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. full angle, or revolution, is equal to 360 degrees. You can calculate the angle of a polygon if you know the type of the polygon and the magnitude of its other angles, or, in the case right triangle, the length of two of its sides.
Steps
Calculating the corners of a polygon
- 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 (octagonal polygon) is 1080 degrees.
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Determine if the polygon is regular. A regular polygon is one in which all sides and all angles are equal to each other. Examples regular polygons can serve as an equilateral triangle and square, while the Pentagon building in Washington was built in the shape of a regular pentagon, and road sign"stop" has the shape of a regular octagon.
Add up the known angles of the polygon, and then subtract that sum from total amount all its corners. Most geometry problems of this kind are about triangles or quadrilaterals, since they require less input, so we will do the same.
- If two angles of a triangle are 60 degrees and 80 degrees, respectively, add those numbers. Get 140 degrees. Then subtract this sum from the total sum of all angles of the triangle, i.e. from 180 degrees: 180 - 140 = 40 degrees. (A triangle, all angles of which are unequal to each other, is called non-equilateral.)
- You can write this solution as a = 180 - (b + c), where a is the angle you want to find, b and c are the known angles. For polygons with more than three sides, replace 180 with the sum of the angles of the given type of polygon, and add one term to the sum in brackets for each known angle.
- Some polygons have their own "tricks" to help you calculate the unknown angle. For example, an isosceles triangle is a triangle with two equal parties and two equal angles. A parallelogram is a quadrilateral opposite sides and whose opposite angles are equal.
Calculating the angles of a right triangle
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Determine what data you know. A right triangle is so called because one of its angles is right. You can find the value of one of the two remaining angles if you know one of the following values:
Determine which trigonometric function to use. Trigonometric functions express the ratios of two of the three sides of a triangle. There are six trigonometric functions, but the following are the most commonly used:
Count the number of corners in the polygon.
Find the sum of all angles of the polygon. The formula for finding the sum of all internal corners The polygon looks like (n - 2) x 180, where n is the number of sides and corners of the polygon. Here are the angle sums of some common polygons:
The first are segments that are adjacent to the right angle, and the hypotenuse is the longest part of the figure and is opposite the 90 degree angle. A Pythagorean triangle is one whose sides are equal natural numbers; their lengths in this case are called the "Pythagorean triple".
egyptian triangle
In order for the current generation to learn geometry in the form in which it is taught at school now, it has been developed for several centuries. The fundamental point is the Pythagorean theorem. The sides of a rectangle are known to the whole world) are 3, 4, 5.
Few people are not familiar with the phrase "Pythagorean pants are equal in all directions." However, in fact, the theorem sounds like this: c 2 (the square of the hypotenuse) \u003d a 2 + b 2 (the sum of the squares of the legs).
Among mathematicians, a triangle with sides 3, 4, 5 (cm, m, etc.) is called "Egyptian". It is interesting 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 on which 12 knots were tied. In this case, the probability of constructing a right-angled triangle increased to 95%.
Signs of equality of figures
- An acute angle in a right triangle and a large side, which are equal to the same elements in the second triangle, is an indisputable sign of the equality of the 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 in the second criterion.
- When two figures are superimposed on each other, we rotate them in such a way 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.
By the first sign, it is very easy to prove that the triangles are really equal, the main thing is that the two smaller sides (i.e., the legs) are equal to each other.
The triangles will be the same according to the II sign, the essence of which is the equality of the leg and the acute angle.
Right angle triangle properties
Height lowered from right angle, splits the figure into two equal parts.
The sides of a right triangle and its median are easy to recognize by the rule: the median, which is lowered to 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 o, 45 o and 60 o apply.
- At an angle that is 30 °, it should be remembered that the opposite leg will be equal to 1/2 of the largest side.
- If the angle is 45 o, then the second sharp corner also 45 o. This suggests that the triangle is isosceles, and its legs are the same.
- The property of an angle of 60 degrees is that the third angle has a measure of 30 degrees.
The area is easy to find by one of three formulas:
- through the height and the side on which it descends;
- according to Heron's formula;
- along the sides and the angle between them.
The sides of a right triangle, or rather the legs, converge with two heights. 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 the ratio of twice the area and the length of the hypotenuse. The most common expression among students is the first, as it requires less calculations.
Theorems that apply to a right triangle
The geometry of a right triangle includes the use of theorems such as:
The circle inscribed in it (r). To do this, increase it six times and divide by Square root from the three: A \u003d r * 6 / √3.
Knowing the radius (R), you can also calculate the length sides(A) correct triangle. This radius is twice the one 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 into three: A = P / 3.
AT isosceles triangle length calculation sides along a known perimeter is a little more difficult - you also need to know the length of at least one of the sides. If the length is known sides And, 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 \u003d (P-A) / 2. And if the lateral side is known, then determine the length of the base by subtracting the double length of the lateral side from the perimeter: A \u003d P-2 * B.
Knowing the area (S) occupied in the plane by a regular triangle is also sufficient to find its length sides(BUT). Take the square root of the ratio of the area and the root of the triple, and double the result: A \u003d 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 desired side is (C), to do this, find the square root of the lengths of the known sides (A and B) squared: C \u003d √ (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 \u003d √ (C²-B²).
Sources:
- how to calculate the side of an equilateral triangle
In the general case, i.e. when there is no data on whether a triangle is equilateral, isosceles, rectangular, 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 so - the theorem of sines, cosines and tangents.
Instruction
One way to calculate the lengths of the sides of an arbitrary triangle assumes the sine theorem. According to it, the ratio of the lengths of the sides of the opposite angles 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 none 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 \u003d A * sin (β) / sin (α).
If one (γ) of two (α and γ) known angles is formed, 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, according to this theorem, can be calculated if we subtract the values of two known angles from 180°. Substitute this value in equality, and you get the formula B \u003d A * sin (180 ° - α - γ) / sin (α).
In geometry, there are often problems related to the sides of triangles. For example, it is often necessary to find the side of a triangle if the other two are known.
Triangles are isosceles, equilateral and equilateral. 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 the legs, and the third is the hypotenuse).
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The length of the sides of a right triangle
The solution of 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 the leg b;
- We put them together;
- From the result obtained, we extract the root of the second degree.
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. This requires a third parameter: it can be an angle, height, area of a triangle, radius of a circle inscribed in it, etc.
If the perimeter is known
In this case, the task is even easier. 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, transferring all known parameters to one side of the equal sign:
2) Substitute 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 the angle and the other two sides, the solution is reduced to calculating the trigonometric equation. Knowing the relationship of 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 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 is not enough.
1) First, we calculate sin γ by 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 sine theorem:
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 line 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 equals 180 degrees.
You will need
- Basic knowledge in geometry and trigonometry.
Instruction
1. Let us denote the lengths of the sides of the triangle a=2, b=3, c=4, and its angles u, v, w, each of which lies on the opposite side of one side. According to the law of cosines, the square of the length of a side of a triangle is equal to the sum of the squares of the lengths of 2 other sides minus twice the product of these sides by the cosine of the angle between them. That is, a^2 = b^2 + c^2 - 2bc*cos(u). We substitute the lengths of the sides into this expression and get: 4 \u003d 9 + 16 - 24cos (u).
2. Let us express cos(u) from the obtained equality. We get the following: cos(u) = 7/8. Next, we find the actual angle u. To do this, we calculate arccos(7/8). That is, the angle u = arccos(7/8).
3. Similarly, expressing the other sides in terms of the rest, we find the remaining angles.
Note!
The value of one angle cannot exceed 180 degrees. The arccos() sign 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 one can be obtained by subtracting the values of the remaining 2 from 180 degrees. This follows from the fact that the sum of all the angles of a triangle is continuous and equals 180 degrees.
