Parting 

How to calculate the length knowing the diameter. How to calculate the circumference of a circle if the diameter and radius of the circle are not specified

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 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:

  1. Using degree measure: CD = \frac(\pi R \alpha ^(\circ))(180^(\circ))
  2. 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.

It can be calculated by knowing the arc size, since it equal to half 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 intersect internal corners figure, 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 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.

A 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?

  1. So, the formula for calculating the circle is p= πd
  2. 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. 1. Calculate the radius of the circular orbit of the Earth satellite: 6370+320=6690 (km)
  2. 2.Calculate the length of the satellite’s circular orbit using the formula: P=2πr
  3. 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, which is denoted 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 a plane removed from a given point, which is called its center, by 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, 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 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.

No matter what sphere of the economy a person works, wittingly or unwittingly he uses mathematical knowledge accumulated over many centuries. We come across devices and mechanisms containing circles every day. A wheel has a round shape, pizza, many vegetables and fruits form a circle when cut, as well as plates, cups, and much more. However, not everyone knows how to correctly calculate the circumference.

To calculate the circumference of a circle, you must first remember what a circle is. This is the set of all points of the plane equidistant from this one. A circle is a geometric locus of points on a plane located inside a circle. From the above it follows that the perimeter of a circle and the circumference are one and the same.

Methods for finding the circumference of a circle

In addition to the mathematical method of finding the perimeter of a circle, there are also practical ones.

  • Take a rope or cord and wrap it around once.
  • Then measure the rope, the resulting number will be the circumference.
  • Roll the round object once and count the length of the path. If the item is very small, you can wrap it with twine several times, then unwind the thread, measure and divide by the number of turns.
  • Find the required value using the formula:

L = 2πr = πD ,

where L is the required length;

π – constant, approximately equal to 3.14 r – radius of the circle, the distance from its center to any point;

D is the diameter, it is equal to two radii.

Applying the formula to find the circumference of a circle

  • Example 1. Treadmill passes around a circle with a radius of 47.8 meters. Find the length of this treadmill, taking π = 3.14.

L = 2πr =2*3.14*47.8 ≈ 300(m)

Answer: 300 meters

  • Example 2. A bicycle wheel, having turned 10 times, has traveled 18.85 meters. Find the radius of the wheel.

18.85: 10 =1.885 (m) is the perimeter of the wheel.

1.885: π = 1.885: 3.1416 ≈ 0.6(m) – required diameter

Answer: wheel diameter 0.6 meters

The amazing number pi

Despite the apparent simplicity of the formula, for some reason it is difficult for many to remember it. Apparently, this is due to the fact that the formula contains an irrational number π, which is not present in the formulas for the area of ​​​​other figures, for example, a square, triangle or rhombus. You just need to remember that this is a constant, that is, a constant meaning the ratio of the circumference to the diameter. About 4 thousand years ago, people noticed that the ratio of the perimeter of a circle to its radius (or diameter) is the same for all circles.

The ancient Greeks approximated the number π with the fraction 22/7. For a long timeπ was calculated as the average between the lengths of inscribed and circumscribed polygons in a circle. In the third century AD, a Chinese mathematician performed a calculation for a 3072-gon and obtained an approximate value of π = 3.1416. It must be remembered that π is always constant for any circle. Its designation with the Greek letter π appeared in the 18th century. This is the first letter of the Greek words περιφέρεια - circle and περίμετρος - perimeter. In the eighteenth century, it was proven that this quantity is irrational, that is, it cannot be represented in the form m/n, where m is an integer and n is a natural number.

And how is it different from a circle? Take a pen or colors and draw a regular circle on a piece of paper. Paint over the entire middle of the resulting figure with a blue pencil. The red outline indicating the boundaries of the shape is a circle. But the blue content inside it is the circle.

The dimensions of a circle and a circle are determined by the diameter. On the red line indicating the circle, mark two points so that they are mirror images of each other. Connect them with a line. The segment will definitely pass through the point in the center of the circle. This segment connecting opposite parts of a circle is called a diameter in geometry.

A segment that does not extend through the center of the circle, but joins it at opposite ends, is called a chord. Consequently, the chord passing through the center point of the circle is its diameter.

Diameter is indicated Latin letter D. You can find the diameter of a circle using values ​​such as area, length and radius of the circle.

The distance from the central point to the point plotted on the circle is called the radius and is denoted by the letter R. Knowing the value of the radius helps to calculate the diameter of the circle in one simple step:

For example, the radius is 7 cm. We multiply 7 cm by 2 and get a value equal to 14 cm. Answer: D of the given figure is 14 cm.

Sometimes you have to determine the diameter of a circle only by its length. Here it is necessary to apply a special formula to help determine Formula L = 2 Pi * R, where 2 is a constant value (constant), and Pi = 3.14. And since it is known that R = D * 2, the formula can be presented in another way

This expression is also applicable as a formula for the diameter of a circle. Substituting the quantities known in the problem, we solve the equation with one unknown. Let's say the length is 7 m. Therefore:

Answer: the diameter is 21.98 meters.

If the area is known, then the diameter of the circle can also be determined. The formula that is used in in this case, looks like this:

D = 2 * (S / Pi) * (1 / 2)

S - in this case. Let's say in the problem it is equal to 30 square meters. m. We get:

D = 2 * (30 / 3, 14) * (1 / 2) D = 9, 55414

When the value indicated in the problem is equal to the volume (V) of the ball, the following formula for finding the diameter is applied: D = (6 V / Pi) * 1 / 3.

Sometimes you have to find the diameter of a circle inscribed in a triangle. To do this, use the formula to find the radius of the represented circle:

R = S / p (S - area given triangle, and p is the perimeter divided by 2).

We double the result obtained, taking into account that D = 2 * R.

Often you have to find the diameter of a circle in everyday life. For example, when determining what is equivalent to its diameter. To do this, you need to wrap the finger of the potential owner of the ring with thread. Mark the points of contact between the two ends. Measure the length from point to point with a ruler. We multiply the resulting value by 3.14, following the formula for determining the diameter with a known length. So, the statement that knowledge of geometry and algebra is not useful in life is not always true. And this is a serious reason for taking school subjects more responsibly.