A circle is a geometric figure, familiar with which occurs 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 angles of 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 side length is known and degree measure the angle opposite it, 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.
First level
Circumscribed circle. Visual guide (2019)
The first question that may arise is: what is described - around what?
Well, actually, sometimes it happens around anything, but we will talk about a circle circumscribed around (sometimes they also say “about”) a triangle. What is it?
And just imagine, an amazing fact takes place:
Why is this fact surprising?
But triangles are different!
And for everyone there is a circle that will go through through all three peaks, that is, the circumscribed circle.
Proof of this amazing fact you can find in the following levels of the theory, but here we only note that if we take, for example, a quadrilateral, then not for everyone there will be a circle passing through the four vertices. For example, a parallelogram is an excellent quadrilateral, but there is no circle passing through all its four vertices!
And there is only for a rectangle:
Here you go, and every triangle always has its own circumscribed circle! And it’s even always quite easy to find the center of this circle.
Do you know what it is perpendicular bisector?
Now let's see what happens if we consider as many as three perpendicular bisectors to the sides of the triangle.
It turns out (and this is precisely what needs to be proven, although we will not) that all three perpendiculars intersect at one point. Look at the picture - all three perpendicular bisectors intersect at one point.
Do you think the center of the circumscribed circle always lies inside the triangle? Imagine - not always!
But if acute-angled, then - inside:
What to do with a right triangle?
And with an additional bonus:
Since we are talking about the radius of the circumscribed circle: what is it equal to for an arbitrary triangle? And there is an answer to this question: the so-called .
Namely:
And, of course,
1. Existence and circumcircle center
Here the question arises: does such a circle exist for every triangle? It turns out that yes, for everyone. And moreover, we will now formulate a theorem that also answers the question of where the center of the circumscribed circle is located.
Look, like this:
Let's be brave and prove this theorem. If you have already read the topic “” and understood why three bisectors intersect at one point, then it will be easier for you, but if you haven’t read it, don’t worry: now we’ll figure it out.
We will carry out the proof using the concept of locus of points (GLP).
Well, for example, is the set of balls the “geometric locus” of round objects? No, of course, because there are round... watermelons. Is it a set of people, a “geometric place”, who can speak? No, either, because there are babies who cannot speak. In life, it is generally difficult to find an example of a real “geometric location of points.” It's easier in geometry. Here, for example, is exactly what we need:
Here the set is the perpendicular bisector, and the property “ ” is “to be equidistant (a point) from the ends of the segment.”
Shall we check? So, you need to make sure of two things:
- Any point that is equidistant from the ends of a segment is located on the perpendicular bisector to it.
Let's connect c and c.Then the line is the median and height b. This means - isosceles - we made sure that any point lying on the perpendicular bisector is equally distant from the points and.
Let's take the middle and connect and. The result is the median. But according to the condition, not only the median is isosceles, but also the height, that is, the perpendicular bisector. This means that the point exactly lies on the perpendicular bisector.
All! We have fully verified the fact that The perpendicular bisector of a segment is the locus of points equidistant from the ends of the segment.
This is all well and good, but have we forgotten about the circumscribed circle? Not at all, we have just prepared ourselves a “springboard for attack.”
Consider a triangle. Let's draw two bisectoral perpendiculars and, say, to the segments and. They will intersect at some point, which we will name.
Now, pay attention!
The point lies on the perpendicular bisector;
the point lies on the perpendicular bisector.
And that means, and.
Several things follow from this:
Firstly, the point must lie on the third bisector perpendicular to the segment.
That is, the perpendicular bisector must also pass through the point, and all three perpendicular bisectors intersect at one point.
Secondly: if we draw a circle with a center at a point and a radius, then this circle will also pass through both the point and the point, that is, it will be a circumscribed circle. This means that it already exists that the intersection of three perpendicular bisectors is the center of the circumscribed circle for any triangle.
And the last thing: about uniqueness. It is clear (almost) that the point can be obtained in a unique way, therefore the circle is unique. Well, we’ll leave “almost” for your reflection. So we proved the theorem. You can shout “Hurray!”
What if the problem asks “find the radius of the circumscribed circle”? Or vice versa, the radius is given, but you need to find something else? Is there a formula that relates the radius of the circumcircle to the other elements of the triangle?
Please note: the sine theorem states that in order to find the radius of the circumscribed circle, you need one side (any!) and the angle opposite to it. That's all!
3. Center of the circle - inside or outside
Now the question is: can the center of the circumscribed circle lie outside the triangle?
Answer: as much as possible. Moreover, this always happens in an obtuse triangle.
And generally speaking:
CIRCULAR CIRCLE. BRIEFLY ABOUT THE MAIN THINGS
1. Circle circumscribed about a triangle
This is the circle that passes through all three vertices of this triangle.
2. Existence and circumcircle center
Well, the topic is over. If you are reading these lines, it means you are very cool.
Because only 5% of people are able to master something on their own. And if you read to the end, then you are in this 5%!
Now the most important thing.
You have understood the theory on this topic. And, I repeat, this... this is just super! You are already better than the vast majority of your peers.
The problem is that this may not be enough...
For what?
For successful passing the Unified State Exam, for admission to college on a budget and, MOST IMPORTANTLY, for life.
I won’t convince you of anything, I’ll just say one thing...
People who received a good education, earn much more than those who did not receive it. This is statistics.
But this is not the main thing.
The main thing is that they are MORE HAPPY (there are such studies). Perhaps because many more opportunities open up before them and life becomes brighter? Don't know...
But think for yourself...
What does it take to be sure to be better than others on the Unified State Exam and ultimately be... happier?
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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 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.
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).
The radius of a circle that is 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.
The topic “Inscribed and circumscribed circles in triangles” is one of the most difficult in the geometry course. She spends very little time in class.
Geometric problems of this topic are included in the second part exam paper Unified State Examination per course high school. To successfully complete these tasks you need solid knowledge basic geometric facts and some experience in solving geometric problems.
For each triangle there is only one circumcircle. This is a circle on which all three vertices of a triangle with given parameters lie. Finding its radius may be needed not only in a geometry lesson. Designers, cutters, mechanics and representatives of many other professions have to constantly deal with this. In order to find its radius, you need to know the parameters of the triangle and its properties. The center of the circumcircle is at the point of intersection of the perpendicular bisectors of the triangle.
I bring to your attention all the formulas for finding the radius of a circumscribed circle and not just a triangle. Formulas for the inscribed circle can be viewed.
a, b. With - sides of the triangle
α -
opposite anglea,
S-area of a triangle,
p- semi-perimeter
Then to find the radius ( R) of the circumcircle using the formulas:
In turn, the area of the triangle can be calculated using one of the following formulas:
Here are a few more formulas.
1. The radius of the circumscribed circle is about regular triangle. If a side of the triangle then
2. The radius of the circumscribed circle about an isosceles triangle. Let a, b- sides of the triangle, then
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, you can also solve whole line tasks. 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 quantities 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 you 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.