The radius of the inscribed circle in a right triangle formula. Formulas for the radii of inscribed and circumscribed circles of regular polygons

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.

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.

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 × 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 circumscribed about an equilateral triangle

To find the radius of a circle circumscribed about an equilateral triangle, there are quite simple formula: R = a/√3, where a is the value 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 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.

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.

Circle inscribed in a triangle

Existence of a circle inscribed in a triangle

Recall the definition angle bisector .

Definition 1 .Angle bisector called a ray that divides an angle into two equal parts.

Theorem 1 (Basic property of the angle bisector) . Each point of the bisector of the angle 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 D.F. on the sides of the corner (Fig. 1).right triangles ADF And ADE equal because they have the same acute anglesDAF And DAE , and the hypotenuse AD - general. Hence,

D.F. = D.E.

Q.E.D.

Theorem 2 (inverse theorem 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 cornerBAC and located at the same distance from the sides of the corner. Drop from pointD perpendiculars DE And D.F. on the sides of the corner (Fig. 2).right triangles ADF And ADE equal , since they have equal legsD.F. 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 is the 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 corner (Fig. 3).

Fig.3

a , b , c - sides of a triangle
 S -square,

r – radius of the inscribed circle,
 p - semiperimeter

.

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 triangler – 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 . I recommend deriving as an exercise 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 in an isosceles triangle.

Theorem 8 . For a right triangle, the equality

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 \u003d CF \u003d r,

By virtue of Theorem 3, the equalities

Therefore, taking also into account , we get

which is what was required.

A selection of tasks on the topic "A circle inscribed in a triangle."

1.

A circle inscribed in an isosceles triangle divides at the point of contact one of the sides into two segments, the lengths of which are equal to 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.

Radius of a circle inscribed in an isosceles right triangle, is equal to 2. Find the hypotenuse c of this triangle. Write c(-1) in your answer.

Here are a number of tasks from the exam with solutions.

The radius of a circle inscribed in an isosceles right triangle is . Find the hypotenuse c of this triangle. Please indicate in your answer.

The triangle is right and isosceles. So his legs are the same. Let each leg be equal. Then the hypotenuse is.

We write the area of ​​triangle ABC in two ways:

Equating these expressions, we get that. Because the, we get that. Then.

In response, write.

Answer:.

Task 2.

1. In any two sides 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: the radii of the 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: V this case the problem can be solved using either the Pythagorean theorem or the formula for R. The second case is 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 is the area of ​​the triangle

We do not know either the sides of the triangle or its area. Let's denote the legs as x, then the hypotenuse will be equal to:

The area of ​​the triangle will be 0.5x 2 .

Means

So the hypotenuse will be:

The answer must be written:

Answer: 4

Task 8.

In triangle ABC, AC = 4, BC = 3, angle C is equal to 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 is the area of ​​the triangle

Two sides are known (these are legs), we can calculate the third (hypotenuse), 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, 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 is the area of ​​the triangle

All sides are known, and the area is calculated. We can find it using Heron's formula:

Then

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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 an inscribed circle in a rhombus can be expressed in several ways

1 way. The 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 the rectangle are equal.

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

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

The area of ​​a rhombus can be expressed in terms of the radius of the inscribed circle
, Where R is the perimeter of the rhombus. Knowing that the perimeter is the sum of all the sides of a quadrilateral, we have P= 4×ha. Then
But the area of ​​a rhombus is also half the product of its diagonals
Equating the right parts 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 length of the diagonals is 30 cm and 40 cm
Let ABCD- rhombus, then AC And BD its diagonals. AC= 30 cm , BD=40 cm
Let the point ABOUT is the center of the inscribed in the rhombus ABCD circle, then it will also be the point of intersection of its diagonals, dividing them in half.


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

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

3 way. The radius of the inscribed circle in the rhombus through the segments m and n

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

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

How to find the radius of a circle? This question is always relevant for schoolchildren studying planimetry. Below we will look at a few examples of how you can cope with the task.

Depending on the condition of the problem, you can find the radius of the circle like this.

Formula 1: R \u003d L / 2π, where L is and π is a constant equal to 3.141 ...

Formula 2: R = √(S / π), where S is the area of ​​the circle.

Formula 1: R = B/2, where B is the hypotenuse.

Formula 2: R \u003d M * B, where B is the hypotenuse, and M is the median drawn to it.

How to find the radius of a circle if it is circumscribed around a regular polygon

Formula: R \u003d A / (2 * sin (360 / (2 * n))), where A is the length of one of the sides of the figure, and n is the number of sides in this geometric figure.

How to find the radius of an inscribed circle

An inscribed circle is called when it touches all sides of the polygon. Let's look at a few examples.

Formula 1: R \u003d S / (P / 2), where - S and P are the area and perimeter of the figure, respectively.

Formula 2: R \u003d (P / 2 - A) * tg (a / 2), where P is the perimeter, A is the length of one of the sides, and is the angle opposite this side.

How to find the radius of a circle if it is inscribed in a right triangle

Formula 1:

Radius of a circle inscribed in a rhombus

A circle can be inscribed in any rhombus, both equilateral and inequilateral.

Formula 1: R \u003d 2 * H, where H is the height of the geometric figure.

Formula 2: R \u003d S / (A * 2), where S is and A is the length of its side.

Formula 3: R \u003d √ ((S * sin A) / 4), where S is the area of ​​\u200b\u200bthe rhombus, and sin A is the sine acute angle this geometric figure.

Formula 4: R \u003d V * G / (√ (V² + G²), where V and G are the lengths of the diagonals of a geometric figure.

Formula 5: R = B * sin (A / 2), where B is the diagonal of the rhombus, and A is the angle at the vertices connecting the diagonal.

Radius of a circle that is inscribed in a triangle

In the event that in the condition of the problem you are given the lengths of all sides of the figure, then first calculate (P), and then the semi-perimeter (p):

P \u003d A + B + C, where A, B, C are the lengths of the sides of the geometric figure.

Formula 1: R = √((p-A)*(p-B)*(p-B)/p).

And if, knowing all the same three sides, you are also given, then you can calculate the desired radius as follows.

Formula 2: R = S * 2(A + B + C)

Formula 3: R \u003d S / p \u003d S / (A + B + C) / 2), where - p is the semi-perimeter of the geometric figure.

Formula 4: R \u003d (n - A) * tg (A / 2), where n is the half-perimeter of the triangle, A is one of its sides, and tg (A / 2) is the tangent of half the angle opposite this side.

And the formula below will help you find the radius of the circle that is inscribed in

Formula 5: R \u003d A * √3/6.

Radius of a circle that is inscribed in a right triangle

If the problem is given the lengths of the legs, as well as the hypotenuse, then the radius of the inscribed circle is found out as follows.

Formula 1: R \u003d (A + B-C) ​​/ 2, where A, B are legs, C is the hypotenuse.

In the event that you are given only two legs, it's time to remember the Pythagorean theorem in order to find the hypotenuse and use the above formula.

C \u003d √ (A² + B²).

Radius of a circle that is inscribed in a square

The circle, which is inscribed in the square, divides all its 4 sides exactly in half at the points of contact.

Formula 1: R \u003d A / 2, where A is the length of the side of the square.

Formula 2: R \u003d S / (P / 2), where S and P are the area and perimeter of the square, respectively.