Very often when deciding 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 looks like in the following way: 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 are quite a few 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 given expression 5 – 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, which will help you in solving problems of 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 – side 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 . I recommend deriving as an exercise the formula for the radius of a circle inscribed in equilateral triangle, directly, i.e. without use general formulas for radii of circles inscribed in arbitrary triangle or into an isosceles triangle.
Theorem 8 . For right triangle equality is true
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.
Let's write down the area triangle ABC 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: V 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 – area of the triangle
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 – area of the triangle
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.
Sides of an isosceles triangle are equal to 5, the base is equal to 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 – area of the triangle
All sides are known, let's calculate the area. We can find it using Heron's formula:
Then
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 – area of the triangle
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 – area of the triangle
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:
This 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 – area of the triangle
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 – area of the triangle
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 – area of the triangle
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 out, 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.
How to find the radius of a circle? This question is always relevant for schoolchildren studying planimetry. Below we will look at several examples of how you can cope with this task.
Depending on the conditions of the problem, you can find the radius of the circle like this.
Formula 1: R = 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 = 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 = 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 = S / (P/2), where - S and P are the area and perimeter of the figure, respectively.
Formula 2: R = (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:
The radius of a circle that is inscribed in a rhombus
A circle can be inscribed in any rhombus, both equilateral and unequal.
Formula 1: R = 2 * H, where H is the height of the geometric figure.
Formula 2: R = S / (A*2), where S is and A is the length of its side.
Formula 3: R = √((S * sin A)/4), where S is the area of the rhombus, and sin A is the sine acute angle of this geometric figure.
Formula 4: R = B*G/(√(B² + G²), where B and G are the lengths of the diagonals of the 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.
The radius of a circle that is inscribed in a triangle
If in the problem statement you are given the lengths of all sides of the figure, then first calculate (P) and then the semi-perimeter (p):
P = 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 one, then you can calculate the required radius as follows.
Formula 2: R = S * 2(A + B + C)
Formula 3: R = S/n = S / (A+B+B)/2), where - n is the semi-perimeter of the geometric figure.
Formula 4: R = (n - A) * tan (A/2), where n is the semi-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 = A * √3/6.
The radius of a circle that is inscribed in a right triangle
If the problem gives the lengths of the legs, as well as the hypotenuse, then the radius of the inscribed circle is found out like this.
Formula 1: R = (A+B-C)/2, where A, B are legs, C is 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 = √(A²+B²).
Radius of a circle inscribed in a square
A circle that is inscribed in a square divides all 4 of its sides exactly in half at the points of contact.
Formula 1: R = A/2, where A is the length of the side of the square.
Formula 2: R = S / (P/2), where S and P are the area and perimeter of the square, respectively.