Diameter of the circumscribed circle of a triangle. Summary of the lesson "circumscribed circle". How to find the radius of a circle circumscribed about a triangle - the general formula
In this part, we will discuss the circumscribed circle around (often said "near") a triangle. First of all, let's give a definition.
1. Existence and center of the circumscribed circle
Here the question arises: does such a circle exist for any triangle? It turns out that yes, for everyone. And moreover, we will now formulate a theorem that also answers the question, where is the center of the circumscribed circle.
Formulas for Finding the Length of the Diagonal of a Rectangle
Formula on the side of a rectangle in diameter and angle β. It is called the diagonal of a rectangle for any segment that connects two vertices of opposite corners of the rectangle. The formula for the diagonal of a rectangle in terms of the two sides of the rectangle.
The formula for the diagonal of a rectangle in area and on both sides. The formula for the diagonal of a rectangle along the perimeter and on both sides. The formula for the diagonal of a rectangle in terms of the radius of the circumscribed circle. The formula for the diagonal of a rectangle along the circumference of a circle.
Look, like this:
Let's muster up the courage and prove this theorem. If you have already read the topic “”, figured out why the three bisectors intersect at one point, then it will be easier for you, but if you haven’t read it, don’t worry: now we’ll figure it all out.
We will carry out the proof using the concept of the locus of points (LPT).
The formula for the diagonal of a rectangle in terms of the sine of the angle adjacent to the diagonal and the length of the side opposite that angle. The formula for the diagonal of a rectangle in terms of the cosine of the angle adjacent to the diagonal and the length of the side adjacent to that angle. The formula for the diagonal of a rectangle in terms of the sine of the acute angle between the diagonals and the area of the rectangle.
Formulas for determining the length of the perimeter of a rectangle
The perimeter of a rectangle is called the sum of the lengths of all sides of the rectangle. The formula for the perimeter of a rectangle in terms of the two sides of the rectangle. The formula for the perimeter of a rectangle given the area and both sides. The formula for the perimeter of a rectangle along the diagonal and on both sides.
Well, for example, is the set of balls a "geometric place" of round objects? No, of course, because there are round ... watermelons. But is a set of people, a “geometric place”, able to speak? Neither, because there are babies who cannot speak. In life, it is generally difficult to find an example of a real “geometric place of points”. Geometry is easier. Here, for example, is just what we need:
The formula for the perimeter of a rectangle in terms of the radius of the circumscribed circle and on both sides. The formula for the perimeter of a rectangle given the diameter of the circumscribed circle and both sides. It is called the area of the rectangle in the space bounded by the sides of the rectangle, that is, inside the perimeter area of the rectangle.
Formulas for determining the area of a rectangle
Formula for the area of a rectangle with two sides. The formula for the area of a rectangle along the perimeter and on both sides. The formula for the area of a rectangle along the diagonal and on both sides. The formula for the area of a rectangle along the diagonal and the sine of the acute angle between the diagonals.
Here the set is the middle perpendicular, and the property "" is "to be equidistant (point) from the ends of the segment."
Let's check? So, you need to make sure of two things:
Connect with and with. Then the line is the median and height in. So, - isosceles, - we made sure that any point lying on the perpendicular bisector is equally distant from the points and.
Circle bounded around a rectangle
The formula for the area of a rectangle given the radius of the circumscribed circle and any side. The formula for the area of a rectangle in a circle is a circle of a circle and on both sides. It is called a circle bounded around a rectangle, to a circle passing through the four vertices of the rectangle, the center of which is the intersection of the diagonals of the rectangle.
Formulas for finding the radius of a circle circumscribed around a rectangle
The formula for the radius of a circle bounded by a rectangle through two sides. The formula for the radius of a circle circumscribed around a rectangle along the perimeter of the square and either side. The formula for the radius of a circle described around a rectangle in terms of the area of the rectangle and the length of one of its sides.
Take - the middle and connect and. Got the median. But - isosceles by condition, not only the median, but also the height, that is, the median perpendicular. This means that the point exactly lies on the perpendicular bisector.
All! We have fully verified the fact that the perpendicular bisector to a segment is the locus of points equidistant from the ends of the segment.
The formula for the radius of a circle circumscribed around a rectangle along the diagonal of the rectangle. The formula for the radius of a circle described around a rectangle by the circumferential diameter of the circle. The formula for the radius of a circle described around a rectangle in terms of the sine of the angle adjacent to the diagonal and the length of the side opposite this angle.
The circle surrounding the given polygon is the circumscribed circle. Most people have heard the terms circumference and radius, but bounded circle is a less familiar term. Imagine a 2D polygon with straight sides, like a triangle. Imagine a circle around a triangle so that it touches all three of its vertices; it's a limited circle. To calculate your radius, just use some simple algebra and a calculator.
That's all well and good, but have we forgotten about the circumscribed circle? Not at all, we just prepared ourselves a "bridgehead for the attack."
Consider a triangle. Let's draw two median perpendiculars and, say, to the segments and. They will intersect at some point, which we will name.
Check all your measurements and make sure the compass says it doesn't change while you're circling.
- It is extremely important to accurately and accurately measure.
- Not all polygons can have a bounded circle.
Thus, the area of a circle inscribed in a hexagon is equal to. Now we turn to the inscribed square. Since the diagonal of the square is equal to the side of the root times, we have. Then the radius of a circle inscribed in a square will have half the measure of the side. Thus, you will see that the radius of a circle inscribed in a triangle is half the radius of a circle circumscribed in a triangle, since the circumference of a circle is described by a divided medium into two parts proportional to 1, and then the radius of the inscribed circle will be.
And now, attention!
The point lies on the perpendicular bisector;
the point lies on the perpendicular bisector.
And that means and.
Several things follow from this:
Firstly, the point must lie on the third perpendicular bisector, to the segment.
Circle measurement and approximation
Thus, the area of the inscribed circle will be. And the problem is to prove that the sum of these areas is equal to the area of the inner circle. So, let's do the sum of the areas of the crowns. This is the sum of the areas of the crowns. You can see that we have 3 circles of rays. This article proposes a hypothesis, a pretext for considering some points of the mathematics of this period in Mesopotamia.
By forming a circle between two hexagons, the perimeter of which can be easily calculated, then successively doubling the number of sides, he obtains a frame with polygons with 96 sides. It's probably easier to evaluate performance by specifying numeric values.
That is, the perpendicular bisector must also pass through the point, and all three perpendicular bisectors intersect at one point.
Secondly: if we draw a circle with a center at a point and a radius, then this circle will also pass through the point and through the point, that is, it will be the circumscribed circle. This means that it already exists that the intersection of the three perpendicular bisectors is the center of the circumscribed circle for any triangle.
We see here the interest and effectiveness of observational methods: on the one hand, they provide an approximation, and on the other hand, they allow you to control a perfect mistake. As far as I know, Archimedes is the first to explicitly justify his results regarding the circle, and gives step by step a series of arguments explaining why what he claims is true. But he is not the first who is interested in the circle and its measure. We have very ancient evidence, one in Egypt and some others in Mesopotamia, going in that direction.
And there are some "Babylonian" clay tablets dating from the same period and along the perimeter or area of the disk. This is the subject of this article. The walk can start but before leaving for Babylon 17th or 18th century BC. A Babylonian clay tablet has been found which gives the ratio of the perimeter of a hexagon to the perimeter of its circumscribed circle.
And the last thing: about uniqueness. It is clear (almost) that the point can be obtained in a unique way, and therefore the circle is also unique. Well, "almost" - we'll leave it up to you. Here we have proved the theorem. You can shout "Hurrah!".
And if the problem is the question "find the radius of the circumscribed circle"? Or vice versa, the radius is given, but you need to find something else? Is there a formula relating the radius of the circumscribed circle to the other elements of a triangle?
And they are often added in one form or another. This was the second one that gave me a problem: could the Babylonians really find this value experimentally? Currently, the experience is easy to use, with a seamstress meter and everyday objects of different diameters: frying pan, saucepan, tin cans. The perimeter and diameter are measured and the division is made. The differences were in the third decimal place. Of course, there are no such precise industrial items, nor seamstress's meters graduated in millimeters.
For measurement, it is more thin: rope, leather strap, can stretch under tension and contract at ease. On the other hand, dried papyrus bark does not elongate. Unfortunately, I didn't have it. Obviously, they are not graduated, but this is not serious: we are interested in the ratio of two lengths: from the perimeter and diameter, and not from the lengths themselves. It is easy to go around an object with rattan and cut it. Curiously, it is less easy to accurately cut rattan thread to match the diameter. Indeed, the upper edge of the pottery is often rounded.
Very often, when solving geometric problems, you have to perform actions with auxiliary figures. For example, find the radius of an inscribed or circumscribed circle, etc. This article will show you how to find the radius of a circle circumscribing a triangle. Or, in other words, the radius of the circle in which the triangle is inscribed.
Therefore, it is necessary to fix the rattan around the perimeter, and then cut the second rattan knife, corresponding to the inner diameter of the first one. It remains to calculate the ratio of the two lengths without knowing them exact values, which can be done by returning to the very origin of division. 
My best performance was 6, 8 times, that is 6 times, with a wonderful rest. However, these experiments convinced me that the Babylonians did not receive experimental value, at least not from this route. But in this case, two questions arise.
How to find the radius of a circle circumscribed about a triangle - the general formula
The general formula is as follows: R = abc/4√p(p - a)(p - b)(p - c), where R is the radius of the circumscribed circle, p is the perimeter of the triangle divided by 2 (half-perimeter). a, b, c are the sides of the triangle.
Find the radius of the circumcircle of the triangle if a = 3, b = 6, c = 7.
Question 1: If it is not experimental, it is theoretical, geometric. 
Thus, the perimeter P of a circle is greater than that of a hexagon, and the ratio between them is equal. It is quite natural for us to estimate the perimeter of the circle, unknown in relation to what is known to the hexagon itself.
Later we will see what to think about it. Hypothesis 1: The Babylonians knew the Pythagorean Theorem a thousand years before. Hypothesis 2: They knew they had found rectangles of integer sides. Maybe not all, but at least those whose hypotenuse and one of the sides are whole integers.
Thus, based on the above formula, we calculate the semi-perimeter:
p = (a + b + c)/2 = 3 + 6 + 7 = 16. => 16/2 = 8.
Substitute the values in the formula and get:
R = 3 x 6 x 7/4√8(8 - 3)(8 - 6)(8 - 7) = 126/4√(8 x 5 x 2 x 1) = 126/4√80 = 126/16 √5.
Answer: R = 126/16√5
How to find the radius of a circle circumscribed about an equilateral triangle
To find the radius of a circle circumscribed about an equilateral triangle, there is a fairly simple formula: R = a/√3, where a is the size of its side.
Of course, there is no evidence that the Babylonians did this. Only the discovery of a new clay tablet can do this. Moreover, this idea is based on hypotheses 1 and did the Babylonians really know how to find such triangles? 
The last 15 lines are divided into 4 columns, the first two lines of which define the content. Column 4 contains the sign followed by the numbers from 1 to.
Example: The side of an equilateral triangle is 5. Find the radius of the circumscribed circle.
Since all sides of an equilateral triangle are equal, to solve the problem, you just need to enter its value in the formula. We get: R = 5/√3.
Answer: R = 5/√3.
How to find the radius of a circle circumscribed about a right triangle
The formula looks like this: R = 1/2 × √(a² + b²) = c/2, where a and b are legs and c is the hypotenuse. If we add the squares of the legs in a right triangle, we get the square of the hypotenuse. As can be seen from the formula, this expression is under the root. By calculating the root of the square of the hypotenuse, we get the length itself. Multiplying the resulting expression by 1/2 eventually leads us to the expression 1/2 × c = c/2.
Columns 1, 2 and 3 refer to right triangles. Columns 2 and 3 respectively give the smallest side and hypotenuse of each triangle. Column 1 gives the square of the ratio on either side of the right corner. For example, line 5 begins in the sexual system.
This is too precise to be true, but you can check that it's the same for other strings. There is controversy about the methods the Babylonians might have used to compile this table. There are also errors in the scribe's copy and little mysteries, such as line 11, which gives 45 and 1 15 as sides, that is, in decimal numbering: 45, and both are multiples of 15, and this triangle is nothing but a triangle of sides . Why is it not given in this form, much simpler? And what could be the advantage to give the square of the slope rather than the slope itself?
Example: Calculate the radius of the circumscribed circle if the legs of the triangle are 3 and 4. Substitute the values into the formula. We get: R = 1/2 × √(3² + 4²) = 1/2 × √25 = 1/2 × 5 = 2.5.
In this expression, 5 is the length of the hypotenuse.
Answer: R = 2.5.
How to find the radius of a circle circumscribed about an isosceles triangle
The formula looks like this: R = a² / √ (4a² - b²), where a is the length of the thigh of the triangle and b is the length of the base.
Example: Calculate the radius of a circle if its hip = 7 and its base = 8.
Solution: We substitute these values \u200b\u200binto the formula and get: R \u003d 7² / √ (4 × 7² - 8²).
R = 49/√(196 - 64) = 49/√132. The answer can be written directly like this.
Answer: R = 49/√132
Online Resources for Calculating the Radius of a Circle
It is very easy to get confused in all these formulas. Therefore, if necessary, you can use online calculators, which will help you in solving problems on finding the radius. The principle of operation of such mini-programs is very simple. Substitute the value of the side in the appropriate field and get a ready-made answer. You can choose several options for rounding the answer: to decimals, hundredths, thousandths, etc.
