Small inscribed circle radius formula. Area of a polygon in terms of the radius of the inscribed circle
In modern mechanical engineering, a lot of elements and spare parts are used, which have both external and internal circles in their structure. The most striking examples are bearing housings, engine parts, hub assemblies and much more. In their production, not only high-tech devices are used, but also knowledge from geometry, in particular information about the circles of a triangle. We will get acquainted with this knowledge in more detail below.
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Which circle is inscribed and which is circumscribed?
First of all, remember that a circle is an infinite set of points at equal distances from the center. If inside a polygon it is possible to construct a circle that has only one common intersection point with each side, then it will be called inscribed. A circumscribed circle (not a circle, these are different concepts) is a geometric locus of points such that the constructed figure with a given polygon has common points only at the vertices of the polygon. Let's get acquainted with these two concepts using a more clear example (see Figure 1.).
Figure 1. Inscribed and circumscribed circles of a triangle
In the image, two figures of large and small diameters are constructed, the centers of which are G and I. The circle of larger value is called the circumscribed circle Δ ABC, and the small one, on the contrary, inscribed in Δ ABC.
In order to describe the surroundings of a triangle, it is required draw a perpendicular line through the middle of each side(i.e. at an angle of 90°) is the point of intersection, it plays a key role. It will be the center of the circumscribed circle. Before finding a circle, its center in a triangle, you need to construct for each angle, and then select the point of intersection of the lines. It, in turn, will be the center of the inscribed neighborhood, and its radius under any conditions will be perpendicular to any of the sides.
To the question: “How many inscribed circles can there be for a polygon with three?” Let us answer right away that a circle can be inscribed in any triangle, and only one. Because there is only one point of intersection of all bisectors and one point of intersection of perpendiculars emanating from the midpoints of the sides.
Property of the circle to which the vertices of a triangle belong
The circumscribed circle, which depends on the lengths of the sides at the base, has its own properties. Let us indicate the properties of the circumcircle:
In order to more clearly understand the principle of the circumscribed circle, let’s solve a simple problem. Let us assume that we are given a triangle Δ ABC, the sides of which are 10, 15 and 8.5 cm. The radius of the circumscribed circle around the triangle (FB) is 7.9 cm. Find the degree measure of each angle and through them the area of the triangle.
Figure 2. Finding the radius of a circle using the ratio of sides and sines of angles
Solution: based on the previously mentioned sine theorem, we will find the value of the sine of each angle separately. By condition, it is known that side AB is 10 cm. Let’s calculate the value of C:
Using the values of the Bradis table, we find out that the degree measure of angle C is 39°. Using the same method, we can find the remaining measures of angles:
How do we know that CAB = 33°, and ABC = 108°. Now, knowing the values of the sines of each of the angles and the radius, let’s find the area by substituting the found values:
Answer: The area of the triangle is 40.31 cm², and the angles are 33°, 108° and 39°, respectively.
Important! When solving problems of this kind, it would be useful to always have Bradis tables or a corresponding application on your smartphone, since the manual process can take a long time. Also, to save more time, it is not necessary to build all three midpoints of the perpendicular or three bisectors. Any third of them will always intersect at the point of intersection of the first two. And for an orthodox construction, the third is usually completed. Maybe this is wrong when it comes to the algorithm, but on the Unified State Exam or other exams it saves a lot of time.
Calculating the radius of an inscribed circle
All points of a circle are equally distant from its center at the same distance. The length of this segment (from and to) is called the radius. Depending on what kind of environment we have, there are two types - internal and external. Each of them is calculated using its own formula and is directly related to the calculation of parameters such as:
- square;
- degree measure of each angle;
- side lengths and perimeter.
Figure 3. Location of the inscribed circle inside the triangle
You can calculate the length of the distance from the center to the point of contact on either side in the following ways: h through the sides, sides and corners(for an isosceles triangle).
Using a semi-perimeter
A semi-perimeter is half the sum of the lengths of all sides. This method is considered the most popular and universal, because no matter what type of triangle is given according to the condition, it is suitable for everyone. The calculation procedure is as follows:
If given "correct"
One of the small advantages of the "ideal" triangle is that inscribed and circumscribed circles have their center at the same point. This is convenient when constructing figures. However, in 80% of cases the answer is “ugly.” What is meant here is that very rarely the radius of the inscribed neighborhood will be whole, rather the opposite. For simplified calculation, use the formula for the radius of the inscribed circle in a triangle:
If the sides are the same length
One of the subtypes of tasks for the state. exams will be finding the radius of the inscribed circle of a triangle, two sides of which are equal to each other and the third is not. In this case, we recommend using this algorithm, which will significantly save time on searching for the diameter of the inscribed region. The radius of an inscribed circle in a triangle with equal “sides” is calculated by the formula:
We will demonstrate a more clear application of these formulas in the following problem. Let us have a triangle (Δ HJI), into which the neighborhood is inscribed at point K. The length of side HJ = 16 cm, JI = 9.5 cm and side HI is 19 cm (Figure 4). Find the radius of the inscribed neighborhood, knowing the sides.
Figure 4. Finding the value of the radius of the inscribed circle
Solution: to find the radius of the inscribed environment, we find the semi-perimeter:
From here, knowing the calculation mechanism, we find out the following value. To do this, you will need the lengths of each side (given according to the condition), as well as half the perimeter, it turns out:
It follows that the required radius is 3.63 cm. According to the condition, all sides are equal, then the desired radius will be equal to:
Provided that the polygon is isosceles (for example, i = h = 10 cm, j = 8 cm), the diameter of the inner circle centered at point K will be equal to:
The problem may contain a triangle with an angle of 90°; in this case, there is no need to memorize the formula. The hypotenuse of the triangle will be equal to the diameter. It looks more clearly like this:
Important! If the task is to find the internal radius, we do not recommend performing calculations using the values of the sines and cosines of angles, the table value of which is not precisely known. If it is impossible to find out the length otherwise, do not try to “pull out” the value from under the root. In 40% of problems, the resulting value will be transcendental (i.e. infinite), and the commission may not count the answer (even if it is correct) due to its inaccuracy or incorrect form of presentation. Pay special attention to how the formula for the circumradius of a triangle can be modified depending on the proposed data. Such “blanks” allow you to “see” the scenario for solving a problem in advance and choose the most economical solution.
Inner circle radius and area
To calculate the area of a triangle inscribed in a circle, use only radius and side lengths of the polygon:
If the problem statement does not directly give the value of the radius, but only the area, then the indicated area formula is transformed into the following:
Let's consider the effect of the last formula using a more specific example. Suppose that we are given a triangle in which a neighborhood is inscribed. The area of the neighborhood is 4π, and the sides are 4, 5 and 6 cm, respectively. Let's calculate the area of a given polygon by calculating the semi-perimeter.
Using the above algorithm, we calculate the area of the triangle through the radius of the inscribed circle:
Due to the fact that a circle can be inscribed in any triangle, the number of variations in finding the area increases significantly. Those. Finding the area of a triangle requires knowing the length of each side, as well as the value of the radius.
Triangle inscribed in a circle geometry grade 7
Right triangles inscribed in a circle
Conclusion
From these formulas you can be sure that the complexity of any problem using inscribed and circumscribed circles lies only in additional actions to find the required values. Problems of this type require only a thorough understanding of the essence of the formulas, as well as the rationality of their application. From the practice of solving, we note that in the future the center of the circumscribed circle will appear in further geometry topics, so it should not be started. Otherwise, the solution may be delayed using unnecessary moves and logical conclusions.
A circle is considered inscribed within the boundaries of a regular polygon if it lies inside it and touches the 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 the 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 diamond
It is considered a parallelogram with sides of equal length. The radius of the circle inscribed in it can be calculated in several ways.
- 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.
- 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 a known angle of 25 degrees, R= v(150*sin(25°)/4) ≈ v(150*0.423/4) ≈ v15.8625 ≈ 3.983 mm.
- 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 circle is inscribed in a triangle. In this article I have collected for you problems in which you are given a triangle with a circle inscribed in it or circumscribed around it. The condition asks the question of finding the radius of a circle or side of a triangle.
It is convenient to solve these tasks using the presented formulas. I recommend learning them, they are very useful not only when solving this type of task. One formula expresses the relationship between the radius of a circle inscribed in a triangle and its sides and area, the other, the radius of a circle inscribed around a triangle, also with its sides and area:
S – triangle area
Let's consider the tasks:
27900. The lateral side of an isosceles triangle is equal to 1, the angle at the vertex opposite the base is equal to 120 0. Find the circumscribed circle diameter of this triangle.
Here a circle is circumscribed about a triangle.
First way:
We can find the diameter if the radius is known. We use the formula for the radius of a circle circumscribed about a triangle:
where a, b, c are the sides of the triangle
S – triangle area
We know two sides (the lateral sides of an isosceles triangle), we can calculate the third using the cosine theorem:
Now let's calculate the area of the triangle:
*We used formula (2) from.
Calculate the radius:
Thus the diameter will be equal to 2.
Second way:
These are mental calculations. For those who have the skill of solving problems with a hexagon inscribed in a circle, they will immediately determine that the sides of the triangle AC and BC “coincide” with the sides of the hexagon inscribed in the circle (the angle of the hexagon is exactly equal to 120 0, as in the problem statement). And then, based on the fact that the side of a hexagon inscribed in a circle is equal to the radius of this circle, it is not difficult to conclude that the diameter will be equal to 2AC, that is, two.
For more information about the hexagon, see the information in (item 5).
Answer: 2
27931. The radius of a circle inscribed in an isosceles right triangle is 2. Find the hypotenuse With this triangle. Please indicate in your answer.
where a, b, c are the sides of the triangle
S – triangle area
We do not know either the sides of the triangle or its area. Let us denote the legs as x, then the hypotenuse will be equal to:
And the area of the triangle will be equal to 0.5x 2.
Means
Thus, the hypotenuse will be equal to:
In your answer you need to write:
Answer: 4
27933. In a triangle ABC AC = 4, BC = 3, angle C equals 90 0 . Find the radius of the inscribed circle.
Let's use the formula for the radius of a circle inscribed in a triangle:
where a, b, c are the sides of the triangle
S – triangle area
Two sides are known (these are the legs), we can calculate the third (the hypotenuse), and we can also calculate the area.
According to the Pythagorean theorem:
Let's find the area:
Thus:
Answer: 1
27934. The sides of an isosceles triangle are 5 and the base is 6. Find the radius of the inscribed circle.
Let's use the formula for the radius of a circle inscribed in a triangle:
where a, b, c are the sides of the triangle
S – triangle area
All sides are known, let's calculate the area. We can find it using Heron's formula:
Then
Thus:
Answer: 1.5
27624. The perimeter of the triangle is 12 and the radius of the inscribed circle is 1. Find the area of this triangle. View solution
27932. The legs of an isosceles right triangle are equal. Find the radius of the circle inscribed in this triangle.
A short summary.
If the condition gives a triangle and an inscribed or circumscribed circle, and we are talking about sides, area, radius, then immediately remember the indicated formulas and try to use them when solving. If it doesn’t work, then look for other solutions.
That's all. Good luck to you!
Sincerely, Alexander Krutitskikh.
P.S: I would be grateful if you tell me about the site on social networks.
Very often, when solving geometric problems, you have to perform actions with auxiliary figures. For example, finding the radius of an inscribed or circumscribed circle, etc. This article will show you how to find the radius of a circle circumscribed by a triangle. Or, in other words, the radius of the circle in which the triangle is inscribed.
How to find the radius of a circle circumscribed about a triangle - 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 (semi-perimeter). a, b, c – sides of the triangle.
Find the circumradius of the triangle if a = 3, b = 6, c = 7.
Thus, based on the above formula, we calculate the semi-perimeter:
p = (a + b + c)/2 = 3 + 6 + 7 = 16. => 16/2 = 8.
We substitute the values into the formula and get:
R = 3 × 6 × 7/4√8(8 – 3)(8 – 6)(8 – 7) = 126/4√(8 × 5 × 2 × 1) = 126/4√80 = 126/16 √5.
Answer: R = 126/16√5
How to find the radius of a circle circumscribing 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.
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 into the formula. We get: R = 5/√3.
Answer: R = 5/√3.
How to find the radius of a circle circumscribing a right triangle
The formula is as follows: R = 1/2 × √(a² + b²) = c/2, where a and b are the legs and c is the hypotenuse. If you add the squares of the legs in a right triangle, you 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 ultimately leads us to the expression 1/2 × c = c/2.
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 circumscribing an isosceles triangle
The formula is as follows: 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 base = 8.
Solution: Substitute these values into the formula and get: R = 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 can be very easy to get confused in all these formulas. Therefore, if necessary, you can use online calculators that will help you in solving problems on finding the radius. The operating principle of such mini-programs is very simple. Substitute the side value into the appropriate field and get a ready-made answer. You can choose several options for rounding your answer: to decimals, hundredths, thousandths, etc.
Circle inscribed in a triangle
Existence of a circle inscribed in a triangle
Let us recall the definition angle bisectors .
Definition 1 .Angle bisector called a ray dividing an angle into two equal parts.
Theorem 1 (Basic property of an angle bisector) . Each point of the angle bisector is at the same distance from the sides of the angle (Fig. 1).
Rice. 1
Proof D , lying on the bisector of the angleBAC , And DE And DF on the sides of the corner (Fig. 1).Right Triangles ADF And ADE equal , since they have equal acute anglesDAF And DAE , and the hypotenuse AD – general. Hence,
DF = DE,
Q.E.D.
Theorem 2 (converse to Theorem 1) . If some, then it lies on the bisector of the angle (Fig. 2).
Rice. 2
Proof . Consider an arbitrary pointD , lying inside the angleBAC and located at the same distance from the sides of the angle. Let's drop from the pointD perpendiculars DE And DF on the sides of the corner (Fig. 2).Right Triangles ADF And ADE equal , since they have equal legsDF And DE , and the hypotenuse AD – general. Hence,
Q.E.D.
Definition 2 . The circle is called circle inscribed in an angle , if it is the sides of this angle.
Theorem 3 . If a circle is inscribed in an angle, then the distances from the vertex of the angle to the points of contact of the circle with the sides of the angle are equal.
Proof . Let the point D – center of a circle inscribed in an angleBAC , and the points E And F – points of contact of the circle with the sides of the angle (Fig. 3).
Fig.3
a , b , c - sides of the triangle, S -square,
r – radius of the inscribed circle, p – semi-perimeter
.
View formula output
a – lateral side of an isosceles triangle , b – base, r – inscribed circle radius
a r – inscribed circle radius
View formula output
,
Where
,
then, in the case of an isosceles triangle, when
we get
which is what was required.
Theorem 7 . For the equality
Where a - side of an equilateral triangle,r – radius of the inscribed circle (Fig. 8).
Rice. 8
Proof .
,
then, in the case of an equilateral triangle, when
b = a,
we get
which is what was required.
Comment . As an exercise, I recommend deriving the formula for the radius of a circle inscribed in an equilateral triangle directly, i.e. without using general formulas for the radii of circles inscribed in an arbitrary triangle or an isosceles triangle.
Theorem 8 . For a right triangle, the following equality holds:
Where a , b – legs of a right triangle, c – hypotenuse , r – radius of the inscribed circle.
Proof . Consider Figure 9.
Rice. 9
Since the quadrilateralCDOF is , which has adjacent sidesDO And OF are equal, then this rectangle is . Hence,
CB = CF= r,
By virtue of Theorem 3, the following equalities are true:
Therefore, also taking into account , we obtain
which is what was required.
A selection of problems on the topic “A circle inscribed in a triangle.”
1.
A circle inscribed in an isosceles triangle divides one of the lateral sides at the point of contact into two segments, the lengths of which are 5 and 3, counting from the vertex opposite the base. Find the perimeter of the triangle.
2.
3
In triangle ABC AC=4, BC=3, angle C is 90º. Find the radius of the inscribed circle.
4.
The legs of an isosceles right triangle are 2+. Find the radius of the circle inscribed in this triangle.
5.
The radius of a circle inscribed in an isosceles right triangle is 2. Find the hypotenuse c of this triangle. Please indicate c(–1) in your answer.
We present a number of problems from the Unified State Exam with solutions.
The radius of a circle inscribed in an isosceles right triangle is equal to . Find the hypotenuse of this triangle. Please indicate in your answer.
The triangle is rectangular and isosceles. This means that its legs are the same. Let each leg be equal. Then the hypotenuse is equal.
We write the area of triangle ABC in two ways:
Equating these expressions, we get that
. Because the, we get that
. Then.
We'll write down in response.
Answer:.
Task 2.
1. In free, there are two sides of 10cm and 6cm (AB and BC). Find the radii of the circumscribed and inscribed circles
The problem is solved independently with commenting.
Solution:
IN.
1) Find: 
2) Prove:
and find CK
3) Find: radii of circumscribed and inscribed circles
Solution:
Task 6.
R the radius of a circle inscribed in a square is. Find the radius of the circle circumscribed about this square.Given :
Find: OS=?
Solution: In this case, the problem can be solved using either the Pythagorean theorem or the formula for R. The second case will be simpler, since the formula for R is derived from the theorem. 
Task 7.
The radius of a circle inscribed in an isosceles right triangle is 2. Find the hypotenuseWith this triangle. Please indicate in your answer.
S – triangle area
We do not know either the sides of the triangle or its area. Let us denote the legs as x, then the hypotenuse will be equal to:
And the area of the triangle will be 0.5x 2 .
Means
Thus, the hypotenuse will be equal to:
In your answer you need to write:
Answer: 4
Task 8.
In triangle ABC AC = 4, BC = 3, angle C equals 90 0. Find the radius of the inscribed circle.
Let's use the formula for the radius of a circle inscribed in a triangle:
where a, b, c are the sides of the triangle
S – triangle area
Two sides are known (these are the legs), we can calculate the third (the hypotenuse), and we can also calculate the area.
According to the Pythagorean theorem:
Let's find the area:
Thus:
Answer: 1
Task 9.
The sides of an isosceles triangle are 5 and the base is 6. Find the radius of the inscribed circle.
Let's use the formula for the radius of a circle inscribed in a triangle:
where a, b, c are the sides of the triangle
S – triangle area
All sides are known, let's calculate the area. We can find it using Heron's formula:
Then