Calculating the radius: how to find the circumference of a circle knowing the diameter. How to find the circumference of a circle: through diameter and radius

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 .

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 × 5 = 31.4 centimeters

Circle is a geometric figure, which is a collection of all points on a 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, 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 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, in the calculations, the degree of it is taken into account, which in a particular case is necessary and sufficient.

Very often, when solving school assignments in physics or science, 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 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*П). Let's substitute the resulting expression for the radius into the formula for the area, we get: s = l^2/(4П). 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 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 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.

The circumference of a circle is indicated by the letter C and is calculated by the formula:

C = 2πR,
Where R - radius of the circle.

Derivation of the formula expressing the circumference

Path C and C’ are the lengths of circles of radii R and R’. Let us inscribe a regular n-gon into each of them and denote their perimeters by P n and P" n, and their sides by a n and a" n. Using the formula for calculating the side of a regular n-gon a n = 2R sin (180°/n) we get:
P n = n a n = n 2R sin (180°/n),
P" n = n · a" n = n · 2R" sin (180°/n).
Hence,
P n / P" n = 2R / 2R". (1)
This equality is valid for any value of n. We will now increase the number n without limit. Since P n → C, P" n → C", n → ∞, then the limit of the ratio P n / P" n is equal to C / C". On the other hand, by virtue of equality (1), this limit is equal to 2R/2R". Thus, C/C" = 2R/2R". From this equality it follows that C/2R = C"/2R", i.e. . The ratio of the circumference of a circle to its diameter is the same number for all circles. This number is usually denoted by the Greek letter π (“pi”).
From the equality C / 2R = π we obtain the formula for calculating the circumference of a circle of radius R:
C = 2πR.

Circular arc length

Since the length of the entire circle is 2πR, then the length l of an arc of 1° is equal to 2πR / 360 = πR / 180.
That's why length l of an arc of a circle with degree measure α expressed by the formula
l = (πR / 180) α.

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 solving school problems in geometry or mathematics, as well as in working calculations in construction or in the repair and decoration of premises, when accurate calculations using this formula are required.