Ability to think mathematically - one of the noblest human abilities.
Irish playwright Bernard Shaw
Heron's formula
In school mathematics, Heron's formula is very popular, the application of which allows you to calculate the area of a triangle along its three sides. At the same time, few of the students know that there is a similar formula for calculating the area of quadrangles inscribed in a circle. Such a formula is called the Brahmagupta formula. There is also a little-known formula for calculating the area of a triangle by its three heights, the derivation of which follows from Heron's formula.
Calculating the area of triangles
Let in a triangle parties, and ... Then the following theorem is true (Heron's formula).
Theorem 1.
where .
Proof. When deriving formula (1), we will use the well-known geomes tric formulas
, (2)
. (3)
From formulas (2) and (3) we obtain and. Since then
. (4)
If we denote then equality (4) implies formula (1). The theorem is proved.
Let us now consider the question of calculating the area of a triangle on condition , that her three heights are known, and .
Theorem 2. The area is calculated by the formula
. (5)
Proof. Since, and, then
In this case, from formula (1) we obtain
or
This implies formula (5). The theorem is proved.
Calculating the area of quadrangles
Consider a generalization of Heron's formula to the case of calculating the area of quadrangles. However, it should be noted right away that such a generalization is possible only for quadrangles that are inscribed in a circle.
Let the quadrangle has sides,, and.
If is a quadrangle, inscribed in a circle, then Theorem 3 (Brahmagupta's formula) holds.
Theorem 3. Square calculated by the formula
where .
Proof. Draw a diagonal in the quadrangle and obtain two triangles and. If we apply the cosine theorem to these triangles, which is equivalent to formula (3), then we can write
Since the quadrilateral is inscribed in a circle, the sum of its opposite angles is equal, i.e. ...
Since or, then from (7) we obtain
Or
. (8)
Since, then. However, and therefore
Since, then from formulas (8) and (9) it follows
If we put, then from this we obtain formula (6). The theorem is proved.
If the inscribed quadrilateralis both described, then formula (6) is greatly simplified.
Theorem 4. The area of a quadrilateral inscribed in one circle and circumscribed around another is calculated by the formula
. (10)
Proof. Since a circle is inscribed in a quadrilateral, the equalities hold
In this case,,,, and formula (6) is easily transformed into formula (10). The theorem is proved.
Let's move on to considering examples of geometry problems, the solution of which is carried out on the basis of the application of the theorems proved.
Examples of problem solving
Example 1. Find area, if .
Solution. Since here, according to Theorem 1, we obtain
Answer: .
Note, if the sides of the triangletake irrational meanings, then calculating its areaby using formula (1), usually , is ineffective. In this case, it is advisable to apply directly formulas (2) and (3).
Example 2. Find the area, if, and.
Solution. Taking into account formulas (2) and (3), we obtain
Since, then or.
Answer: .
Example 3. Find the area, if, and.
Solution. Insofar as ,
then it follows from Theorem 2 that.
Answer: .
Example 4. The triangle has sides, and. Find and, where are the radii of the circumcircle and the incircle, respectively.
Solution. Let's first calculate the area. Since, then from formula (1) we obtain.
It is known that and. That's why .
Example 5. Find the area of a quadrilateral inscribed in a circle, if,, and.
Solution. It follows from the conditions of the example that. Then, according to Theorem 3, we obtain.
Example 6. Find the area of a quadrilateral inscribed in a circle, the sides of which are,, and.
Solution. Since and, equality holds in the quadrangle. However, it is known that the existence of such an equality is a necessary and sufficient condition for a circle to be inscribed in a given quadrangle. In this regard, to calculate the area, you can use the formula (10), from which it follows.
For independent and high-quality preparation for entrance examinations in the field of solving problems of school geometry, you can effectively use teaching aids, listed in the list of recommended reading.
1. Gotman E.G. Planimetry problems and methods for their solution. - M .: Education, 1996 .-- 240 p.
2. Kulagin E. D. , Fedin S.N. Geometry of a triangle in problems. - M .: KD "Librokom" / URSS, 2009 .-- 208 p.
3. Collection of problems in mathematics for applicants to technical colleges / Ed. M.I. Skanavi. - M .: Peace and Education, 2013 .-- 608 p.
4. Suprun V.P. Mathematics for high school students: additional sections of the school curriculum. - M .: Lenand / URSS, 2014 .-- 216 p.
Still have questions?
To get help from a tutor -.
blog. site, with full or partial copying of the material, a link to the source is required.
Ability to think mathematically - one of the noblest human abilities.
Irish playwright Bernard Shaw
Heron's formula
In school mathematics, Heron's formula is very popular, the application of which allows you to calculate the area of a triangle along its three sides. At the same time, few of the students know that there is a similar formula for calculating the area of quadrangles inscribed in a circle. Such a formula is called the Brahmagupta formula. There is also a little-known formula for calculating the area of a triangle by its three heights, the derivation of which follows from Heron's formula.
Calculating the area of triangles
Let in a triangle parties, and ... Then the following theorem is true (Heron's formula).
Theorem 1.
where .
Proof. When deriving formula (1), we will use the well-known geomes tric formulas
, (2)
. (3)
From formulas (2) and (3) we obtain and. Since then
. (4)
If we denote then equality (4) implies formula (1). The theorem is proved.
Let us now consider the question of calculating the area of a triangle on condition , that her three heights are known, and .
Theorem 2. The area is calculated by the formula
. (5)
Proof. Since, and, then
In this case, from formula (1) we obtain
or
This implies formula (5). The theorem is proved.
Calculating the area of quadrangles
Consider a generalization of Heron's formula to the case of calculating the area of quadrangles. However, it should be noted right away that such a generalization is possible only for quadrangles that are inscribed in a circle.
Let the quadrangle has sides,, and.
If is a quadrangle, inscribed in a circle, then Theorem 3 (Brahmagupta's formula) holds.
Theorem 3. Square calculated by the formula
where .
Proof. Draw a diagonal in the quadrangle and obtain two triangles and. If we apply the cosine theorem to these triangles, which is equivalent to formula (3), then we can write
Since the quadrilateral is inscribed in a circle, the sum of its opposite angles is equal, i.e. ...
Since or, then from (7) we obtain
Or
. (8)
Since, then. However, and therefore
Since, then from formulas (8) and (9) it follows
If we put, then from this we obtain formula (6). The theorem is proved.
If the inscribed quadrilateralis both described, then formula (6) is greatly simplified.
Theorem 4. The area of a quadrilateral inscribed in one circle and circumscribed around another is calculated by the formula
. (10)
Proof. Since a circle is inscribed in a quadrilateral, the equalities hold
In this case,,,, and formula (6) is easily transformed into formula (10). The theorem is proved.
Let's move on to considering examples of geometry problems, the solution of which is carried out on the basis of the application of the theorems proved.
Examples of problem solving
Example 1. Find area, if .
Solution. Since here, according to Theorem 1, we obtain
Answer: .
Note, if the sides of the triangletake irrational meanings, then calculating its areaby using formula (1), usually , is ineffective. In this case, it is advisable to apply directly formulas (2) and (3).
Example 2. Find the area, if, and.
Solution. Taking into account formulas (2) and (3), we obtain
Since, then or.
Answer: .
Example 3. Find the area, if, and.
Solution. Insofar as ,
then it follows from Theorem 2 that.
Answer: .
Example 4. The triangle has sides, and. Find and, where are the radii of the circumcircle and the incircle, respectively.
Solution. Let's first calculate the area. Since, then from formula (1) we obtain.
It is known that and. That's why .
Example 5. Find the area of a quadrilateral inscribed in a circle, if,, and.
Solution. It follows from the conditions of the example that. Then, according to Theorem 3, we obtain.
Example 6. Find the area of a quadrilateral inscribed in a circle, the sides of which are,, and.
Solution. Since and, equality holds in the quadrangle. However, it is known that the existence of such an equality is a necessary and sufficient condition for a circle to be inscribed in a given quadrangle. In this regard, to calculate the area, you can use the formula (10), from which it follows.
For independent and high-quality preparation for entrance examinations in the field of solving problems of school geometry, you can effectively use teaching aids, listed in the list of recommended reading.
1. Gotman E.G. Planimetry problems and methods for their solution. - M .: Education, 1996 .-- 240 p.
2. Kulagin E. D. , Fedin S.N. Geometry of a triangle in problems. - M .: KD "Librokom" / URSS, 2009 .-- 208 p.
3. Collection of problems in mathematics for applicants to technical colleges / Ed. M.I. Skanavi. - M .: Peace and Education, 2013 .-- 608 p.
4. Suprun V.P. Mathematics for high school students: additional sections of the school curriculum. - M .: Lenand / URSS, 2014 .-- 216 p.
Still have questions?
To get help from a tutor - register.
site, with full or partial copying of the material, a link to the source is required.
This formula allows you to calculate the area of a triangle along its sides a, b and c:
S = √ (p (p-a) (p-b) (p-c),where p is the semiperimeter of the triangle, i.e. p = (a + b + c) / 2.
The formula is named after the ancient Greek mathematician Heron of Alexandria (around the 1st century). Heron considered triangles with integer sides, the areas of which are also integers. Such triangles are called geron triangles. For example, these are triangles with sides 13, 14, 15 or 51, 52, 53.
There are analogues of Heron's formula for quadrangles. Due to the fact that the problem of constructing a quadrilateral along its sides a, b, c and d has more than a unique solution, to calculate in the general case the area of a quadrilateral it is not enough just to know the lengths of the sides. You have to introduce additional parameters or impose restrictions. For example, the area of an inscribed quadrangle is found by the formula: S = √ (p-a) (p-b) (p-c) (p-d)
If the quadrilateral is both inscribed and circumscribed at the same time, its area is by a simpler formula: S = √ (abcd).
Heron of Alexandria - Greek mathematician and mechanic.
He was the first to invent automatic doors, an automatic puppet theater, a vending machine, a rapid-fire self-loading crossbow, a steam turbine, automatic decorations, a device for measuring the length of roads (ancient odometer), etc. He was the first to create programmable devices (a shaft with pins with a rope ).
He was engaged in geometry, mechanics, hydrostatics, optics. Major works: Metrica, Pneumatics, Automatopoetics, Mechanics (the work is preserved in its entirety in Arabic translation), Catoptrika (the science of mirrors; preserved only in Latin translation), etc. land survey, in fact based on the use of rectangular coordinates. Heron used the achievements of his predecessors: Euclid, Archimedes, Straton of Lampsac. Many of his books are irretrievably lost (the scrolls were kept in the Library of Alexandria).
In the treatise "Mechanics" Heron described five types of the simplest machines: lever, gate, wedge, screw and block.
In the treatise "Pneumatics" Heron described various siphons, ingeniously arranged vessels, automata, driven by compressed air or steam. This is eolipil, which was the first steam turbine - a ball rotated by the force of jets of water vapor; door opener, holy water vending machine, fire pump, water organ, mechanical puppet theater.
The book "On the Diopter" describes the diopter - the simplest device used for geodetic work. Geron sets out in his treatise the rules for land surveying based on the use of rectangular coordinates.
In "Catoptrik" Heron substantiates the straightness of light rays by the infinitely high speed of their propagation. Heron examines various types of mirrors, with particular attention to cylindrical mirrors.
Heron's "Metric" and the "Geometrics" and "Stereometrics" extracted from it are reference books on applied mathematics. Among the information contained in the "Metric":
Formulas for the areas of regular polygons.
Volumes of regular polyhedra, pyramid, cone, truncated cone, torus, spherical segment.
Heron's formula for calculating the area of a triangle by the lengths of its sides (discovered by Archimedes).
Rules for the numerical solution of quadratic equations.
Algorithms for extracting square and cube roots.
Heron's book "Definitions" is an extensive collection of geometric definitions, for the most part coinciding with the definitions of Euclid's "Principles".
Heron's formula Gerona formula
expresses area s a triangle through the lengths of its three sides a, b and with and semi-perimeter R = (a + b + with) / 2:. Named for Heron of Alexandria.
HERONA FORMULAHERONA FORMULA, expresses area S a triangle through the lengths of its three sides a, b and c and semi-perimeter P = (a + b + c)/2
Named for Heron of Alexandria.
encyclopedic Dictionary. 2009 .
See what "Herona's formula" is in other dictionaries:
Expresses the area S of a triangle in terms of the lengths of its three sides a, b and c and the semiperimeter P = (a + b + c) / 2 Named after Heron of Alexandria ... Big Encyclopedic Dictionary
Formula expressing the area of a triangle through its three sides. Namely, if a, b, with the lengths of the sides of the triangle, and S is its area, then G. f. has the form: where p denotes the semiperimeter of the triangle of the G. f. ... ...
Formula expressing the area of a triangle through its sides a, b, c: where Named for Heron (c. 1st century AD), A.B. Ivanov ... Encyclopedia of mathematics
Expresses the area 5 of a triangle in terms of the lengths of its three sides a, b and c and the semiperimeter p = (a + b + c) / 2: s = sq. root p (p a) (p b) (p c). Named after the Hero of Alexandria ... Natural science. encyclopedic Dictionary
- ... Wikipedia
Allows you to calculate the area of a triangle (S) by its sides a, b, c: where p is the triangle semiperimeter:. Proof where the angle of the triangle is ... Wikipedia
Expresses the area of a quadrilateral inscribed in a circle as a function of the lengths of its sides. If an inscribed quadrilateral has side lengths and a semi-perimeter, then its area is ... Wikipedia
This article is missing links to sources of information. The information must be verifiable, otherwise it can be questioned and deleted. You can edit this article by adding links to authoritative sources. This mark ... ... Wikipedia
- (Heronus Alexandrinus) (years of birth and death unknown, probably 1st century), an ancient Greek scientist who worked in Alexandria. The author of works in which he systematically outlined the main achievements of the ancient world in the field of applied mechanics, In ... ... Great Soviet Encyclopedia
Alexandrian (Heronus Alexandrinus) (years of birth and death unknown, probably 1st century), an ancient Greek scientist who worked in Alexandria. The author of works in which he systematically outlined the main achievements of the ancient world in the field of ... ... Great Soviet Encyclopedia
