First, let's understand the difference between a circle and a circle. To see this difference, it is enough to consider what both figures are. These are an infinite number of points on the plane located on equal distance from the only one center point. But, if the circle also consists of internal space, then it does not belong to the circle. It turns out that a circle is both a circle that limits it (circle(r)), and an innumerable number of points that are inside the circle.
For any point L lying on the circle, the equality OL=R applies. (The length of the segment OL is equal to the radius of the circle).
A segment that connects two points on a circle is its chord.
A chord passing directly through the center of a circle is diameter this circle (D). The diameter can be calculated using the formula: D=2R
Circumference calculated by the formula: C=2\pi R
Area of a circle: S=\pi R^(2)
Arc of a circle is called that part of it that is located between its two points. These two points define two arcs of a circle. The chord CD subtends two arcs: CMD and CLD. Identical chords subtend equal arcs.
Central angle An angle that lies between two radii is called.
Arc length can be found using the formula:
- Using degree measure: CD = \frac(\pi R \alpha ^(\circ))(180^(\circ))
- Using radian measure: CD = \alpha R
The diameter, which is perpendicular to the chord, divides the chord and the arcs contracted by it in half.
If the chords AB and CD of the circle intersect at the point N, then the products of the segments of the chords separated by the point N are equal to each other.
AN\cdot NB = CN\cdot ND
Tangent to a circle
Tangent to a circle It is customary to call a straight line that has one common point with a circle.
If a straight line has two common points, they call her secant.
If you draw the radius to the tangent point, it will be perpendicular to the tangent to the circle.
Let's draw two tangents from this point to our circle. It turns out that the tangent segments will be equal to one another, and the center of the circle will be located on the bisector of the angle with the vertex at this point.
AC = CB
Now let’s draw a tangent and a secant to the circle from our point. We find that the square of the length of the tangent segment will be equal to the product the entire segment secant to its outer part.
AC^(2) = CD \cdot BC
We can conclude: the product of an entire segment of the first secant and its external part is equal to the product of an entire segment of the second secant and its external part.
AC\cdot BC = EC\cdot DC
Angles in a circle
Degree measures central angle and the arc on which it rests are equal.
\angle COD = \cup CD = \alpha ^(\circ)
Inscribed angle is an angle whose vertex is on a circle and whose sides contain chords.
It can be calculated by knowing the arc size, since it equal to half this arc.
\angle AOB = 2 \angle ADB
Based on a diameter, inscribed angle, right angle.
\angle CBD = \angle CED = \angle CAD = 90^ (\circ)
Inscribed angles that subtend the same arc are identical.
Inscribed angles resting on one chord are identical or their sum is equal to 180^ (\circ) .
\angle ADB + \angle AKB = 180^ (\circ)
\angle ADB = \angle AEB = \angle AFB
On the same circle are the vertices of triangles with identical angles and a given base.
An angle with a vertex inside a circle and located between two chords is identical to half the sum angular values arcs of a circle that are contained within a given and vertical angle.
\angle DMC = \angle ADM + \angle DAM = \frac(1)(2) \left (\cup DmC + \cup AlB \right)
An angle with a vertex outside the circle and located between two secants is identical to half the difference in the angular values of the arcs of the circle that are contained inside the angle.
\angle M = \angle CBD - \angle ACB = \frac(1)(2) \left (\cup DmC - \cup AlB \right)
Inscribed circle
Inscribed circle is a circle tangent to the sides of a polygon.
At the point where the bisectors of the corners of a polygon intersect, its center is located.
A circle may not be inscribed in every polygon.
The area of a polygon with an inscribed circle is found by the formula:
S = pr,
p is the semi-perimeter of the polygon,
r is the radius of the inscribed circle.
It follows that the radius of the inscribed circle is equal to:
r = \frac(S)(p)
The sums of the lengths of opposite sides will be identical if the circle is inscribed in convex quadrilateral. And vice versa: a circle fits into a convex quadrilateral if the sums of the lengths of opposite sides are identical.
AB + DC = AD + BC
It is possible to inscribe a circle in any of the triangles. Only one single one. At the point where the bisectors intersect internal corners figure, the center of this inscribed circle will lie.
The radius of the inscribed circle is calculated by the formula:
r = \frac(S)(p) ,
where p = \frac(a + b + c)(2)
Circumcircle
If a circle passes through each vertex of a polygon, then such a circle is usually called described about a polygon.
At the point of intersection of the perpendicular bisectors of the sides of this figure will be the center of the circumcircle.
The radius can be found by calculating it as the radius of the circle that is circumscribed about the triangle defined by any 3 vertices of the polygon.
Eat next condition: a circle can be described around a quadrilateral only if its sum opposite corners is equal to 180^( \circ) .
\angle A + \angle C = \angle B + \angle D = 180^ (\circ)
Around any triangle you can describe a circle, and only one. The center of such a circle will be located at the point where the perpendicular bisectors of the sides of the triangle intersect.
The radius of the circumscribed circle can be calculated using the formulas:
R = \frac(a)(2 \sin A) = \frac(b)(2 \sin B) = \frac(c)(2 \sin C)
R = \frac(abc)(4 S)
a, b, c are the lengths of the sides of the triangle,
S is the area of the triangle.
Ptolemy's theorem
Finally, consider Ptolemy's theorem.
Ptolemy's theorem states that the product of diagonals is identical to the sum of the products of opposite sides of a cyclic quadrilateral.
AC \cdot BD = AB \cdot CD + BC \cdot AD
.png)
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 the same straight line (according to Newton’s theorem). A line segment whose ends are where they intersect opposite sides regular quadrilateral 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.
- 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.
- Famous sharp corner one of the peaks. 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.
- 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.
MKOU "Volchikhinskaya Secondary School No. 2"
Teacher Bakuta E.P.
9th grade
Lesson on the topic “Formulas of radii of inscribed and circumscribed circles regular polygons"
Lesson objectives:
Educational: study of formulas for the radii of inscribed and circumscribed circles of regular polygons;
Developmental: activation cognitive activity students through the solution practical problems, ability to choose correct solution, concisely express your thoughts, analyze and draw conclusions.
Educational: organization joint activities, instilling in students interest in the subject, goodwill, and the ability to listen to the answers of their comrades.
Equipment: Multimedia computer, multimedia projector, exposure screen
Lesson progress:
To argue the right thing,
And the motto of our lesson will be these words:
Think collectively!
Solve quickly!
Answer with evidence!
Fight hard!
2. Lesson motivation.
3. Update background knowledge. Checking d/z.
Frontal survey:
What shape is called a polygon?
Which polygon is called regular?
What other name regular triangle?
What is another name for a regular quadrilateral?
Formula for the sum of angles of a convex polygon.
Regular polygon angle formula.
4. Studying new material. (slides)
A circle is said to be inscribed in a polygon if all sides of the polygon touch the circle.
A circle is called circumscribed about a polygon if all the vertices of the polygon lie on the circle.
A circle can be inscribed or circumscribed about any triangle, and the center of a circle inscribed in a triangle lies at the intersection of the bisectors of the triangle, and the center of a circle circumscribed about a triangle lies at the intersection of the perpendicular bisectors.
A circle can be circumscribed around any regular polygon, and a circle can be inscribed into any regular polygon, and the center of the circle circumscribed around the regular polygon coincides with the center of the circle inscribed in the same polygon.
Formulas for the radii of inscribed and circumscribed circles of a regular triangle, regular quadrilateral, regular hexagon.
Radius of the inscribed circle in a regular polygon (r):
a - side of the polygon, N - number of sides of the polygon
Circumradius of a regular polygon (R):
a is the side of the polygon, N is the number of sides of the polygon.
Let's fill out the table for regular triangle, regular quadrilateral, regular hexagon.
5. Consolidation of new material.
Solve No. 1088, 1090, 1092, 1099.
6. Physical exercise . One - stretch Two - bend down
Three - look around Four - sit down
Five - hands up Six - forward
Seven - lowered Eight - sat down
Nine - stood up Ten - sat down again
7. Independent work students (work in groups)
Solve No. 1093.
8. Lesson summary. Reflection. D/z.
What impression did you get? (Like it – didn’t like it)
– How are you feeling after the lesson? (Joyful - sad)
– How are you feeling? (Tired - not tired)
– What is your attitude towards the material covered? (Got it - didn't get it)
– What is your self-esteem after the lesson? (Satisfied – not satisfied)
– Evaluate your activity in class. (I tried - I didn’t try).
repeat paragraphs 105-108;
learn formulas;
№ 1090, 1091, 1087(3)
Mathematics has a rumor
That she puts her mind in order,
Because Nice words
People often talk about her.
You give us geometry
Hardening is important for victory.
Young people study with you
Develop both will and ingenuity.
Note The presentation contains sections:
Repetition theoretical material
Examination homework
Derivation of basic formulas, i.e. new material
Consolidation: solving problems in groups and independently
View presentation content
"9_klass_pravilnye_mnogougolniki_urok_2"
- To argue the right thing,
- So as not to know failures in life,
- Let's go boldly into the world of mathematics,
- Into the world of examples and different tasks.
LESSON MOTTO
Think collectively!
Solve quickly!
Answer with evidence!
Fight hard!
And discoveries are definitely waiting for us!
Repetition.
- What geometric figure
shown in the picture?
D
E
2.What polygon is called
correct?
ABOUT
3.What circle is called
inscribed in a polygon?
F
WITH
4.What circle is called
described about a polygon?
5.Name the radius of the inscribed circle.
A
IN
N
6.Name the radius of the circumscribed circle.
7.How to find the center of the inscribed in the correct
circle polygon?
8. How to find the center of a circle circumscribed about
regular polygon?
Checking progress
homework ..
№ 1084.
β – angle corresponding
the arc that is pulled together
side of a polygon .
ABOUT
A P
A 2
β
Answers:
a) 6;
b) 12;
A
A 1
at 4;
d) 8;
d) 10
e) 20;
e) 7.
e) 5.
REGULAR POLYGON
A regular polygon is called convex polygon, in which all angles are equal and all sides are equal.
Sum of right angles n -square
Angle correct n - square
A circle is said to be inscribed in a polygon
if all sides of the polygon touch this circle.
A circle is called circumscribed about a polygon if all its vertices lie on this
circles.
Inscribed and circumscribed circle
A circle inscribed in a regular polygon touches the sides of the polygon at their midpoints.
The center of a circle circumscribed about a regular polygon coincides with the center of a circle inscribed in the same polygon.
Let us derive the formula for the inscribed and circumscribed circle radii of a regular polygon.
Let r be the radius of the inscribed circle,
R – radius of the circumscribed circle,
n – the number of sides and angles of the polygon.
Consider a regular n-gon.
Let a be the side of the n-gon,
α – angle.
Let's construct point O - the center of the inscribed and circumscribed circle.
OS – height ∆AOB.
∟ С = 90 º - (by construction),
Let's consider ∆AOC:
∟ OAS = α /2 - (OA is the bisector of the angle of the p-gon),
AC = a/2 – (OS – median to the base isosceles triangle),
∟ AOB = 360 º: p,
let ∟AOC = β.
then β = 0.5 ∙ ∟AOB
0.5∙(360º:p)
2 sin (180º:n)
2 tg (180º:p)
Area of a regular polygon
Side of a regular polygon
Inscribed circle radius
Group 1 Given: R , n =3 Find: a
Group 2 Given: R , n =4 Find: a
Group 3 Given: R , n =6 Find: a
Group 4 Given: r , n =3 Find: a
Group 5 Given: r , n = 4 Find: a
Group 6 Given: r , n = 6 Find: a
Group 1 Given: R , n =3 Find: a
Group 2 Given: R , n =4 Find: a
Group 3 Given: R , n =6 Find: a
Group 4 Given: r , n =3 Find: a
Group 5 Given: r , n = 4 Find: a
Group 6 Given: r , n = 6 Find: a
n = 3
n = 4
n = 6
2 tg (180º:p)
2 sin (180º:n)
then 180 º: p
A regular triangle has n = 3,
whence 2 sin 60 º =
then 180 º: p
A regular quadrilateral has n = 4,
whence 2 sin 45 º =
A regular hexagon has n = 6,
then 180 º: p
whence 2 sin 30 º =
Using formulas for the radii of inscribed and circumscribed circles of some regular polygons, derive formulas for finding the dependence of the sides of regular polygons on the radii of inscribed and circumscribed circles and fill out the table:
2 R ∙ sin (180 º: n)
2 r ∙ tg (180 º: p)
triangle
hexagon
pp. 105 – 108;
№ 1087;
№ 1088 – prepare a table.
n=4
R
r
a 4
P
2
6
4
S
28
16
3
3√2
24
32
2√2
4
16
16
16√2
32
4√2
2√2
7
3,5√2
3,5
49
4
2√2
16
2
№ 1087(5)
Given: S=16 , n =4
Find: a, r, R, P
We know the formulas:
№ 1088( 5 )
Given: P=6 , n = 3
Find: R, a, r, S
We know the formulas:
№ 108 9
Given:
Find:
Summarize
We know the formulas:
- repeat paragraphs 105-108;
- learn formulas;
- № 1090, 1091, 1087(3)
Maintaining your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please review our privacy practices and let us know if you have any questions.
Collection and use of personal information
Personal information refers to data that can be used to identify or contact a specific person.
You may be asked to provide your personal information at any time when you contact us.
Below are some examples of the types of personal information we may collect and how we may use such information.
What personal information do we collect:
- When you submit an application on the site, we may collect various information, including your name, telephone number, address Email etc.
How we use your personal information:
- Collected by us personal information allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
- From time to time, we may use your personal information to send important notices and communications.
- We may also use personal information for internal purposes such as auditing, data analysis and various studies in order to improve the services we provide and provide you with recommendations regarding our services.
- If you participate in a prize draw, contest or similar promotion, we may use the information you provide to administer such programs.
Disclosure of information to third parties
We do not disclose the information received from you to third parties.
Exceptions:
- If necessary, in accordance with the law, judicial procedure, V trial, and/or based on public requests or requests from government agencies on the territory of the Russian Federation - disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public importance purposes.
- In the event of a reorganization, merger, or sale, we may transfer the personal information we collect to the applicable successor third party.
Protection of personal information
We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as unauthorized access, disclosure, alteration and destruction.
Respecting your privacy at the company level
To ensure that your personal information is secure, we communicate privacy and security standards to our employees and strictly enforce privacy practices.
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:
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 – 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. 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 – 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. Legs of an isosceles right triangle 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.