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 right triangle 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, solving the 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) Substitute the values ​​instead of them 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.

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.

The lengths of the sides (a, b, c) are known, use the cosine theorem. It states that the square of the length of any of the sides is equal to the sum of the squares of the lengths of the other two, from which twice the product of the lengths of the same two sides by the cosine of the angle between them is subtracted. You can use this theorem to calculate the angle at any of the vertices; it is important to know only its location relative to the sides. For example, to find the angle α that lies between sides b and c, the theorem must be written as follows: a² = b² + c² - 2*b*c*cos(α).

Express the cosine of the desired angle from the formula: cos(α) = (b²+c²-a²)/(2*b*c). To both sides of the equality, apply the inverse function of cosine - arc cosine. It allows you to restore the angle in degrees using the cosine value: arccos(cos(α)) = arccos((b²+c²-a²)/(2*b*c)). The left side can be simplified and the calculation of the angle between sides b and c will take the final form: α = arccos((b²+c²-a²)/2*b*c).

When finding the values ​​of acute angles in a right triangle, knowing the lengths of all sides is not necessary; two of them are sufficient. If these two sides are legs (a and b), divide the length of the one opposite the desired angle (α) by the length of the other. This way you will get the tangent value of the desired angle tg(α) = a/b, and applying equalities to both sides inverse function- arctangent - and simplifying, as in the previous step, left side, derive the final formula: α = arctan(a/b).

If the known sides are the leg (a) and the hypotenuse (c), to calculate the angle (β) formed by these sides, use the cosine function and its inverse - arc cosine. The cosine is determined by the ratio of the length of the leg to the hypotenuse, and the formula in its final form can be written as follows: β = arccos(a/c). To calculate using the same initial acute angle(α), lying opposite famous leg, use the same relationship, replacing arccosine with arcsine: α = arcsin(a/c).

Sources:

• triangle formula with 2 sides

Tip 2: How to find the angles of a triangle by the lengths of its sides

There are several options for finding the values ​​of all angles in a triangle if the lengths of its three are known parties. One way is to use two different formulas area calculations triangle. To simplify calculations, you can also apply the sine theorem and the sum of angles theorem triangle.

Instructions

Use, for example, two formulas for calculating area triangle, one of which involves only three of his known parties s (Heron), and in the other - two parties s and the sine of the angle between them. Using in the second formula different couples parties, you can determine the magnitude of each of the angles triangle.

Solve the problem in general form. Heron's formula determines the area triangle, How Square root from the product of the semi-perimeter (half of all parties) on the difference between the semi-perimeter and each of parties. If we replace with the sum parties, then the formula can be written in this form: S=0.25∗√(a+b+c)∗(b+c-a)∗(a+c-b)∗(a+b-c).C other parties s area triangle can be expressed as half the product of its two parties by the sine of the angle between them. For example, for parties a and b with an angle γ between them, this formula can be written as follows: S=a∗b∗sin(γ). Replace the left side of the equality with Heron's formula: 0.25∗√(a+b+c)∗(b+c-a)∗(a+c-b)∗(a+b-c)=a∗b∗sin(γ). Derive from this equality the formula for

In geometry, an angle is a figure formed by two rays emanating from one point (the vertex of the angle). Most often, angles are measured in degrees, while full angle, or revolution, is equal to 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.

Steps

Calculating Polygon Angles

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:

• 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 regular polygons can serve as an equilateral triangle and a square, while the Pentagon building in Washington is built in the shape regular pentagon, A road sign“stop” has the shape of 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 the majority geometric problems of such 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 a formula a = 180 - (b + c), where a is the angle whose value you need to find, b and c are the values 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, isosceles triangle is a triangle with two equal sides and two equal angles. A parallelogram is a quadrilateral opposite sides And opposite angles which 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:

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