How to find the length of the hypotenuse

06.06.2023 | Education

The hypotenuse is the longest side of a right triangle. It is located opposite an angle of ninety degrees and is calculated, as usual, according to the theorem of the ancient Greek scientist, Pythagoras, known from the seventh grade. It sounds like this: "square hypotenuse, is equal to the sum of the squares of the legs. It looks menacing, but is solved primitively. There are other ways to find the length of a given side of a triangle.

You will need

Bradis table, calculator.

Instruction

1. If you need to calculate the hypotenuse using the Pythagorean theorem, use the following algorithm: - Determine in the triangle which sides are the legs and which are the hypotenuse. The two sides that form an angle of ninety degrees are the legs, the remaining third side of the triangle is the hypotenuse. (see figure) - Raise the entire leg of this triangle to the second power, that is, multiply their value by itself. Example 1. Suppose you need to calculate the hypotenuse if one leg in a triangle is 12 cm and the other is 5 cm. Firstly, the squares of the legs are: 12 * 12 = 144 cm and 5 * 5 = 25 cm. - Next, determine the sum of the squares legs. A certain number is a square hypotenuse, then you need to get rid of the 2nd power of the number in order to find length this side of the triangle. To do this, extract from under the square root the value of the sum of the squares of the legs. Example 1. 144+25=169. The square root of 169 will be 13. Consequently, the length of this hypotenuse equal to 13 cm.

2. Another method for calculating length hypotenuse lies in the terminology of the sine and cosine of angles in a triangle. By definition: the sine of angle alpha is the ratio of the opposite leg to the hypotenuse. That is, looking at the figure, sin a \u003d CB / AB. Otsel, hypotenuse AB \u003d CB / sin a. Example 2. Let the angle a be 30 degrees, and the opposite leg - 4 cm. It is necessary to detect the hypotenuse. Solution: AB \u003d 4 cm / sin 30 \u003d 4 cm / 0.5 \u003d 8 cm. Result: length hypotenuse equal to 8 cm.

3. A similar method of finding hypotenuse from the definition of the cosine of an angle. The cosine of an angle is the ratio of the leg adjacent to it and hypotenuse. That is, cos a \u003d AC / AB, otsel AB \u003d AC / cos a. Example 3. In the triangle ABC, AB is the hypotenuse, the angle BAC is 60 degrees, the leg AC is 2 cm. Find AB. Solution: AB \u003d AC / cos 60 \u003d 2 / 0.5 \u003d 4 cm. in length.

Tip 2: How to find the length of the hypotenuse in a right triangle

The hypotenuse is called the longest of the sides in a right triangle, therefore it is not miraculous that this word is translated from Greek as “stretched”. This side invariably lies opposite the angle of 90 °, and the sides forming this angle are called the legs. Knowing the lengths of these sides and the magnitude of acute angles in various combinations of these values, it is possible to calculate the length of the hypotenuse.

Instruction

1. If the lengths of both legs of the triangle (A and B) are known, then use the most probably known mathematical postulate on our planet, the Pythagorean theorem, to find the length of the hypotenuse (C). It says that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs, from which it follows that you should calculate the square root of the sum of the squared lengths of the 2 famous sides: C \u003d? (A? + B?). Say, if the length of one leg is 15 centimeters, and the other is 10 centimeters, then the length of the hypotenuse will be approximately 18.0277564 centimeters, because? (15? + 10?) \u003d? (225 + 100) \u003d? .

2. If the length of only one of the legs (A) in a right triangle is known, as well as the value of the angle lying opposite it (?), then the length of the hypotenuse (C) can be determined with the support of one of the trigonometric functions - the sine. To do this, divide the length of the known side by the sine of the known angle: C \u003d A / sin (?). Say, if the length of one of the legs is 15 centimeters, and the angle at the opposite vertex of the triangle is 30 °, then the length of the hypotenuse will be 30 centimeters, because 15 / sin (30 °) \u003d 15 / 0.5 \u003d 30.

3. If in a right triangle the value of one of the acute angles (?) and the length of the leg adjacent to it (B) are known, then to calculate the length of the hypotenuse (C) it is allowed to use another trigonometric function - cosine. You should divide the length of the driven leg by the cosine of the famous angle: C \u003d B / cos (?). Say, if the length of this leg is 15 centimeters, and the value of the acute angle adjacent to it is 30 °, then the length of the hypotenuse will be approximately 17.3205081 centimeters, because 15 / cos (30 °) \u003d 15 / (0.5 * ?3)=30/?3?17.3205081.

Length is the distance between two points on a line segment. It can be a straight line, a broken line, or a closed line. It is possible to calculate the length in a fairly simple way, if you know some other indicators of the segment.

Instruction

1. If you need to find the length of a side of a square, then it will not be difficult if you know its area S. Due to the fact that all sides of a square have identical lengths, you can calculate the value of one of them using the formula: a \u003d? S.

2. In the case when you need to calculate the length of a side of a rectangle, use the values of its area s and the length of the other side b. From the formula a=S/b you get the desired value.

3. In order to determine the length of a circle, that is, a closed line that forms a circle, use the values: r - its radius and D - diameter. The diameter can be calculated by multiplying the radius of the circle by 2. Substitute the values you know into the formula for determining the circumference: C=2?r=?D, where?=3.14.

4. To calculate the length of an ordinary segment, use the experimental method. That is, take a ruler and measure.

5. In order to calculate the length of a side of a shape such as a triangle, you will need the dimensions of the other 2 sides, as well as the angles. If you are dealing with a right triangle, and one of its angles is equal to 60 degrees, then the value of its leg can be determined by the formula a=c*cos?, where c is the hypotenuse of the triangle, huh? is the angle between the hypotenuse and the leg.

6. In addition, if you have such famous quantities as the height b and area S of a triangle, then the length of the side that is the base can be found out due to the formula a \u003d 2? S /?? b.

7. As for a positive polygon, the length of its side can be calculated using the formula an=2R*sin(?/2)=2r*tg(?/2), where R is the radius of the circumscribed circle, r is the radius of the inscribed circle, n is the number corners.

8. If you want to calculate the length of an equilateral figure around which a circle is described, then this can be done using the formula an=R?3, where R is the radius of the circle, n is the number of corners of the figure.

Related videos