How to find the radius of an inscribed circle? Inscribed and excircle circle. Visual guide with examples (2019)

The circle is considered to be inscribed within the boundaries regular polygon, in case it lies inside it, touching the straight lines that pass through all sides. Let's look at how to find the center and radius of a circle. The center of the circle will be the point at which the bisectors of the corners of the polygon intersect. Radius is calculated: R=S/P; S is the area of ​​the polygon, P is the semi-perimeter of the circle.

In a triangle

Only one circle is inscribed in a regular triangle, the center of which is called the incenter; it is located the same distance from all sides and is the intersection of the bisectors.

In a quadrangle

Often you have to decide how to find the radius of the inscribed circle in this geometric figure. It must be convex (if there are no self-intersections). A circle can be inscribed in it only if the sums of opposite sides are equal: AB+CD=BC+AD.

In this case, the center of the inscribed circle, the midpoints of the diagonals, are located on one straight line (according to Newton’s theorem). A segment whose ends are located where the opposite sides of a regular quadrilateral intersect lies on the same straight line, called the Gaussian straight line. The center of the circle will be the point at which the altitudes of the triangle intersect with the vertices and diagonals (according to Brocard’s theorem).

In a rhombus

It is considered a parallelogram with sides of equal length. The radius of the circle inscribed in it can be calculated in several ways.

1. To do this correctly, find the radius of the inscribed circle of the rhombus, if the area of ​​the rhombus and the length of its side are known. The formula r=S/(2Xa) is used. For example, if the area of ​​a rhombus is 200 mm square, the side length is 20 mm, then R = 200/(2X20), that is, 5 mm.
2. The acute angle of one of the vertices is known. Then you need to use the formula r=v(S*sin(α)/4). For example, with an area of ​​150 mm and known coal at 25 degrees, R= v(150*sin(25°)/4) ≈ v(150*0.423/4) ≈ v15.8625 ≈ 3.983 mm.
3. All angles in a rhombus are equal. In this situation, the radius of a circle inscribed in a rhombus will be equal to half the length of one side of this figure. If we reason according to Euclid, who states that the sum of the angles of any quadrilateral is 360 degrees, then one angle will be equal to 90 degrees; those. it will turn out to be a square.

A radius is a line segment that connects any point on a circle to its center. This is one of the most important characteristics of this figure, since on its basis all other parameters can be calculated. If you know how to find the radius of a circle, you can calculate its diameter, length, and area. In the case when a given figure is inscribed or described around another, a number of other problems can be solved. Today we will look at the basic formulas and the features of their application.

Known quantities

If you know how to find the radius of a circle, which is usually denoted by the letter R, then it can be calculated using one characteristic. These values ​​include:

• circumference (C);
• diameter (D) - a segment (or rather, a chord) that passes through the central point;
• area (S) - the space that is limited by a given figure.

Circumference

If the value of C is known in the problem, then R = C / (2 * P). This formula is a derivative. If we know what the circumference is, then we no longer need to remember it. Let's assume that in the problem C = 20 m. How to find the radius of the circle in this case? We simply substitute the known value into the above formula. Note that in such problems knowledge of the number P is always implied. For convenience of calculations, we take its value as 3.14. The solution in this case looks like this: we write down what values ​​are given, derive the formula and carry out the calculations. In the answer we write that the radius is 20 / (2 * 3.14) = 3.19 m. It is important not to forget what we calculated and mention the name of the units of measurement.

By diameter

Let us immediately emphasize that this is the simplest type of problem, which asks how to find the radius of a circle. If you came across such an example on a test, then you can rest assured. You don't even need a calculator here! As we have already said, diameter is a segment or, more correctly, a chord that passes through the center. In this case, all points of the circle are equidistant. Therefore, this chord consists of two halves. Each of them is a radius, which follows from its definition as a segment that connects a point on a circle and its center. If the diameter is known in the problem, then to find the radius you simply need to divide this value by two. The formula is as follows: R = D / 2. For example, if the diameter in the problem is 10 m, then the radius is 5 meters.

By area of ​​a circle

This type of problem is usually called the most difficult. This is primarily due to ignorance of the formula. If you know how to find the radius of a circle in this case, then the rest is a matter of technique. In the calculator, you just need to find the square root calculation icon in advance. The area of ​​a circle is the product of the number P and the radius multiplied by itself. The formula is as follows: S = P * R 2. By isolating the radius on one side of the equation, you can easily solve the problem. It will be equal to the square root of the quotient of the area divided by the number P. If S = 10 m, then R = 1.78 meters. As in previous problems, it is important to remember the units of measurement used.

How to find the circumradius of a circle

Let's assume that a, b, c are the sides of the triangle. If you know their values, you can find the radius of the circle described around it. To do this, you first need to find the semi-perimeter of the triangle. To make it easier to understand, let's denote it with the small letter p. It will be equal to half the sum of the sides. Its formula: p = (a + b + c) / 2.

We also calculate the product of the lengths of the sides. For convenience, let's denote it by the letter S. The formula for the radius of the circumscribed circle will look like this: R = S / (4 * √(p * (p - a) * (p - b) * (p - c)).

Let's look at an example task. We have a circle circumscribed around a triangle. The lengths of its sides are 5, 6 and 7 cm. First, we calculate the semi-perimeter. In our problem it will be equal to 9 centimeters. Now let's calculate the product of the lengths of the sides - 210. We substitute the results of intermediate calculations into the formula and find out the result. The radius of the circumscribed circle is 3.57 centimeters. We write down the answer, not forgetting about the units of measurement.

How to find the radius of an inscribed circle

Let's assume that a, b, c are the lengths of the sides of the triangle. If you know their values, you can find the radius of the circle inscribed in it. First you need to find its semi-perimeter. To make it easier to understand, let's denote it with the small letter p. The formula for calculating it is as follows: p = (a + b + c) / 2. This type of problem is somewhat simpler than the previous one, so no more intermediate calculations are needed.

The radius of the inscribed circle is calculated using the following formula: R = √((p - a) * (p - b) * (p - c) / p). Let's look at this specific example. Suppose the problem describes a triangle with sides 5, 7 and 10 cm. A circle is inscribed in it, the radius of which needs to be found. First we find the semi-perimeter. In our problem it will be equal to 11 cm. Now we substitute it into the main formula. The radius will be equal to 1.65 centimeters. Write down the answer and don’t forget about correct units measurements.

Circle and its properties

Each geometric figure has its own characteristics. The correctness of problem solving depends on their understanding. The circle also has them. They are often used when solving examples with described or inscribed figures, since they provide a clear picture of such a situation. Among them:

• A straight line can have zero, one or two points of intersection with a circle. In the first case it does not intersect with it, in the second it is a tangent, in the third it is a secant.
• If you take three points that do not lie on the same line, then only one circle can be drawn through them.
• A straight line can be tangent to two figures at once. In this case, it will pass through a point that lies on the segment connecting the centers of the circles. Its length is equal to the sum of the radii of these figures.
• An infinite number of circles can be drawn through one or two points.

This article popularly explains how to find the radius of a circle inscribed in a square. Theoretical material will help you understand all the nuances related to the topic. After reading this text, you will be able to easily solve similar problems in the future.

Basic theory

Before moving directly to finding the radius of a circle inscribed in a square, it is worth familiarizing yourself with some fundamental concepts. They may seem too simple and obvious, but they are necessary to understand the issue.

A square is a quadrilateral, all sides of which are equal to each other, and the degree measure of all angles is 90 degrees.

A circle is a two-dimensional closed curve located at a certain distance from a certain point. A segment, one end of which lies in the center of the circle, and the other on any of its surfaces, is called a radius.

We got acquainted with the terms, only the main question remained. We need to find the radius of a circle inscribed in a square. But what does the last phrase mean? Nothing complicated here either. If all sides of a polygon touch a curved line, then it is considered inscribed in this polygon.

Radius of a circle inscribed in a square

WITH theoretical material finished. Now we need to figure out how to put it into practice. Let's use a drawing for this.

The radius is obviously perpendicular to AB. This means that at the same time it is parallel to AD and BC. Roughly speaking, you can “overlay” it on the side of the square to further determine the length. As you can see, segment BK will correspond to it.

One of its ends r lies in the center of the circle, which is the intersection point of the diagonals. The latter divide each other in half based on one of their properties. Using the Pythagorean theorem, we can prove that they also divide the side of the figure into two equal parts.

Taking these arguments, we draw a conclusion.

A rhombus is a parallelogram with all sides equal. Therefore, it inherits all the properties of a parallelogram. Namely:

• The diagonals of a rhombus are mutually perpendicular.
• The diagonals of a rhombus are the bisectors of its interior angles.

A circle can be inscribed in a quadrilateral if and only if the sums of opposite sides are equal.
Therefore, a circle can be inscribed in any rhombus. The center of the inscribed circle coincides with the center of intersection of the diagonals of the rhombus.
The radius of the inscribed circle in a rhombus can be expressed in several ways

1 way. Radius of the inscribed circle in a rhombus through the height

The height of a rhombus is equal to the diameter of the inscribed circle. This follows from the property of a rectangle, which is formed by the diameter of the inscribed circle and the height of the rhombus - the opposite sides of a rectangle are equal.

Therefore, the formula for the radius of an inscribed circle in a rhombus in terms of height:

Method 2. Radius of the inscribed circle in a rhombus through diagonals

The area of ​​a rhombus can be expressed in terms of the radius of the inscribed circle
, Where R– perimeter of a rhombus. Knowing that the perimeter is the sum of all sides of the quadrilateral, we have P= 4×a. Then
But the area of ​​a rhombus is also equal to half the product of its diagonals
Equating the right-hand sides of the area formulas, we have the following equality
As a result, we obtain a formula that allows us to calculate the radius of the inscribed circle in a rhombus through the diagonals

An example of calculating the radius of a circle inscribed in a rhombus if the diagonals are known
Find the radius of a circle inscribed in a rhombus if it is known that the lengths of the diagonals are 30 cm and 40 cm
Let ABCD-rhombus, then A.C. And BD its diagonals. AC= 30 cm ,BD=40 cm
Let the point ABOUT– is the center of the inscribed in rhombus ABCD circle, then it will also be the point of intersection of its diagonals, dividing them in half.


since the diagonals of a rhombus intersect at right angles, then the triangle AOB rectangular. Then, by the Pythagorean theorem
, substitute the previously obtained values ​​into the formula

AB= 25 cm
Applying the previously derived formula for the radius of the circumscribed circle in a rhombus, we obtain

3 way. Radius of the inscribed circle in a rhombus through segments m and n

Dot F– the point of contact of the circle with the side of the rhombus, which divides it into segments A.F. And B.F.. Let AF=m, BF=n.
Dot O– the center of intersection of the diagonals of a rhombus and the center of the circle inscribed in it.
Triangle AOB– rectangular, since the diagonals of a rhombus intersect at right angles.
, because is the radius drawn to the tangent point of the circle. Hence OF– height of the triangle AOB to the hypotenuse. Then A.F. And BF projections of the legs onto the hypotenuse.
Height in right triangle, lowered to the hypotenuse is the average proportional between the projections of the legs onto the hypotenuse.

The formula for the radius of an inscribed circle in a rhombus through segments is equal to the square root of the product of these segments into which the point of tangency of the circle divides the side of the rhombus