrees: 180 - 140 = 40 degrees. (A triangle whose angles are all unequal is called equilateral.)
   • You can write this solution as the formula a = 180 - (b + c), where a is the angle whose value needs to be found, b and c are the values ​​of the known angles. For polygons with more than three sides, replace 180 with the sum of the angles of the polygon of that type and add one term to the sum in parentheses for each known angle.
   • Some polygons have their own "tricks" that will help you calculate an unknown angle. For example, an isosceles triangle is a triangle with two equal sides and two equal angles. A parallelogram is a quadrilateral whose opposite sides and opposite angles are equal.

   Calculating the angles of a right triangle

   1. 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 following:

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 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.

Online calculator.
Solving triangles.

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 your math or algebra homework done as quickly as possible? 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.

Solve triangle
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A little theory.

Theorem of sines

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 of sines
Cosine 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 triangles

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 life, we will often have to deal with mathematical problems: at school, at university, and then helping our child with completing homework. People in certain professions will encounter mathematics on a daily basis. Therefore, it is useful to memorize or recall mathematical rules. In this article we will look at one of them: finding the side of a right triangle.

What is a right triangle

First, let's remember what a right triangle is. A right triangle is geometric figure of three segments that connect points that do not lie on the same straight line, and one of the angles of this figure is 90 degrees. The sides forming a right angle are called legs, and the side that lies opposite the right angle is called the hypotenuse.

Finding the leg of a right triangle

There are several ways to find out the length of the leg. I would like to consider them in more detail.

Pythagorean theorem to find the side of a right triangle

If we know the hypotenuse and the leg, then we can find the length of the unknown leg using the Pythagorean theorem. It sounds like this: “The square of the hypotenuse is equal to the sum of the squares of the legs.” Formula: c²=a²+b², where c is the hypotenuse, a and b are the legs. We transform the formula and get: a²=c²-b².

Example. The hypotenuse is 5 cm, and the leg is 3 cm. We transform the formula: c²=a²+b² → a²=c²-b². Next we solve: a²=5²-3²; a²=25-9; a²=16; a=√16; a=4 (cm).

Trigonometric ratios to find the leg of a right triangle

You can also find an unknown leg if any other side and any acute angle of a right triangle are known. There are four options for finding a leg using trigonometric functions: sine, cosine, tangent, cotangent. To solve problems, the table below will help us. Let's consider these options.

Find the leg of a right triangle using sine

The sine of an angle (sin) is the ratio of the opposite side to the hypotenuse. Formula: sin=a/c, where a is the leg opposite the given angle, and c is the hypotenuse. Next, we transform the formula and get: a=sin*c.

Example. The hypotenuse is 10 cm and angle A is 30 degrees. Using the table, we calculate the sine of angle A, it is equal to 1/2. Then, using the transformed formula, we solve: a=sin∠A*c; a=1/2*10; a=5 (cm).

Find the leg of a right triangle using cosine

The cosine of an angle (cos) is the ratio of the adjacent leg to the hypotenuse. Formula: cos=b/c, where b is the leg adjacent to a given angle, and c is the hypotenuse. Let's transform the formula and get: b=cos*c.

Example. Angle A is equal to 60 degrees, the hypotenuse is equal to 10 cm. Using the table, we calculate the cosine of angle A, it is equal to 1/2. Next we solve: b=cos∠A*c; b=1/2*10, b=5 (cm).

Find the leg of a right triangle using tangent

Tangent of an angle (tg) is the ratio of the opposite side to the adjacent side. Formula: tg=a/b, where a is the side opposite to the angle, and b is the adjacent side. Let's transform the formula and get: a=tg*b.

Example. Angle A is equal to 45 degrees, the hypotenuse is equal to 10 cm. Using the table, we calculate the tangent of angle A, it is equal to Solve: a=tg∠A*b; a=1*10; a=10 (cm).

Find the leg of a right triangle using cotangent

Angle cotangent (ctg) is the ratio of the adjacent side to the opposite side. Formula: ctg=b/a, where b is the side adjacent to the angle, and is the opposite side. In other words, cotangent is an “inverted tangent.” We get: b=ctg*a.

Example. Angle A is 30 degrees, the opposite leg is 5 cm. According to the table, the tangent of angle A is √3. We calculate: b=ctg∠A*a; b=√3*5; b=5√3 (cm).

So now you know how to find a leg in a right triangle. As you can see, it’s not that difficult, the main thing is to remember the formulas.

A triangle is a primitive polygon bounded on a plane by three points and three segments connecting these points in pairs. The angles in a triangle are acute, obtuse and right. The sum of the angles in a triangle is continuous and equal to 180 degrees.

You will need

• Basic knowledge of geometry and trigonometry.

Instructions

1. Let us denote the lengths of the sides of the triangle as a=2, b=3, c=4, and its angles as u, v, w, each of which lies opposite to one of the sides. According to the cosine theorem, the square of the length of a side of a triangle is equal to the sum of the squares of the lengths of the other 2 sides minus twice the product of these sides and the cosine of the angle between them. That is, a^2 = b^2 + c^2 – 2bc*cos(u). Let's substitute the lengths of the sides into this expression and get: 4 = 9 + 16 – 24cos(u).

2. Let us express cos(u) from the resulting equality. We get the following: cos(u) = 7/8. Next we will find the actual angle u. To do this, let's calculate arccos(7/8). That is, angle u = arccos(7/8).

3. Similarly, expressing the other sides in terms of the others, we find the remaining angles.

Note!
The value of one angle cannot exceed 180 degrees. The sign arccos() cannot contain a number larger than 1 and smaller than -1.

Helpful advice
In order to detect all three angles, it is not necessary to express all three sides, it is allowed to detect only 2 angles, and the 3rd is obtained by subtracting the value of the remaining 2 from 180 degrees. This follows from the fact that the sum of all angles of a triangle is continuous and equal to 180 degrees.