1. Harder to find circumference through diameter, so let’s look at this option first.
Example: Find the circumference of a circle whose diameter is 6 cm. We use the circle circumference formula above, but first we need to find the radius. To do this, we divide the diameter of 6 cm by 2 and get the radius of the circle 3 cm.
After that, everything is extremely simple: Multiply the number Pi by 2 and by the resulting radius of 3 cm.
2 * 3.14 * 3 cm = 6.28 * 3 cm = 18.84 cm.
2. Now let’s look at the simple option again find the circumference of the circle, the radius is 5 cm
Solution: Multiply the radius of 5 cm by 2 and multiply by 3.14. Don’t be alarmed, because rearranging the multipliers does not affect the result, and circumference formula can be used in any order.
5cm * 2 * 3.14 = 10 cm * 3.14 = 31.4 cm - this is the found circumference for a radius of 5 cm!
Online circumference calculator
Our circumference calculator will perform all these simple calculations instantly and write the solution in a line and with comments. We will calculate the circumference for a radius of 3, 5, 6, 8 or 1 cm, or the diameter is 4, 10, 15, 20 dm; our calculator does not care for which radius value to find the circumference.
All calculations will be accurate, tested by specialist mathematicians. The results can be used in the solution school tasks in geometry or mathematics, as well as for working calculations in construction or in the repair and decoration of premises, when accurate calculations using this formula are required.
Very often when deciding school assignments in physics, the question arises - how to find the circumference of a circle, knowing the diameter? In fact, there are no difficulties in solving this problem; you just need to clearly imagine what formulas,concepts and definitions are required for this.
In contact with
Basic concepts and definitions
- Radius is the line connecting the center of the circle and its arbitrary point. It is designated Latin letter r.
- A chord is a line connecting two arbitrary points lying on a circle.
- Diameter is the line connecting two points of a circle and passing through its center. It is denoted by the Latin letter d.
- is a line consisting of all points located at equal distances from one selected point, called its center. We will denote its length by the Latin letter l.
The area of a circle is the entire territory enclosed within a circle. It is measured V square units and is denoted by the Latin letter s.
Using our definitions, we come to the conclusion that the diameter of a circle is equal to its largest chord.
Attention! From the definition of what the radius of a circle is, you can find out what the diameter of a circle is. These are two radii laid out in opposite directions!
Diameter of a circle.
Finding the circumference and area of a circle
If we are given the radius of a circle, then the diameter of the circle is described by the formula d = 2*r. Thus, to answer the question of how to find the diameter of a circle, knowing its radius, the last one is enough multiply by two.
The formula for the circumference of a circle, expressed in terms of its radius, has the form l = 2*P*r.
Attention! The Latin letter P (Pi) denotes the ratio of the circumference of a circle to its diameter, and this is a non-periodic decimal. IN school math it is considered a previously known tabular value equal to 3.14!
Now let's rewrite the previous formula to find the circumference of a circle through its diameter, remembering what its difference is in relation to the radius. It will turn out: l = 2*P*r = 2*r*P = P*d.
From the mathematics course we know that the formula describing the area of a circle has the form: s = П*r^2.
Now let's rewrite the previous formula to find the area of a circle through its diameter. We get,
s = П*r^2 = П*d^2/4.
One of the most difficult tasks in this topic is determining the area of a circle through the circumference and vice versa. Let's take advantage of the fact that s = П*r^2 and l = 2*П*r. From here we get r = l/(2*P). Let's substitute the resulting expression for the radius into the formula for the area, we get: s = l^2/(4P). In a completely similar way, the circumference is determined through the area of the circle.
Determining radius length and diameter
Important! First of all, let's learn how to measure the diameter. It's very simple - draw any radius, extend it by the opposite side until it intersects with the arc. We measure the resulting distance with a compass and use any metric tool to find out what we are looking for!
Let us answer the question of how to find out the diameter of a circle, knowing its length. To do this, we express it from the formula l = П*d. We get d = l/P.
We already know how to find its diameter from the circumference of a circle, and we can also find its radius in the same way.
l = 2*P*r, hence r = l/2*P. In general, to find out the radius, it must be expressed in terms of the diameter and vice versa.
Suppose now you need to determine the diameter, knowing the area of the circle. We use the fact that s = П*d^2/4. Let us express d from here. It will work out d^2 = 4*s/P. To determine the diameter itself, you will need to extract square root of the right side. It turns out d = 2*sqrt(s/P).
Solving typical tasks
- Let's find out how to find the diameter if the circumference is given. Let it be equal to 778.72 kilometers. Required to find d. d = 778.72/3.14 = 248 kilometers. Let's remember what a diameter is and immediately determine the radius; to do this, we divide the value d determined above in half. It will work out r = 248/2 = 124 kilometer
- Let's consider how to find the length of a given circle, knowing its radius. Let r have a value of 8 dm 7 cm. Let's convert all this into centimeters, then r will be equal to 87 centimeters. Let's use the formula to find the unknown length of a circle. Then our desired value will be equal to l = 2*3.14*87 = 546.36 cm. Let's convert our obtained value into integer numbers of metric quantities l = 546.36 cm = 5 m 4 dm 6 cm 3.6 mm.
- Let us need to determine the area of a given circle using the formula through its known diameter. Let d = 815 meters. Let's remember the formula for finding the area of a circle. Let's substitute the values given to us here, we get s = 3.14*815^2/4 = 521416.625 sq. m.
- Now we will learn how to find the area of a circle, knowing the length of its radius. Let the radius be 38 cm. We use the formula known to us. Let us substitute here the value given to us by condition. You get the following: s = 3.14*38^2 = 4534.16 sq. cm.
- The last task is to determine the area of a circle based on the known circumference. Let l = 47 meters. s = 47^2/(4P) = 2209/12.56 = 175.87 sq. m.
Circumference
A circle is a closed curve, all points of which are at the same distance from the center. This figure is flat. Therefore, the solution to the problem, the question of which is how to find the circumference, is quite simple. We will look at all available methods in today's article.
Figure descriptions
In addition to a fairly simple descriptive definition, there are three more mathematical characteristics of a circle, which in themselves contain the answer to the question of how to find the circumference:
- Consists of points A and B and all others from which AB can be seen at right angles. Diameter of this figure equal to length the segment under consideration.
- Includes only those points X such that the ratio AX/BX is constant and not equal to one. If this condition is not met, then it is not a circle.
- Consists of points, for each of which the following equality holds: the sum of the squares of the distances to the other two is set value, which is always more than half the length of the segment between them.
Terminology
Not everyone at school had good teacher mathematics. Therefore, the answer to the question of how to find the circumference is further complicated by the fact that not everyone knows the basic geometric concepts. Radius is a segment that connects the center of a figure to a point on a curve. A special case in trigonometry is unit circle. A chord is a segment that connects two points on a curve. For example, the already discussed AB falls under this definition. The diameter is the chord passing through the center. The number π is equal to the length of a unit semicircle.
Basic formulas
From the definitions it follows directly geometric formulas, which allow you to calculate the main characteristics of a circle:
- The length is equal to the product of the number π and the diameter. The formula is usually written in the following way: C = π*D.
- Radius equal to half diameter It can also be calculated by calculating the quotient of dividing the circumference by twice the number π. The formula looks like this: R = C/(2* π) = D/2.
- The diameter is equal to the quotient of the circumference divided by π or twice the radius. The formula is quite simple and looks like this: D = C/π = 2*R.
- The area of a circle is equal to the product of π and the square of the radius. Similarly, diameter can be used in this formula. In this case, the area will be equal to the quotient of the product of π and the square of the diameter divided by four. The formula can be written as follows: S = π*R 2 = π*D 2 /4.
How to find the circumference of a circle by diameter
For simplicity of explanation, let us denote by letters the characteristics of the figure necessary for the calculation. Let C be the desired length, D its diameter, and π approximately equal to 3.14. If we only have one known quantity, then the problem can be considered solved. Why is this necessary in life? Suppose we decide to surround a round pool with a fence. How to calculate required amount columns? And here the ability to calculate the circumference comes to the rescue. The formula is as follows: C = π D. In our example, the diameter is determined based on the radius of the pool and the required distance from the fence. For example, suppose that our home artificial pond is 20 meters wide, and we are going to place the posts at a ten-meter distance from it. The diameter of the resulting circle is 20 + 10*2 = 40 m. Length is 3.14*40 = 125.6 meters. We will need 25 posts if the gap between them is about 5 m.
Length through radius
As always, let's start by assigning letters to the characteristics of the circle. In fact, they are universal, so mathematicians from different countries It is not at all necessary to know each other's language. Let's assume that C is the circumference of the circle, r is its radius, and π is approximately equal to 3.14. The formula in this case looks like this: C = 2*π*r. Obviously, this is an absolutely correct equation. As we have already figured out, the diameter of a circle is equal to twice its radius, so this formula looks like this. In life, this method can also often come in handy. For example, we bake a cake in a special sliding form. To prevent it from getting dirty, we need a decorative wrapper. But how to cut a circle the right size. This is where mathematics comes to the rescue. Those who know how to find out the circumference of a circle will immediately say that you need to multiply the number π by twice the radius of the shape. If its radius is 25 cm, then the length will be 157 centimeters.
Sample problems
We have already looked at several practical cases of the knowledge gained on how to find out the circumference of a circle. But often we are not concerned about them, but about real math problems which are contained in the textbook. After all, the teacher gives points for them! So let's look at the problem increased complexity. Let's assume that the circumference of the circle is 26 cm. How to find the radius of such a figure?
Example solution
First, let's write down what we are given: C = 26 cm, π = 3.14. Also remember the formula: C = 2* π*R. From it you can extract the radius of the circle. Thus, R= C/2/π. Now let's proceed to the actual calculation. First, divide the length by two. We get 13. Now we need to divide by the value of the number π: 13/3.14 = 4.14 cm. It is important not to forget to write the answer correctly, that is, with units of measurement, otherwise the entire practical meaning similar tasks. In addition, for such inattention you can receive a grade one point lower. And no matter how annoying it may be, you will have to put up with this state of affairs.
The beast is not as scary as it is painted
So we have dealt with such a difficult task at first glance. As it turns out, you just need to understand the meaning of the terms and remember a few simple formulas. Math is not that scary, you just need to put in a little effort. So geometry is waiting for you!
Circle concept
Definition 1
Circle-- a geometric figure consisting of all points located at equal distances from given point.
Definition 2
For the purposes of Definition 1, the given point is called the center of the circle.
Definition 3
The segment connecting the center of the circle with any of its points is called the radius of the circle $(r)$ (Fig. 1).
Figure 1. Circle with center at point $O$ and radius $r$
Equation of a circle
Let us derive the equation of a circle in Cartesian system coordinates $xOy$. Let the center of the circle $C$ have coordinates $(x_0,y_0)$, and the radius of the circle be equal to $r$. Let a point $M$ with coordinates $(x,y)$ -- arbitrary point this circle (Fig. 2).
Figure 2. Circle in Cartesian coordinate system
The distance from the center of the circle to the point $M$ is calculated as follows
But, since $M$ lies on the circle, then by definition 3, we get $CM=r$. Then we get the following
Equation (1) is the equation of a circle with center at point $(x_0,y_0)$ and radius $r$.
In particular, if the center of the circle coincides with the origin. That equation of a circle has the form
Circumference
Let us derive the formula for the circumference of a circle $C$ in terms of its radius. To do this, consider two circles with lengths $C$ and $C"$ and radii $R$ and $R"$. Let us inscribe in it regular $n-gons$ with perimeters $P$ and $P"$ and side lengths $a$ and $a"$, respectively. As we know, the side of an inscribed triangle is equal to
Then we get
Hence
Unlimitedly increasing the number of sides of regular polygons $n$ we get that
From here we get
We found that the ratio of the circumference of a circle to its diameter constant number for any circle. This constant is usually denoted by the number $\pi \approx 3.14$. Thus, we get
Formula (2) is the formula for calculating the circumference.
Area of a circle
Definition 4
Circle-- part of a plane bounded by a circle.
Let us derive a formula for calculating the area of a circle.
Consider the following situation. Let us be given a circle with radius $R$. Let's denote its area by $S$. A regular -gon with area $S_n$ is inscribed into it, into which, in turn, a circle with area $(S")_n$ is inscribed (Fig. 3).
Figure 3.
From the figure it is obvious that
We use the following well-known formula For regular polygon:
We will now increase the number of sides of a regular polygon without limit. Then, for $n\to \infty $, we get
According to the formula, the area of a regular polygon is equal to $S_n=\frac(1)(2)P_nr$, $P_n\to 2\pi R$, therefore
Formula (3) is the formula for calculating the area of a circle.
Example problem on the concept of a circle
Example 1
Find the equation of a circle with center at point $(1,\ 1)$. passing through the origin, find the length of the given circle and the area of the circle bounded by the given circle.
Solution.
Let's first find the equation of this circle. For this we will use formula (1). Since the center of the circle lies at the point $(1,\ 1)$, we get
\[((x-1))^2+((y-1))^2=r^2\]
Let's find the radius of the circle as the distance from the point $(1,\ 1)$ to the point $(0,0)$
We find that the equation of a circle has the form:
\[((x-1))^2+((y-1))^2=2\]
Let's find the circumference using formula (2). We get
Let's find the area using formula (3)
Answer:$((x-1))^2+((y-1))^2=2$, $C=2\sqrt(2)\pi $, $S=2\pi $
A circle is a series of points equidistant from one point, which, in turn, is the center of this circle. A circle also has its own radius, equal to distance these points from the center.
The ratio of the length of a circle to its diameter is the same for all circles. This ratio is a number that is a mathematical constant and is denoted by the Greek letter π .
Determining the circumference
You can calculate the circle using the following formula:
L= π D=2 π r
r- circle radius
D- circle diameter
L- circumference
π - 3.14
Task:
Calculate circumference, having a radius of 10 centimeters.
Solution:Formula for calculating the circumference of a circle has the form:
L= π D=2 π r
where L is the circumference, π is 3.14, r is the radius of the circle, D is the diameter of the circle.
Thus, the length of a circle having a radius of 10 centimeters is:
L = 2 × 3.14 × 10 = 62.8 centimeters
Circle is a geometric figure, which is a collection of all points on the plane removed from a given point, which is called its center, at a certain distance, not equal to zero and called the radius. Determine its length with varying degrees Scientists were able to achieve accuracy already in ancient times: historians of science believe that the first formula for calculating the circumference of a circle was compiled around 1900 BC in ancient Babylon.
With such geometric shapes, like circles, we encounter every day and everywhere. It is its shape that has the outer surface of the wheels that are equipped with various vehicles. This detail, despite its external simplicity and unpretentiousness, is considered one of greatest inventions humanity, and it is interesting that the aborigines of Australia and American Indians Until the arrival of Europeans, they had absolutely no idea what it was.
In all likelihood, the very first wheels were pieces of logs that were mounted on an axle. Gradually, the design of the wheel was improved, their design became more and more complex, and their manufacture required the use of a lot of different tools. First, wheels appeared consisting of a wooden rim and spokes, and then, in order to reduce wear on their outer surface, they began to cover it with metal strips. In order to determine the lengths of these elements, it is necessary to use a formula for calculating the circumference (although in practice, most likely, the craftsmen did this “by eye” or simply by encircling the wheel with a strip and cutting off the required section).
It should be noted that wheel is not only used in vehicles. For example, its shape is shaped like a potter's wheel, as well as elements of gears of gears, widely used in technology. Wheels have long been used in the construction of water mills (the oldest structures of this kind known to scientists were built in Mesopotamia), as well as spinning wheels, which were used to make threads from animal wool and plant fibers.
Circles can often be found in construction. Their shape is shaped by fairly widespread round windows, very characteristic of the Romanesque architectural style. The manufacture of these structures is a very difficult task and requires high skill, as well as the availability special tool. One of the varieties of round windows are portholes installed in ships and aircraft.
Thus, design engineers who develop various machines, mechanisms and units, as well as architects and designers, often have to solve the problem of determining the circumference of a circle. Since the number π , necessary for this, is infinite, then with absolute accuracy it is not possible to determine this parameter, and therefore the calculations take into account its degree, which in one or another specific case is necessary and sufficient.