You will need

• Triangle with given parameters
• Compass
• Ruler
• Square
• Table of sines and cosines
• Mathematical concepts
• Determining the height of a triangle
• Sine and cosine formulas
• Triangle area formula

Instructions

Draw a triangle with the necessary parameters. A triangle has either three sides, or two sides and an angle between them, or a side and two adjacent angles. Label the vertices of the triangle as A, B, and C, the angles as α, β, and γ, and the sides opposite the vertices as a, b, and c.

Draw to all sides of the triangle and find their intersection point. Denote the heights as h with the corresponding indices for the sides. Find the point of their intersection and label it O. It will be the center of the circle. Thus, the radii of this circle will be the segments OA, OB and OS.

Find the radius using two formulas. For one, you need to first calculate . It is equal to all the sides of the triangle by the sine of any of the angles divided by 2.

In this case, the radius of the circumscribed circle is calculated by the formula

For the other, the length of one of the sides and the sine of the opposite angle are sufficient.

Calculate the radius and describe the circumference of the triangle.

Helpful advice

Remember what the height of a triangle is. This is a perpendicular drawn from a corner to the opposite side.

The area of ​​a triangle can also be represented as the product of the square of one of the sides and the sines of two adjacent angles, divided by twice the sine of the sum of these angles.
S=а2*sinβ*sinγ/2sinγ

Sources:

• table with circumscribed circle radii
• Radius of a circle circumscribed about an equilateral

It is considered circumscribed around a polygon if it touches all its vertices. What is noteworthy is that the center of such circle coincides with the intersection point of perpendiculars drawn from the midpoints of the sides of the polygon. Radius described circle completely depends on the polygon around which it is described.

You will need

• Know the sides of a polygon and its area/perimeter.

Instructions

note

A circle can be drawn around a polygon only if it is regular, i.e. all its sides are equal and all its angles are equal.
The thesis stating that the center of a circle circumscribed around a polygon is its intersection perpendicular bisectors, is valid for all regular polygons.

Sources:

• how to find the radius of a polygon

If it is possible to construct a circumcircle for a polygon, then the area of ​​this polygon is less area circumscribed circle, but more area inscribed circle. For some polygons, formulas are known to find radius inscribed and circumscribed circles.

Instructions

A circle inscribed in a polygon that touches all sides of the polygon. For a triangle radius circles: r = ((p-a)(p-b)(p-c)/p)^1/2, where p is the semi-perimeter; a, b, c - sides of the triangle. For the formula is simplified: r = a/(2*3^1/2), a is the side of the triangle.

A circle circumscribed around a polygon is a circle on which all the vertices of the polygon lie. For a triangle, the radius is found by the formula: R = abc/(4(p(p-a)(p-b)(p-c))^1/2), where p is the semi-perimeter; a, b, c - sides of the triangle. For the correct one it’s easier: R = a/3^1/2.

For polygons, it is not always possible to find out the ratio of the inscribed radii and the lengths of its sides. More often they are limited to constructing such circles around the polygon, and then physical radius circles using measuring instruments or vector space.
To construct the circumcircle of a convex polygon, the bisectors of its two corners are constructed; at their intersection lies the center of the circumscribed circle. The radius will be the distance from the point of intersection of the bisectors to the vertex of any corner of the polygon. The center of the inscribed at the intersection of perpendiculars built inside the polygon from the centers of the sides (these perpendiculars are median). It is enough to construct two such perpendiculars. The radius of the inscribed circle is equal to the distance from the point of intersection of the median perpendiculars to the side of the polygon.

Video on the topic

note

It is impossible to inscribe a circle in an arbitrarily given polygon and describe a circle around it.

Helpful advice

A circle can be inscribed in a quadrilateral if a+c = b+d, where a, b, c, d are the sides of the quadrilateral in order. A circle can be described around a quadrilateral if its opposite angles add up to 180 degrees;

For a triangle, such circles always exist.

Tip 4: How to find the area of ​​a triangle based on three sides

Finding the area of ​​a triangle is one of the most common problems in school planimetry. Knowing the three sides of a triangle is enough to determine the area of ​​any triangle. In special cases of equilateral triangles, it is enough to know the lengths of two and one side, respectively.

You will need

• lengths of sides of triangles, Heron's formula, cosine theorem

Instructions

Heron's formula for the area of ​​a triangle is as follows: S = sqrt(p(p-a)(p-b)(p-c)). If we write the semi-perimeter p, we get: S = sqrt(((a+b+c)/2)((b+c-a)/2)((a+c-b)/2)((a+b-c)/2) ) = (sqrt((a+b+c)(a+b-c)(a+c-b)(b+c-a)))/4.

You can derive a formula for the area of ​​a triangle from considerations, for example, by applying the cosine theorem.

By the cosine theorem, AC^2 = (AB^2)+(BC^2)-2*AB*BC*cos(ABC). Using the introduced notations, these can also be written in the form: b^2 = (a^2)+(c^2)-2a*c*cos(ABC). Hence, cos(ABC) = ((a^2)+(c^2)-(b^2))/(2*a*c)

The area of ​​a triangle is also found by the formula S = a*c*sin(ABC)/2 using two sides and the angle between them. The sine of angle ABC can be expressed in terms of it using the basic trigonometric identity: sin(ABC) = sqrt(1-((cos(ABC))^2) Substituting the sine into the formula for the area and writing it out, you can arrive at the formula for the area triangle ABC.

Video on the topic

The three points that uniquely define a triangle in the Cartesian coordinate system are its vertices. Knowing their position relative to each of the coordinate axes, you can calculate any parameters of this flat figure, including those limited by its perimeter square. This can be done in several ways.

Instructions

Use Heron's formula to calculate area triangle. It involves the dimensions of the three sides of the figure, so start your calculations with . The length of each side must be equal to the root of the sum of the squares of the lengths of its projections onto the coordinate axes. If we denote the coordinates A(X₁,Y₁,Z₁), B(X₂,Y₂,Z₂) and C(X₃,Y₃,Z₃), the lengths of their sides can be expressed as follows: AB = √((X₁-X₂)² + (Y₁ -Y₂)² + (Z₁-Z₂)²), BC = √((X₂-X₃)² + (Y₂-Y₃)² + (Z₂-Z₃)²), AC = √((X₁-X₃)² + (Y₁-Y₃)² + (Z₁-Z₃)²).

To simplify calculations, introduce an auxiliary variable - semiperimeter (P). From the fact that this is half the sum of the lengths of all sides: P = ½*(AB+BC+AC) = ½*(√((X₁-X₂)² + (Y₁-Y₂)² + (Z₁-Z₂)²) + √ ((X₂-X₃)² + (Y₂-Y₃)² + (Z₂-Z₃)²) + √((X₁-X₃)² + (Y₁-Y₃)² + (Z₁-Z₃)²).

Calculate square(S) using Heron's formula - take the root of the product of the semi-perimeter and the difference between it and the length of each side. IN general view it can be written as follows: S = √(P*(P-AB)*(P-BC)*(P-AC)) = √(P*(P-√((X₁-X₂)² + (Y₁-Y₂ )² + (Z₁-Z₂)²))*(P-√((X₂-X₃)² + (Y₂-Y₃)² + (Z₂-Z₃)²))*(P-√((X₁-X₃) ² + (Y₁-Y₃)² + (Z₁-Z₃)²)).

For practical calculations, it is convenient to use specialized calculators. These are scripts hosted on the servers of some sites that will do all the necessary calculations based on the coordinates you entered into the appropriate form. The only such service is that it does not provide explanations and justifications for each step of the calculations. Therefore, if you are only interested in the final result, and not in general calculations, go, for example, to the page http://planetcalc.ru/218/.

In the form fields, enter each coordinate of each vertex triangle- they are here as Ax, Ay, Az, etc. If the triangle is specified by two-dimensional coordinates, write zero in the fields Az, Bz and Cz. In the “Calculation accuracy” field, set the required number of decimal places by clicking the plus or minus mouse. It is not necessary to press the orange “Calculate” button located under the form; the calculations will be made without it. You will find the answer next to the inscription “Area triangle" - it is located immediately below the orange button.

Sources:

• find the area of ​​a triangle with vertices at points

Sometimes around a convex polygon you can draw it in such a way that the vertices of all the corners lie on it. Such a circle in relation to the polygon should be called circumscribed. Her center does not have to be inside the perimeter of the inscribed figure, but using the properties of the described circle, finding this point is usually not very difficult.

You will need

• Ruler, pencil, protractor or square, compass.

Instructions

If the polygon around which you need to describe a circle is drawn on paper, to find center and a circle is enough with a ruler, pencil and protractor or square. Measure the length of any side of the figure, determine its middle and place an auxiliary point in this place in the drawing. Using a square or protractor, draw a segment inside the polygon perpendicular to this side until it intersects with the opposite side.

Do the same operation with any other side of the polygon. The intersection of the two constructed segments will be the desired point. This follows from the main property of the described circle- her center in a convex polygon with any side always lies at the point of intersection of the bisector perpendiculars drawn to these

A radius is a line segment that connects any point on a circle to its center. This is one of the most important characteristics of this figure, since on its basis all other parameters can be calculated. If you know how to find the radius of a circle, you can calculate its diameter, length, and area. In the case when a given figure is inscribed or described around another, a number of other problems can be solved. Today we will look at the basic formulas and the features of their application.

Known quantities

If you know how to find the radius of a circle, which is usually denoted by the letter R, then it can be calculated using one characteristic. These values ​​include:

• circumference (C);
• diameter (D) - a segment (or rather, a chord) that passes through the central point;
• area (S) - the space that is limited by a given figure.

Circumference

If the value of C is known in the problem, then R = C / (2 * P). This formula is a derivative. If we know what the circumference is, then we no longer need to remember it. Let's assume that in the problem C = 20 m. How to find the radius of the circle in this case? We simply substitute the known value into the above formula. Note that in such problems knowledge of the number P is always implied. For convenience of calculations, we take its value as 3.14. The solution in this case looks like this: we write down what values ​​are given, derive the formula and carry out the calculations. In the answer we write that the radius is 20 / (2 * 3.14) = 3.19 m. It is important not to forget what we calculated and mention the name of the units of measurement.

By diameter

Let us immediately emphasize that this is the simplest type of problem, which asks how to find the radius of a circle. If you came across such an example on a test, then you can rest assured. You don't even need a calculator here! As we have already said, diameter is a segment or, more correctly, a chord that passes through the center. In this case, all points of the circle are equidistant. Therefore, this chord consists of two halves. Each of them is a radius, which follows from its definition as a segment that connects a point on a circle and its center. If the diameter is known in the problem, then to find the radius you simply need to divide this value by two. The formula is as follows: R = D / 2. For example, if the diameter in the problem is 10 m, then the radius is 5 meters.

By area of ​​a circle

This type of problem is usually called the most difficult. This is primarily due to ignorance of the formula. If you know how to find the radius of a circle in this case, then the rest is a matter of technique. In the calculator, you just need to find the square root calculation icon in advance. The area of ​​a circle is the product of the number P and the radius multiplied by itself. The formula is as follows: S = P * R 2. By isolating the radius on one side of the equation, you can easily solve the problem. It will be equal to the square root of the quotient of the area divided by the number P. If S = 10 m, then R = 1.78 meters. As in previous problems, it is important to remember the units of measurement used.

How to find the circumradius of a circle

Let's assume that a, b, c are the sides of the triangle. If you know their values, you can find the radius of the circle described around it. To do this, you first need to find the semi-perimeter of the triangle. To make it easier to understand, let's denote it with the small letter p. It will be equal to half the sum of the sides. Its formula: p = (a + b + c) / 2.

We also calculate the product of the lengths of the sides. For convenience, let's denote it by the letter S. The formula for the radius of the circumscribed circle will look like this: R = S / (4 * √(p * (p - a) * (p - b) * (p - c)).

Let's look at an example task. We have a circle circumscribed around a triangle. The lengths of its sides are 5, 6 and 7 cm. First, we calculate the semi-perimeter. In our problem it will be equal to 9 centimeters. Now let's calculate the product of the lengths of the sides - 210. We substitute the results of intermediate calculations into the formula and find out the result. The radius of the circumscribed circle is 3.57 centimeters. We write down the answer, not forgetting about the units of measurement.

How to find the radius of an inscribed circle

Let's assume that a, b, c are the lengths of the sides of the triangle. If you know their values, you can find the radius of the circle inscribed in it. First you need to find its semi-perimeter. To make it easier to understand, let's denote it with the small letter p. The formula for calculating it is as follows: p = (a + b + c) / 2. This type of problem is somewhat simpler than the previous one, so no more intermediate calculations are needed.

The radius of the inscribed circle is calculated using the following formula: R = √((p - a) * (p - b) * (p - c) / p). Let's look at this specific example. Suppose the problem describes a triangle with sides of 5, 7 and 10 cm. A circle is inscribed in it, the radius of which needs to be found. First we find the semi-perimeter. In our problem it will be equal to 11 cm. Now we substitute it into the main formula. The radius will be equal to 1.65 centimeters. We write down the answer and do not forget about the correct units of measurement.

Circle and its properties

Each geometric figure has its own characteristics. The correctness of problem solving depends on their understanding. The circle also has them. They are often used when solving examples with described or inscribed figures, since they provide a clear picture of such a situation. Among them:

• A straight line can have zero, one or two points of intersection with a circle. In the first case it does not intersect with it, in the second it is a tangent, in the third it is a secant.
• If we take three points that do not lie on the same line, then only one circle can be drawn through them.
• A straight line can be tangent to two figures at once. In this case, it will pass through a point that lies on the segment connecting the centers of the circles. Its length is equal to the sum of the radii of these figures.
• An infinite number of circles can be drawn through one or two points.

Circumference – geometric figure, acquaintance with which occurs back in preschool age. Later you will learn its properties and characteristics. If the vertices of an arbitrary polygon lie on a circle, and the figure itself is located inside it, then you have a geometric figure inscribed in the circle.

The concept of radius characterizes the distance from any point on a circle to its center. The latter is located at the intersection of perpendiculars to each side of the polygon. Having decided on the terminology, let's consider expressions that will help find the radius for any type of polygon.

How to find the radius of a circumscribed circle - regular polygon

This figure can have any number of vertices, but all its sides are equal. To find the radius of a circle in which a regular polygon is placed, it is enough to know the number of sides of the figure and their length.
R = b/2sin(180°/n),
b – side length,
n is the number of vertices (or sides) of the figure.
The given relationship for the case of a hexagon will have the following form:
R = b/2sin(180°/6) = b/2sin30°,
R = b.

How to find the circumradius of a rectangle

When a quadrilateral is located in a circle, having 2 pairs of parallel sides and internal corners 90°, the point of intersection of the diagonals of the polygon will be its center. Using the Pythagorean relation, as well as the properties of a rectangle, we obtain the expressions necessary to find the radius:
R = (√m 2 + l 2)/2,
R = d/2,
m, l – sides of the rectangle,
d is its diagonal.

How to find the radius of a circumscribed circle - square

Place a square in the circle. The latter is a regular polygon with 4 sides. Because Since a square is a special case of a rectangle, its diagonals are also divided in half at their intersection point.
R = (√m 2 + l 2)/2 = (√m 2 + m 2)/2 = m√2/2 = m/√2,
R = d/2,
m – side of the square,
d is its diagonal.

How to find the radius of a circumscribed circle - an isosceles trapezoid

If a trapezoid is placed in a circle, then to determine the radius you will need to know the lengths of its sides and the diagonal.
R = m*l*d/4√p(p – m)*(p – l)*(p – d),
p = (m + l + d)/2,
m, l – sides of the trapezoid,
d is its diagonal.

How to find the radius of a circumscribed circle - a triangle

Free Triangle

• To determine the radius of a circle describing a triangle, it is enough to know the size of its sides.
  R = m*l*k/4√p(p – m)*(p – l)*(p – k),
  p = (m + l + k)/2,
  m, l, k – sides of the triangle.
• If the length of the side and the degree measure of the angle opposite it are known, then the radius is determined as follows:
  For triangle MLK
  R = m/2sinM = l/2sinL = k/2sinK,
  
  M, L, K – its angles (vertices).
• Given the area of ​​a figure, you can also calculate the radius of the circle in which it is placed:
  R = m*l*k/4S,
  m, l, k – sides of the triangle,
  S is its area.

Isosceles triangle

If a triangle is isosceles, then its 2 sides are equal to each other. When describing such a figure, the radius can be found using the following relationship:
R = m*l*k/4√p(p – m)*(p – l)*(p – k), but m = l
R = m 2 /√(4m 2 – k 2),
m, k – sides of the triangle.

Right triangle

If one of the angles of the triangle is right, and a circle is circumscribed around the figure, then to determine the length of the radius of the latter, the presence of known sides of the triangle will be required.
R = (√m 2 + l 2)/2 = k/2,
m, l – legs,
k – hypotenuse.

A triangle is called inscribed if all its vertices lie on the circle. In this case the circle is called described around the triangle. The distance from its center to each vertex of the triangle will be the same and equal to the radius this circle. Any triangle can be surrounded by a circle, but only one.

The center of the circumcircle will lie at the intersection point of the perpendicular bisectors drawn to each side of the triangle. If a circle is circumscribed around a right triangle, then its center will lie at the middle of the hypotenuse. For any triangle around which a circle is circumscribed, the formula for the area of ​​a triangle in terms of the radius of the circumscribed circle applies:

in which a, b, c are the sides of the triangle, and R is the radius of the circumscribed circle.

An example of calculating the area of ​​a triangle using the radius of the circumscribed circle:
Let a triangle be given with sides a = 5 cm, b = 6 cm, c = 4 cm. A circle with R = 3 cm is circumscribed around it. Find the area.
Having all the required data, we simply substitute the values ​​into the formula:

The area of ​​the triangle will be 10 square meters. cm

Quite often, according to the conditions, you can find a given area of ​​the circumscribed circle, which must be used to find the area of ​​the inscribed triangle. The formula for the area of ​​a triangle through the area of ​​the circumcircle is found after calculating the radius. It can be calculated in several ways. First, consider the formula for the area of ​​a circle:
Transforming this formula, we get that the radius is:
Using this formula, we find that knowing the area of ​​the circumscribed circle, we can find the area of ​​the triangle in the following way:

Knowing all three sides of a given triangle can be used to find the area. From it you can also find the radius of the circumscribed circle. That is, if all the sides of a triangle are given in the conditions and we need to find the area through the radius of the circumscribed circle, we must first calculate it using the formula:

That is, knowing the lengths of all sides of the triangle, we can find the area of ​​the triangle through the radius of the circumscribed circle.

An example of calculating the area of ​​a triangle using the area of ​​the circumcircle:
Given a triangle around which a circle with an area of ​​8 square meters is circumscribed. cm. The sides of the triangle are a = 4 cm, b = 3 cm, c = 5 cm. First, let’s find the radius of the circle through its area:

Let's try to find the radius using another formula, which we derived from the method of finding

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 equilateral triangle, there are quite 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, 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.