How to calculate the length knowing the diameter. How to find and what will be the circumference of a circle?

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.

A circle is found in everyday life no less often than a rectangle. And for many people, the problem of how to calculate the circumference is difficult. And all because it has no corners. If they were available, everything would become much easier.

What is a circle and where does it occur?

This flat figure represents a number of points that are located at the same distance from another one, which is the center. This distance is called the radius.

In everyday life, it is not often necessary to calculate the circumference of a circle, except for people who are engineers and designers. They create designs for mechanisms that use, for example, gears, portholes and wheels. Architects create houses with round or arched windows.

Each of these and other cases requires its own precision. Moreover, it turns out to be impossible to calculate the circumference absolutely accurately. This is due to the infinity of the main number in the formula. "Pi" is still being refined. And the rounded value is most often used. The degree of accuracy is chosen to give the most correct answer.

Designations of quantities and formulas

Now it’s easy to answer the question of how to calculate the circumference of a circle by radius; for this you will need the following formula:

Since radius and diameter are related to each other, there is another formula for calculations. Since the radius is two times smaller, the expression will change slightly. And the formula for how to calculate the circumference of a circle, knowing the diameter, will be as follows:

l = π * d.

What if you need to calculate the perimeter of a circle?

Just remember that a circle includes all the points inside the circle. This means that its perimeter coincides with its length. And after calculating the circumference, put an equal sign with the perimeter of the circle.

By the way, their designations are the same. This applies to radius and diameter, and the perimeter is the Latin letter P.

Examples of tasks

Task one

Condition. Find out the length of a circle whose radius is 5 cm.

Solution. Here it is not difficult to understand how to calculate the circumference. You just need to use the first formula. Since the radius is known, all you need to do is substitute the values and calculate. 2 multiplied by a radius of 5 cm gives 10. All that remains is to multiply it by the value of π. 3.14 * 10 = 31.4 (cm).

Answer: l = 31.4 cm.

Task two

Condition. There is a wheel whose circumference is known and equal to 1256 mm. It is necessary to calculate its radius.

Solution. In this task you will need to use the same formula. But only the known length will need to be divided into the product of 2 and π. It turns out that the product will give the result: 6.28. After division, the number left is: 200. This is the desired value.

Answer: r = 200 mm.

Task three

Condition. Calculate the diameter if the circumference of the circle is known, which is 56.52 cm.

Solution. Similar to the previous problem, you will need to divide the known length by the value of π, rounded to the nearest hundredth. As a result of this action, the number 18 is obtained. The result is obtained.

Answer: d = 18 cm.

Problem four

Condition. The clock hands are 3 and 5 cm long. You need to calculate the lengths of the circles that describe their ends.

Solution. Since the arrows coincide with the radii of the circles, the first formula is required. You need to use it twice.

For the first length, the product will consist of factors: 2; 3.14 and 3. The result will be 18.84 cm.

For the second answer, you need to multiply 2, π and 5. The product will give the number: 31.4 cm.

Answer: l 1 = 18.84 cm, l 2 = 31.4 cm.

Task five

Condition. A squirrel runs in a wheel with a diameter of 2 m. How far does it run in one full revolution of the wheel?

Solution. This distance is equal to the circumference. Therefore, you need to use a suitable formula. Namely, multiply the value of π and 2 m. Calculations give the result: 6.28 m.

Answer: The squirrel runs 6.28 m.

We are surrounded by many objects. And many of them are round in shape. It is given to them for convenient use. Take, for example, a wheel. If it were made in the shape of a square, how would it roll along the road?

In order to make a round object, you need to know what the formula for circumference through diameter looks like. To do this, we first define what this concept is.

Circle and circumference

A circle is a set of points that are located at equal distances from the main point - the center. This distance is called the radius.

The distance between two points on a given line is called a chord. In addition, if a chord passes through the main point (center), then it is called a diameter.

Now let's look at what a circle is. The set of all points that are inside the outline is called a circle.

What is circumference?

After we have covered all the definitions, we can calculate the diameter of a circle. The formula will be discussed a little later.

First, we will try to measure the length of the outline of the glass. To do this, we will wrap it with thread, then measure it with a ruler and find out the approximate length of the imaginary line around the glass. Because the size depends on the correct measurement of the item, and this method is not reliable. But nevertheless, it is quite possible to make accurate measurements.

To do this, let us again remember the wheel. We have repeatedly seen that if you increase the spoke in the wheel (radius), the length of the wheel rim (circumference) will also increase. And also, as the radius of the circle decreases, the length of the rim also decreases.

If we carefully follow these changes, we will see that the length of an imaginary circular line is proportional to its radius. And this number is constant. Next, let's look at how the diameter of a circle is determined: the formula for this will be used in the example below. And let's look at it step by step.

Circle formula through diameter

Since the length of the outline is proportional to the radius, it is correspondingly proportional to the diameter. Therefore, we will conventionally denote its length by the letter C, and its diameter by d. Since the ratio of the length of the outline and the diameter is a constant number, it can be determined.

Having done all the calculations, we will determine a number that is approximately equal to 3.1415... For the reason that during the calculations a specific number did not work out, we will denote it with the letter π . This icon will be useful to us in order to derive the formula for the circumference of a circle through its diameter.

Let's draw an imaginary line through the central point and measure the distance between the two extreme ones. This will be the diameter. If we know the diameter of a circle, the formula for determining its length will look like this: C = d * π.

If we determine the length of different outlines, then if their diameter is known, the same formula will be applied. Because the sign π - this is an approximate calculation, it was decided to multiply the diameter by 3.14 (a number rounded to hundredths).

How to calculate diameter: formula

This time, let's try using this formula to calculate other quantities besides the length of the outline. To calculate the diameter from the circumference, the same formula is used. Only for this purpose we divide its length by π . It will look like this d = C / π.

Let's look at how this formula works in practice. For example, we know the length of the outline of a well, we need to calculate its diameter. It is impossible to measure it because there is no access to it due to weather conditions. Our task is to make a lid. What should we do in this case?

You need to use the formula. Let's take the length of the well outline - for example, 600 cm. We put a specific number in the formula, namely C = 600 / 3.14. As a result, we get approximately 191 cm. Let's round the result to 200 cm. Then, using a compass, draw a round line with a radius of 100 cm.

Since an outline with a large diameter must be drawn with an appropriate compass, you can make such a tool yourself. To do this, take a strip of the required length and drive a nail at each end. We install one nail into the workpiece and drive it in lightly so that it does not move from the intended place. And with the help of the second we draw a line. The device is very simple and convenient.

Modern technologies allow you to use an online calculator to calculate the length of the outline. To do this, you just need to enter the diameter of the circle. The formula will be applied automatically. You can also calculate the circumference of a circle using the radius. Also, if you know the circumference of a circle, the online calculator will calculate the radius and diameter using this formula.

First, let's understand the difference between a circle and a circle. To see this difference, it is enough to consider what both figures are. These are an infinite number of points on the plane, located at an equal distance from a single central point. But, if the circle also consists of internal space, then it does not belong to the circle. It turns out that a circle is both a circle that limits it (circle(r)), and an innumerable number of points that are inside the circle.

For any point L lying on the circle, the equality OL=R applies. (The length of the segment OL is equal to the radius of the circle).

A segment that connects two points on a circle is its chord.

A chord passing directly through the center of a circle is diameter this circle (D). The diameter can be calculated using the formula: D=2R

Circumference calculated by the formula: C=2\pi R

Area of a circle: S=\pi R^(2)

Arc of a circle is called that part of it that is located between its two points. These two points define two arcs of a circle. The chord CD subtends two arcs: CMD and CLD. Identical chords subtend equal arcs.

Central angle An angle that lies between two radii is called.

Arc length can be found using the formula:

Using degree measure: CD = \frac(\pi R \alpha ^(\circ))(180^(\circ))
Using radian measure: CD = \alpha R

The diameter, which is perpendicular to the chord, divides the chord and the arcs contracted by it in half.

If the chords AB and CD of the circle intersect at the point N, then the products of the segments of the chords separated by the point N are equal to each other.

AN\cdot NB = CN\cdot ND

Tangent to a circle

Tangent to a circle It is customary to call a straight line that has one common point with a circle.

If a line has two common points, it is called secant.

If you draw the radius to the tangent point, it will be perpendicular to the tangent to the circle.

Let's draw two tangents from this point to our circle. It turns out that the tangent segments will be equal to one another, and the center of the circle will be located on the bisector of the angle with the vertex at this point.

AC = CB

Now let’s draw a tangent and a secant to the circle from our point. We obtain that the square of the length of the tangent segment will be equal to the product of the entire secant segment and its outer part.

AC^(2) = CD \cdot BC

We can conclude: the product of an entire segment of the first secant and its external part is equal to the product of an entire segment of the second secant and its external part.

AC\cdot BC = EC\cdot DC

Angles in a circle

The degree measures of the central angle and the arc on which it rests are equal.

\angle COD = \cup CD = \alpha ^(\circ)

Inscribed angle is an angle whose vertex is on a circle and whose sides contain chords.

You can calculate it by knowing the size of the arc, since it is equal to half of this arc.

\angle AOB = 2 \angle ADB

Based on a diameter, inscribed angle, right angle.

\angle CBD = \angle CED = \angle CAD = 90^ (\circ)

Inscribed angles that subtend the same arc are identical.

Inscribed angles resting on one chord are identical or their sum is equal to 180^ (\circ) .

\angle ADB + \angle AKB = 180^ (\circ)

\angle ADB = \angle AEB = \angle AFB

On the same circle are the vertices of triangles with identical angles and a given base.

An angle with a vertex inside the circle and located between two chords is identical to half the sum of the angular values of the arcs of the circle that are contained within the given and vertical angles.

\angle DMC = \angle ADM + \angle DAM = \frac(1)(2) \left (\cup DmC + \cup AlB \right)

An angle with a vertex outside the circle and located between two secants is identical to half the difference in the angular values of the arcs of the circle that are contained inside the angle.

\angle M = \angle CBD - \angle ACB = \frac(1)(2) \left (\cup DmC - \cup AlB \right)

Inscribed circle

Inscribed circle is a circle tangent to the sides of a polygon.

At the point where the bisectors of the corners of a polygon intersect, its center is located.

A circle may not be inscribed in every polygon.

The area of a polygon with an inscribed circle is found by the formula:

S = pr,

p is the semi-perimeter of the polygon,

r is the radius of the inscribed circle.

It follows that the radius of the inscribed circle is equal to:

r = \frac(S)(p)

The sums of the lengths of opposite sides will be identical if the circle is inscribed in a convex quadrilateral. And vice versa: a circle fits into a convex quadrilateral if the sums of the lengths of opposite sides are identical.

AB + DC = AD + BC

It is possible to inscribe a circle in any of the triangles. Only one single one. At the point where the bisectors of the internal angles of the figure intersect, the center of this inscribed circle will lie.

The radius of the inscribed circle is calculated by the formula:

r = \frac(S)(p) ,

where p = \frac(a + b + c)(2)

Circumcircle

If a circle passes through each vertex of a polygon, then such a circle is usually called described about a polygon.

At the point of intersection of the perpendicular bisectors of the sides of this figure will be the center of the circumscribed circle.

The radius can be found by calculating it as the radius of the circle that is circumscribed about the triangle defined by any 3 vertices of the polygon.

There is the following condition: a circle can be described around a quadrilateral only if the sum of its opposite angles is equal to 180^( \circ) .

\angle A + \angle C = \angle B + \angle D = 180^ (\circ)

Around any triangle you can describe a circle, and only one. The center of such a circle will be located at the point where the perpendicular bisectors of the sides of the triangle intersect.

The radius of the circumscribed circle can be calculated using the formulas:

R = \frac(a)(2 \sin A) = \frac(b)(2 \sin B) = \frac(c)(2 \sin C)

R = \frac(abc)(4 S)

a, b, c are the lengths of the sides of the triangle,

S is the area of the triangle.

Ptolemy's theorem

Finally, consider Ptolemy's theorem.

Ptolemy's theorem states that the product of diagonals is identical to the sum of the products of opposite sides of a cyclic quadrilateral.

AC \cdot BD = AB \cdot CD + BC \cdot AD

A circle consists of many points that are at equal distances from the center. This is a flat geometric figure, and finding its length is not difficult. A person encounters a circle and a circle every day, regardless of what field he works in. Many vegetables and fruits, devices and mechanisms, dishes and furniture are round in shape. A circle is the set of points that lies within the boundaries of the circle. Therefore, the length of the figure is equal to the perimeter of the circle.

Characteristics of the figure

In addition to the fact that the description of the concept of a circle is quite simple, its characteristics are also easy to understand. With their help you can calculate its length. The inner part of the circle consists of many points, among which two - A and B - can be seen at right angles. This segment is called the diameter, it consists of two radii.

Within the circle there are points X such, which does not change and is not equal to unity, the ratio AX/BX. In a circle, this condition must be met; otherwise, this figure does not have the shape of a circle. Each point that makes up a figure is subject to the following rule: the sum of the squared distances from these points to the other two always exceeds half the length of the segment between them.

Basic circle terms

In order to be able to find the length of a figure, you need to know the basic terms relating to it. The main parameters of the figure are diameter, radius and chord. The radius is the segment connecting the center of the circle with any point on its curve. The magnitude of a chord is equal to the distance between two points on the curve of the figure. Diameter - distance between points, passing through the center of the figure.

Basic formulas for calculations

The parameters are used in the formulas for calculating the dimensions of a circle:

Diameter in calculation formulas

In economics and mathematics there is often a need to find the circumference of a circle. But in everyday life you may encounter this need, for example, when building a fence around a round pool. How to calculate the circumference of a circle by diameter? In this case, use the formula C = π*D, where C is the desired value, D is the diameter.

For example, the width of the pool is 30 meters, and the fence posts are planned to be placed at a distance of ten meters from it. In this case, the formula for calculating the diameter is: 30+10*2 = 50 meters. The required value (in this example, the length of the fence): 3.14*50 = 157 meters. If the fence posts stand at a distance of three meters from each other, then a total of 52 of them will be needed.

Radius calculations

How to calculate the circumference of a circle from a known radius? To do this, use the formula C = 2*π*r, where C is the length, r is the radius. The radius in a circle is half the diameter, and this rule can be useful in everyday life. For example, in the case of preparing a pie in a sliding form.

To prevent the culinary product from getting dirty, it is necessary to use a decorative wrapper. How to cut a paper circle of the appropriate size?

Those who are a little familiar with mathematics understand that in this case you need to multiply the number π by twice the radius of the shape used. For example, the diameter of the shape is 20 centimeters, respectively, its radius is 10 centimeters. Using these parameters, the required circle size is found: 2*10*3, 14 = 62.8 centimeters.

Handy calculation methods

If it is not possible to find the circumference using the formula, then you should use available methods for calculating this value:

If a round object is small, its length can be found using a rope wrapped around it once.
The size of a large object is measured as follows: a rope is laid out on a flat surface, and a circle is rolled along it once.
Modern students and schoolchildren use calculators for calculations. Online, you can find out unknown quantities using known parameters.

Round objects in the history of human life

The first round-shaped product that man invented was the wheel. The first structures were small round logs mounted on an axle. Then came wheels made of wooden spokes and rims. Gradually, metal parts were added to the product to reduce wear. It was in order to find out the length of the metal strips for the wheel upholstery that scientists of past centuries were looking for a formula for calculating this value.

A potter's wheel has the shape of a wheel, most parts in complex mechanisms, designs of water mills and spinning wheels. Round objects are often found in construction - frames of round windows in the Romanesque architectural style, portholes in ships. Architects, engineers, scientists, mechanics and designers every day in their professional activities are faced with the need to calculate the dimensions of a circle.