Formula for the circumference of a circle based on a known diameter. How to find the circumference of a circle: through diameter and radius

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. The diameter of this figure is equal to the length of 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.
It consists of points, for each of which the following equality holds: the sum of the squares of the distances to the other two is a given value, which is always more than half the length of the segment between them.

Terminology

Not everyone at school had a good math teacher. Therefore, the answer to the question of how to find the circumference of a circle 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 the 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

The definitions directly follow geometric formulas that 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 as follows: C = π*D.
The radius is equal to half the 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 have only 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 the required number of 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 do not necessarily need to know each other’s languages. 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 of the required 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 acquired knowledge on how to find out the circumference of a circle. But often we are not concerned about them, but about the real mathematical problems contained in the textbook. After all, the teacher gives points for them! So let's look at a more complex problem. 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 of such problems is lost. In addition, for such inattention you can get 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!

Many objects in the world around us are round in shape. These are wheels, round window openings, pipes, various dishes and much more. You can calculate the length of a circle by knowing its diameter or radius.

There are several definitions of this geometric figure.

This is a closed curve consisting of points that are located at the same distance from a given point.
This is a curve consisting of points A and B, which are the ends of the segment, and all points from which A and B are visible at right angles. In this case, the segment AB is the diameter.
For the same segment AB, this curve includes all points C such that the ratio AC/BC is constant and not equal to 1.
This is a curve consisting of points for which the following is true: if you add the squares of the distances from one point to two given other points A and B, you get a constant number greater than 1/2 of the segment connecting A and B. This definition is derived from the Pythagorean theorem.

Pay attention! There are other definitions. A circle is an area within a circle. The perimeter of a circle is its length. According to different definitions, a circle may or may not include the curve itself, which is its boundary.

Definition of a circle

Formulas

How to calculate the circumference of a circle using the radius? This is done using a simple formula:

where L is the desired value,

π is the number pi, approximately equal to 3.1413926.

Usually, to find the required value, it is enough to use π to the second digit, that is, 3.14, this will provide the required accuracy. On calculators, in particular engineering ones, there may be a button that automatically enters the value of the number π.

Designations

To find through the diameter there is the following formula:

If L is already known, the radius or diameter can be easily found out. To do this, L must be divided by 2π or π, respectively.

If a circle has already been given, you need to understand how to find the circumference from this data. The area of the circle is S = πR2. From here we find the radius: R = √(S/π). Then

L = 2πR = 2π√(S/π) = 2√(Sπ).

Calculating the area in terms of L is also easy: S = πR2 = π(L/(2π))2 = L2/(4π)

To summarize, we can say that there are three basic formulas:

through the radius – L = 2πR;
through diameter – L = πD;
through the area of the circle – L = 2√(Sπ).

Pi

Without the number π it will not be possible to solve the problem under consideration. The number π was first found as the ratio of the circumference of a circle to its diameter. This was done by the ancient Babylonians, Egyptians and Indians. They found it quite accurately - their results differed from the currently known value of π by no more than 1%. The constant was approximated by such fractions as 25/8, 256/81, 339/108.

Further, the value of this constant was calculated not only from the point of view of geometry, but also from the point of view of mathematical analysis through sums of series. The designation of this constant by the Greek letter π was first used by William Jones in 1706, and it became popular after the work of Euler.

It is now known that this constant is an infinite non-periodic decimal fraction; it is irrational, that is, it cannot be represented as a ratio of two integers. Using supercomputer calculations, the 10-trillionth sign of the constant was discovered in 2011.

This is interesting! Various mnemonic rules have been invented to remember the first few digits of the number π. Some allow you to store a large number of numbers in memory, for example, one French poem will help you remember pi up to the 126th digit.

If you need the circumference, an online calculator will help you with this. There are many such calculators; you just need to enter the radius or diameter. Some of them have both of these options, others calculate the result only through R. Some calculators can calculate the desired value with different precision, you need to specify the number of decimal places. You can also calculate the area of a circle using online calculators.

Such calculators are easy to find with any search engine. There are also mobile applications that will help you solve the problem of how to find the circumference of a circle.

Useful video: circumference

Practical Application

Solving such a problem is most often necessary for engineers and architects, but in everyday life, knowledge of the necessary formulas can also be useful. For example, you need to wrap a paper strip around a cake baked in a mold with a diameter of 20 cm. Then it will not be difficult to find the length of this strip:

L = πD = 3.14 * 20 = 62.8 cm.

Another example: you need to build a fence around a round pool at a certain distance. If the radius of the pool is 10 m, and the fence needs to be placed at a distance of 3 m, then R for the resulting circle will be 13 m. Then its length is:

L = 2πR = 2 * 3.14 * 13 = 81.68 m.

Useful video: circle - radius, diameter, circumference

Bottom line

The perimeter of a circle can be easily calculated using simple formulas involving diameter or radius. You can also find the desired quantity through the area of a circle. Online calculators or mobile applications, in which you need to enter a single number - diameter or radius, will help you solve this problem.

Very often, when solving school assignments in or 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.

Basic concepts and definitions

Radius is the line connecting the center of the circle and its arbitrary point. It is denoted by the 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 in 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 fraction. In school mathematics, 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 in the opposite direction until it intersects with the arc. We measure the resulting distance with a compass and use any metric instrument 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 curved line that encloses a circle. In geometry, shapes are flat, so the definition refers to a two-dimensional image. It is assumed that all points of this curve are located at an equal distance from the center of the circle.

The circle has several characteristics on the basis of which calculations related to this geometric figure are made. These include: diameter, radius, area and circumference. These characteristics are interrelated, that is, to calculate them, information about at least one of the components is sufficient. For example, knowing only the radius of a geometric figure, you can use the formula to find the circumference, diameter, and area.

The radius of a circle is the segment inside the circle connected to its center.
A diameter is a segment inside a circle connecting its points and passing through the center. Essentially, the diameter is two radii. This is exactly what the formula for calculating it looks like: D=2r.
There is one more component of a circle - a chord. This is a straight line that connects two points on a circle, but does not always pass through the center. So the chord that passes through it is also called the diameter.

How to find out the circumference? Let's find out now.

Circumference: formula

The Latin letter p was chosen to denote this characteristic. Archimedes also proved that the ratio of the circumference of a circle to its diameter is the same number for all circles: this is the number π, which is approximately equal to 3.14159. The formula for calculating π is: π = p/d. According to this formula, the value of p is equal to πd, that is, the circumference: p= πd. Since d (diameter) is equal to two radii, the same formula for the circumference can be written as p=2πr. Let's consider the application of the formula using simple problems as an example:

Problem 1

At the base of the Tsar Bell the diameter is 6.6 meters. What is the circumference of the base of the bell?

So, the formula for calculating the circle is p= πd
Substitute the existing value into the formula: p=3.14*6.6= 20.724

Answer: The circumference of the bell base is 20.7 meters.

Problem 2

The artificial satellite of the Earth rotates at a distance of 320 km from the planet. The radius of the Earth is 6370 km. What is the length of the satellite's circular orbit?

1. Calculate the radius of the circular orbit of the Earth satellite: 6370+320=6690 (km)
2.Calculate the length of the satellite’s circular orbit using the formula: P=2πr
3.P=2*3.14*6690=42013.2

Answer: the length of the circular orbit of the Earth satellite is 42013.2 km.

Methods for measuring circumference

Calculating the circumference of a circle is not often used in practice. The reason for this is the approximate value of the number π. In everyday life, to find the length of a circle, a special device is used - a curvimeter. An arbitrary starting point is marked on the circle and the device is led from it strictly along the line until they reach this point again.

How to find the circumference of a circle? You just need to keep simple calculation formulas in your head.

A circle is a series of points equidistant from one point, which, in turn, is the center of this circle. The circle also has its own radius, equal to the distance of 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 .

Determination of 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 × 5 = 31.4 centimeters

Circle is a geometric figure, which is the collection of all points on the plane removed from a given point, which is called its center, by a certain distance not equal to zero and called the radius. Scientists were able to determine its length with varying degrees of accuracy already in ancient times: historians of science believe that the first formula for calculating the circumference was compiled around 1900 BC in ancient Babylon.

We encounter geometric shapes such as circles 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 apparent simplicity and unpretentiousness, is considered one of the greatest inventions of mankind, and it is interesting that the Australian aborigines and American Indians, until the arrival of Europeans, 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 It is not only used in vehicles. For example, it 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 of special tools. 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, it is not possible to determine this parameter with absolute accuracy, and therefore, the calculations take into account the degree of it that in a particular case is necessary and sufficient.