Hmm, yoda_daro claims that this is a test to determine their sexuality...
Instruction The frames of the film merge for us into continuous movement due to ...
In this article, we will talk about how to express the area of a polygon in which a circle can be inscribed in terms of the radius of this circle. It is immediately worth noting that not every polygon can be inscribed in a circle. However, if this is possible, then the formula by which the area of such a polygon is calculated becomes very simple. Read this article to the end or watch the attached video tutorial and you will learn how to express the area of a polygon in terms of the radius of a circle inscribed in it.
The formula for the area of a polygon in terms of the radius of the inscribed circle
Let's draw a polygon A 1 A 2 A 3 A 4 A 5 , not necessarily correct, but one in which a circle can be inscribed. Let me remind you that an inscribed circle is a circle that touches all sides of the polygon. In the figure, this is a green circle centered at a point O:
We have taken a 5-gon here as an example. But in fact, this is not essential, since the further proof is valid for both the 6-gon and the 8-gon, and in general for any "gon" arbitrarily.
If you connect the center of the inscribed circle with all the vertices of the polygon, then it will be divided into as many triangles as there are vertices in the given polygon. In our case: 5 triangles. If we connect the dot O with all points of contact of the inscribed circle with the sides of the polygon, you get 5 segments (in the figure below, these are the segments Oh 1 , Oh 2 , Oh 3 , Oh 4 and Oh 5), which are equal to the radius of the circle and are perpendicular to the sides of the polygon to which they are drawn. The latter is true, since the radius drawn to the point of contact is perpendicular to the tangent:
How to find the area of our circumscribed polygon? The answer is simple. It is necessary to add up the areas of all the triangles obtained as a result of splitting:
Consider what is the area of a triangle. In the picture below, it is highlighted in yellow:
It is equal to half the product of the base A 1 A 2 to the height Oh 1 drawn to this base. But, as we have already found out, this height is equal to the radius of the inscribed circle. That is, the formula for the area of a triangle takes the form: 
, where r is the radius of the inscribed circle. Similarly, the areas of all the remaining triangles are found. As a result, the desired area of the polygon is equal to:
It can be seen that in all terms of this sum, common factor, which can be bracketed. The result is the following expression:
That is, in brackets there was simply the sum of all sides of the polygon, that is, its perimeter P. Most often, in this formula, the expression is simply replaced by p and call this letter "half-perimeter". As a result, the final formula becomes:
That is, the area of a polygon in which a circle of known radius is inscribed is equal to the product of this radius and the semiperimeter of the polygon. This is the result we were aiming for.
Finally, he notes that a circle can always be inscribed in a triangle, which is a special case of a polygon. Therefore, for a triangle, this formula can always be applied. For other polygons with more than 3 sides, you first need to make sure that a circle can be inscribed in them. If so, you can safely use this simple formula and find the area of \u200b\u200bthis polygon from it.
Prepared by Sergey Valerievich
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 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 = 2 * H, where H is the height 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 of the acute angle of 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 required 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.
A circle is considered to be inscribed in the boundaries of a regular polygon if it lies inside it, while touching the lines that pass through all sides. Consider how to find the center and radius of a circle. The center of the circle will be the point where 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 semiperimeter 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 the same distance from all sides and is the intersection of the bisectors.
In a quadrilateral
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). The segment whose ends are located where the opposite sides of a regular quadrilateral intersect lies on the same line, called the Gauss line. The center of the circle will be the point at which the heights of the triangle intersect with the vertices, the diagonals (according to Brocard's theorem).
In a rhombus
It is considered a parallelogram with the same side length. The radius of a 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 is known, the length of its side. The formula r=S/(2Xa) is applied. 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.
- Known sharp corner one of the peaks. Then it is necessary 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 the circle inscribed in the rhombus will be half the length of one side of the figure. If we argue according to Euclid, who claims that the sum of the angles of any quadrilateral is 360 degrees, then one angle will be equal to 90 degrees; those. get a square.
MKOU "Volchikhinsky secondary school No. 2"
Teacher Bakuta E.P.
Grade 9
Lesson on the topic “Formulas for the radii of inscribed and circumscribed circles regular polygons"
Lesson Objectives:
Educational: the study of the formulas for the radii of inscribed and circumscribed circles of regular polygons;
Developing: activation cognitive activity students through solving practical problems, the ability to choose correct solution, concisely express their thoughts, analyze and draw conclusions.
Educational: organization joint activities, education in students of interest in the subject, goodwill, the ability to listen to the answers of comrades.
Equipment: Multimedia computer, multimedia projector, exposure screen
Course of the lesson:
To argue the right thing,
And the motto of our lesson will be these words:
Think collectively!
Decide quickly!
Answer - proof!
Fight hard!
2. Lesson motivation.
3. Update basic knowledge. Checking d / z.
Front poll:
What shape is called a polygon?
What is a regular polygon?
What is another name for a right triangle?
What is another name for a regular quadrilateral?
The formula for the sum of the angles of a convex polygon.
Angle formula for a regular polygon.
4. Learning 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 said to be 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 the circle inscribed in the triangle lies at the intersection of the bisectors of the triangle, and the center of the circle circumscribed about the triangle lies at the intersection of the perpendicular bisectors.
A circle can be circumscribed about any regular polygon, and a circle can be inscribed into any regular polygon, and the center of the circle circumscribed about the regular polygon coincides with the center of the circle inscribed in the same polygon.
Formulas for the radii of the inscribed and circumscribed circles of a regular triangle, regular quadrilateral, regular hexagon.
Radius of an inscribed circle in a regular polygon (r):
a - side of the polygon, N - number of sides of the polygon
Radius of the circumscribed circle of a regular polygon (R):
a is the side of the polygon, N is the number of sides of the polygon.
Let's fill in the table for a regular triangle, a regular quadrangle, a regular hexagon.
5. Consolidation of new material.
Solve #1088, 1090, 1092, 1099.
6. Physical Minute . 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 (group work)
Solve #1093.
8. The results of the lesson. Reflection. D / s.
What is your impression? (liked - disliked)
What is your mood after class? (joyful - sad)
- How are you feeling? (tired - not tired)
- What is the relationship to the material covered? (understood - did not understand)
- What is your self-esteem after the lesson? (Satisfied - not satisfied)
- Evaluate your activity in the lesson. (I tried - I didn’t try).
p.105-108 repeat;
learn formulas;
№ 1090, 1091, 1087(3)
Mathematics has a rumor
That she puts her mind in order,
Because good words
People often talk about her.
You give us geometry
To win an important hardening.
Youth is learning with you
Develop both will and ingenuity.
Note The presentation contains sections:
Repetition of 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,
- In order not to know failures in life,
- Let's go boldly into the world of mathematics,
- In the world of examples and different tasks.
THE MOTTO OF THE LESSON
Think collectively!
Decide quickly!
Answer - proof!
Fight hard!
And discoveries are waiting for us!
Repetition.
- What geometric figure
shown in the picture?
D
E
2. What polygon is called
right?
O
3. What circle is called
inscribed in a polygon?
F
With
4. What circle is called
circumscribed about a polygon?
5. Name the radius of the inscribed circle.
BUT
AT
H
6. Name the radius of the circumscribed circle.
7. How to find the center inscribed in the correct
circle polygon?
8. How to find the center of the circumscribed circle
regular polygon?
Execution check
homework ..
№ 1084.
β is the angle corresponding to
arc that pulls
polygon side .
O
BUT P
BUT 2
β
Answers:
a) 6;
b) 12;
BUT
BUT 1
at 4;
d) 8;
d) 10
e) 20;
e) 7.
e) 5.
REGULAR POLYGON
A regular polygon is a convex polygon in which all angles are equal and all sides are equal.
The sum of the right angles n -gon
Correct angle n - square
A circle is said to be inscribed in a polygon,
if all sides of the polygon touch the circle.
A circle is said to be 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.
We derive the formula for the radius of the inscribed and the radius of the circumscribed circle of a regular polygon.
Let r be the radius of the inscribed circle,
R is the radius of the circumscribed circle,
n is the number of sides and angles of the polygon.
Consider a regular n-gon.
Let a be the side of the n-gon,
α is the angle.
Let's construct a point O - the center of the inscribed and circumscribed circle.
OS is the height ∆AOB.
∟ C \u003d 90 º - (by construction),
Consider ∆AOC:
∟ ОАС = α / 2 - (ОА is the bisector of the angle of the n-gon),
AC \u003d a / 2 - (OS - median to the base of an isosceles triangle),
∟ AOB = 360 º: p,
let ∟AOC = β .
then β = 0.5 ∙ ∟AOB
0.5∙(360º:p)
2 sin(180º:n)
2tg (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
2tg (180º: p)
2 sin(180º:n)
then 180º :p
A regular triangle has n = 3,
whence 2 sin 60 º =
then 180º :p
A regular quadrilateral has n = 4,
whence 2 sin 45 º =
A regular hexagon has n = 6,
then 180º :p
whence 2 sin 30 º =
Using the formulas for the radii of the inscribed and circumscribed circles of some regular polygons, derive formulas for finding the dependence of the sides of regular polygons on the radii of the inscribed and circumscribed circles and fill in the table:
2 R ∙ sin (180 º: p)
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
To find: a, r, R, P
We know formulas:
№ 1088( 5 )
Given: P=6 , n = 3
To find: R, a, r, S
We know formulas:
№ 108 9
Given:
To find:
Summarize
We know formulas:
- p.105-108 repeat;
- learn formulas;
- № 1090, 1091, 1087(3)
If a circle is located inside an angle and touches its sides, it is called inscribed in this angle. The center of such an inscribed circle is located at bisector of this angle.
If it lies inside a convex polygon and is in contact with all its sides, it is called inscribed in a convex polygon.
A circle inscribed in a triangle touches each side of this figure at only one point. Only one circle can be inscribed in one triangle.
The radius of such a circle will depend on the following parameters triangle:
- The length of the sides of a triangle.
- His area.
- Its perimeter.
- The angles of a triangle.
In order to calculate the radius of the inscribed circle in a triangle, it is not always necessary to know all the parameters listed above, since they are interconnected through trigonometric functions.
Calculation using the semi-perimeter
- If the lengths of all sides of a geometric figure are known (we denote them by the letters a, b and c), then the radius will have to be calculated by extracting square root.
- Starting calculations, it is necessary to add one more variable to the initial data - the semi-perimeter (p). It can be calculated by adding all the lengths and dividing the resulting amount by 2. p = (a+b+c)/2. Thus, the formula for finding the radius can be significantly simplified.
- In general, the formula should include the sign of the radical under which the fraction is placed, the denominator of this fraction will be the value of the semi-perimeter p.
- The numerator of this fraction will be the product of the differences (p-a)*(p-b)*(p-c)
- Thus, the full form of the formula will be presented as follows: r = √(p-a)*(p-b)*(p-c)/p).
Calculation considering the area of a triangle
If we know area of a triangle and the lengths of all its sides, this will allow us to find the radius of the circle of interest to us without resorting to extracting roots.
- First you need to double the size of the area.
- The result is divided by the sum of the lengths of all sides. Then the formula will look like this: r = 2*S/(a+b+c).
- If you use the value of the semi-perimeter, you can get a very simple formula: r \u003d S / p.
Calculation using trigonometric functions
If the condition of the problem contains the length of one of the sides, the value of the opposite angle and the perimeter, you can use trigonometric function- tangent. In this case, the calculation formula will look like this:
r \u003d (P / 2- a) * tg (α / 2), where r is the desired radius, P is the perimeter, a is the value of the length of one of the sides, α is the value opposite side, but the angle.
The radius of the circle, which will need to be inscribed in a regular triangle, can be found by the formula r = a*√3/6.
Circle inscribed in a right triangle
You can inscribe in a right triangle only one circle. The center of such a circle simultaneously serves as the intersection point of all bisectors. This geometric figure has some distinctive features that must be taken into account when calculating the radius of the inscribed circle.
- First you need to build a right triangle with given parameters. You can build such a figure by the size of its one side and the values of two angles, or by two sides and the angle between these sides. All these parameters must be specified in the task statement. The triangle is denoted as ABC, with C being the vertex right angle. The legs are denoted by variables, a and b, and the hypotenuse is a variable with.
- To build a classical formula and calculate the radius of a circle, it is necessary to find the dimensions of all sides of the figure described in the condition of the problem and calculate the semiperimeter from them. If the conditions give the dimensions of two legs, they can be used to calculate the value of the hypotenuse, based on the Pythagorean theorem.
- If the size of one leg and one angle is given in the condition, it is necessary to understand whether this angle is adjacent or opposite. In the first case, the hypotenuse is found using the sine theorem: с=a/sinСАВ, in the second case, the cosine theorem is applied с=a/cosCBA.
- When all calculations are completed and the dimensions of all sides are known, the semi-perimeter is found using the formula described above.
- Knowing the value of the semi-perimeter, you can find the radius. The formula is a fraction. Its numerator is the product of the differences of the semi-perimeter and each of the sides, and the denominator is the value of the semi-perimeter.
It should be noted that the numerator of this formula is an indicator of the area. In this case, the formula for finding the radius is much simpler - it is enough to divide the area by a half-perimeter.
It is also possible to determine the area of a geometric figure if both legs are known. The sum of the squares of these legs is the hypotenuse, then the semi-perimeter is calculated. You can calculate the area by multiplying the lengths of the legs by each other and dividing the result by 2.
If the conditions give the lengths of both the legs and the hypotenuse, you can determine the radius using a very simple formula: for this, the lengths of the legs are added, the length of the hypotenuse is subtracted from the resulting number. The result must be divided in half.
Video
From this video you will learn how to find the radius of a circle inscribed in a triangle.
Didn't get an answer to your question? Suggest a topic to the authors.
