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 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, which is denoted by the Greek letter π .
Determining the circumference of a circle
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 with a radius of 10 centimeters.
Decision:Formula for calculating the dyne of a circle looks like:
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 circumference of a circle with 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, remote from a given point, which is called its center, at some distance, not equal to zero and called the radius. Scientists knew how 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 of a circle was compiled around 1900 BC in ancient Babylon.
With such geometric figures as circles, we encounter daily and everywhere. It is its shape that has the outer surface of the wheels that are equipped with various vehicles. This detail, despite its outward simplicity and unpretentiousness, is considered one of the greatest inventions of mankind, and it is interesting that the natives of Australia and the American Indians, until the arrival of the 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 improved, their design became more and more complex, and for their manufacture it was necessary to use mass various 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 upholster it with metal strips. In order to determine the lengths of these elements, it is necessary to use the formula for calculating the circumference (although in practice, most likely, the craftsmen did this “by eye” or simply girding the wheel with a strip and cutting off the required section of it).
It should be noted that wheel is used not only in vehicles. For example, a potter's wheel has its shape, as well as elements of gears of gears widely used in technology. Since ancient times, wheels have 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 used to make threads from animal wool and plant fibers.
circles often found in construction. Their shape is quite 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 round windows are portholes installed in ships and aircraft.
Thus, design engineers often have to solve the problem of determining the circumference of a circle, developing various machines, mechanisms and assemblies, as well as architects and designers. Since the number π necessary for this is infinite, then it is not possible to determine this parameter with absolute accuracy, and therefore the calculations take into account that degree of it, which in a particular case is necessary and sufficient.
Its diameter. To do this, you just need to apply the formula for the circumference of a circle. L \u003d p DHere: L - circumference, p- the number Pi, equal to 3.14, D - the diameter of the circle. Rearrange the formula for the circumference of the circle to the left side and get: D \u003d L / p
Let's analyze a practical problem. Suppose you need to make a cover for a round country well, access to which is this moment no. No, and unsuitable weather conditions. But do you have data on length its circumference. Suppose it is 600 cm. We substitute the values \u200b\u200bin the indicated formula: D \u003d 600 / 3.14 \u003d 191.08 cm. So, 191 cm is your diameter. Increase the diameter to 2, taking into account the allowance for the edges. Set the compass to a radius of 1 m (100 cm) and draw a circle.
Circles comparatively large diameters at home it is convenient to draw with a compass, which can be quickly made. It is done like this. Two nails are driven into the rail at a distance from each other equal to the radius of the circle. Drive one nail shallowly into the workpiece. And use the other, rotating the rail, as a marker.
A circle is a geometric figure on a plane, which consists of all points of this plane that are at the same distance from a given point. Set point is called the center circles, and the distance at which the points circles are from its center - radius circles. The area of the plane bounded by a circle is called a circle. There are several calculation methods diameter circles, the choice of a specific envy from the available initial data.
Instruction
In the simplest case, if a circle of radius R, then it will be equal to
D=2*R
If the radius circles is not known, but it is known, then the diameter can be calculated using the length formula circles
D = L/P, where L is the length circles, P - P.
Same diameter circles can be calculated, knowing the area bounded by it
D \u003d 2 * v (S / P), where S is the area of \u200b\u200bthe circle, P is the number of P.
Sources:
- circle diameter calculation
In the course of high school planimetry, the concept circle is defined as a geometric figure consisting of all points of a plane lying at a radius distance from a point called its center. Inside the circle, you can draw many segments connecting its points in various ways. Depending on the construction of these segments, circle can be divided into several parts different ways.
Instruction
Finally, circle can be divided into segments. A segment is a part of a circle made up of a chord and an arc of a circle. A chord in this case is a line segment joining any two points on the circle. Using segments circle can be divided into an infinite number of parts with or without education in its center.
Related videos
note
The figures obtained by the listed methods - polygons, segments and sectors, can also be divided using appropriate methods, for example, polygon diagonals or angle bisectors.
A circle is called a flat geometric figure, and the line that limits it is usually called a circle. The main property is that each point on this line is the same distance from the center of the figure. A segment starting at the center of the circle and ending at any of the points on the circle is called the radius, and a segment connecting two points of the circle and passing through the center is called the diameter.
Instruction
Use pi to find the length of a diameter given the circumference of a circle. This constant expresses a constant ratio between these two parameters of the circle - regardless of the size of the circle, dividing its circumference by the length of the diameter always gives the same number. From this it follows that to find the length of the diameter, the circumference should be divided by the number Pi. As a rule, for practical calculations of the length of the diameter, accuracy up to hundredths of a unit, that is, up to two decimal places, is sufficient, so the number Pi can be considered equal to 3.14. But since this constant is an irrational number, it has an infinite number of decimal places. If there is a need for more exact definition, then the required number of characters for pi can be found, for example, at this link - http://www.math.com/tables/constants/pi.htm.
Given the lengths of the sides (a and b) of a rectangle inscribed in a circle, the length of the diameter (d) can be calculated by finding the length of the diagonal of this rectangle. Since the diagonal here is the hypotenuse in a right triangle, the legs of which form sides of a known length, then, according to the Pythagorean theorem, the length of the diagonal, and with it the length of the diameter of the circumscribed circle, can be calculated by finding from the sum of the squares of the lengths famous parties: d=√(a² + b²).
Dividing into several equal parts is a common task. So you can build regular polygon, draw a star, or prepare the basis for a diagram. There are several ways to solve this interesting problem.
You will need
- - a circle with a marked center (if the center is not marked, you will have to find it in any way);
- - protractor;
- - compasses with lead;
- - pencil;
- - ruler.
Instruction
The easiest way to share circle into equal parts - with the help of a protractor. By dividing 360° into the required number of parts, you get the angle. Start at any point on the circle - the radius corresponding to it will be the zero mark. Starting from there, make marks on the protractor corresponding to the calculated angle. This method is recommended if you need to divide circle by five, seven, nine, etc. parts. For example, to build a regular pentagon, its vertices must be located every 360/5 = 72°, that is, at 0°, 72°, 144°, 216°, 288°.
To share circle into six parts, you can use the property of a regular one - its longest diagonal is equal to twice the side. A regular hexagon is, as it were, made up of six equilateral triangles. Set the compass opening equal to the radius of the circle, and make serifs with it, starting from any arbitrary point. The serifs form a regular hexagon, one of the vertices of which will be at this point. By connecting the vertices through one, you will build a regular triangle inscribed in circle, that is, it into three equal parts.
To share circle into four parts, start with an arbitrary diameter. Its ends will give two of the required four points. To find the rest, set the compass solution, equal to the circle. Putting the compass needle on one of the ends of the diameter, make notches outside the circle and below. Repeat the same with the other end of the diameter. Draw an auxiliary line between the intersection points of the serifs. It will give you a second diameter strictly perpendicular to the original. Its ends will become the other two vertices of the square inscribed in circle.
Using the method described above, you can find the midpoint of any segment. As a consequence, this method can double the number of equal parts that you circle. Finding the midpoint of each side of a regular n- inscribed in circle, you can draw perpendiculars to them, find their point of intersection with circle yu and thus construct the vertices of a regular 2n-gon. This procedure can be repeated any time. So, the square turns into , that one - into, etc. Starting with a square, you can, for example, divide circle into 256 equal parts.
note
To divide the circle into equal parts, dividing heads or dividing tables are usually used, which allow dividing the circle into equal parts with high precision. When it is necessary to divide the circle into equal parts, use the table below. To do this, multiply the diameter of the divisible circle by the coefficient given in the table: K x D.
Helpful advice
Division of a circle into three, six and twelve equal parts. Two perpendicular axes are drawn, which, crossing the circle at points 1,2,3,4, divide it into four equal parts; Using the well-known method of division right angle bisectors of right angles are built into two equal parts using a compass or square, which intersect with the circle at points 5, 6, 7, and 8 and divide each fourth part of the circle in half.
When building various geometric shapes sometimes you need to determine their characteristics: length, width, height, and so on. If we are talking about a circle or a circle, then it is often necessary to determine their diameter. Diameter is a line segment that connects two points on a circle that are farthest from each other.
You will need
- - yardstick;
- - compass;
- - calculator.
Instruction
If only the diameter is known, then the formula will look like "R = D / 2".
If length circles is unknown, but there is data on the length of a certain one, then the formula will look like “R \u003d (h ^ 2 * 4 + L ^ 2) / 8 * h”, where h is the height of the segment (is the distance from the middle of the chord to the most protruding part of the specified arc), and L is the length of the segment (which is not the length of the chord). Chord is a segment that connects two points circles.
note
It is necessary to distinguish between the concepts of "circumference" and "circle". A circle is part of a plane, which, in turn, is limited by a circle of a certain radius. To find the radius, you need to know the area of a circle. In this case, the equation will look like "R = (S/π)^1/2", where S is the area. To calculate the area, in turn, you should know the radius (“S = πr^2”).
Knowing only the length diameter circles, you can calculate not only square circle, but also the area of some other geometric shapes. This follows from the fact that the diameters of the circles inscribed or described around such figures coincide with the lengths of their sides or diagonals.
Instruction
If you need to find square(S) according to the known length of it diameter(D), multiply the number pi (π) by the length diameter, and divide the result by four: S=π ² * D² / 4. For example, a circle is equal to twenty centimeters, then its square can be calculated as follows: 3.14² * 20² / 4 \u003d 9.86 * 400 / 4 \u003d 986 centimeters.
If you need to find square square (S) by the diameter of the circle (D) around it, raise the length diameter squared, and divide the result in half: S = D² / 2. For example, if the diameter of the circumscribed circle is twenty centimeters, then square square can be calculated as follows: 20² / 2 \u003d 400 / 2 \u003d 200 square centimeters.
If a square square (S) must be found by the diameter of the circle inscribed in it (D), it is enough to build the length diameter squared: S=D². For example, if the diameter of the inscribed circle is 20 cm, then square square can be calculated as follows: 20² \u003d 400 square centimeters.
If you need to find square(S) by known diameter m inscribed (d) and circumscribed (D) circles around it, then build the length diameter the inscribed circle into a square and divide by four, and add half the product of the lengths of the inscribed and circumscribed circles to the result: S = d² / 4 + D * d / 2. For example, if the diameter of the circumscribed circle is twenty centimeters, and the inscribed circle is ten centimeters, then square triangles can be calculated like this: 10² / 4 + 20 * 10/2 \u003d 25 + 100 \u003d 125 square centimeters.
Use the built-in Google search engine to conduct necessary calculations. For example, to use this search engine square right triangle according to the example from fourth step, you need to enter such a search query: "10 ^ 2 / 4 + 20 * 10/2", and press the Enter key.
Sources:
- how to find the area of a circle given its diameter
A circle is a flat geometric figure, all points of which are at the same and non-zero distance from the selected point, which is called the center of the circle. A straight line connecting any two points of a circle and passing through the center is called it. diameter. The total length of all the boundaries of a two-dimensional figure, which is usually called the perimeter, for a circle is more often denoted as the "circumference". Knowing the circumference of a circle, you can calculate its diameter.
Instruction
Use one of the basic properties of a circle to find the diameter, which is that the ratio of the length of its perimeter to the diameter is the same for absolutely all circles. Of course, constancy did not go unnoticed by mathematicians, and this proportion has long since received its own - this is the number Pi (π is the first Greek word " circle" and "perimeter"). The numerical value of this is determined by the circumference of a circle whose diameter is equal to one.
Divide the known circumference of a circle by pi to calculate its diameter. Since this number is "", it does not have a finite value - it is a fraction. Round pi according to the accuracy of the result you need to get.
Use any to calculate the length of the diameter if you can’t do it in your mind. For example, you can use the one that is built into the Nigma or Google search engine - it is mathematical operations entered on a "human". For example, if the known circumference is four meters, then to find the diameter, you can “humanly” ask the search engine: “4 meters divided by pi.” But if you enter, for example, “4/pi” in the search query field, then the search engine will also understand such a statement of the problem. In any case, the answer is "1.27323954 meters".
The question of the diameter of the globe is not as simple as it might seem at first glance, because the very concept of " Earth” is very conditional. For a real sphere, the diameter will always be the same, no matter where a segment is drawn connecting two points on the surface of the sphere and passing through the center.
With regard to the Earth, it is not possible, since its sphericity is far from ideal (in nature, there are no ideal geometric shapes and bodies at all, they are abstract geometric concepts). To accurately designate the Earth, scientists even had to introduce a special concept - "geoid".
Earth official diameter
The diameter of the Earth is determined by where it will be measured. For convenience, two indicators are taken as the officially recognized diameter: the diameter of the Earth along the equator and the distance between the North and South Poles. The first indicator is 12,756.274 km, and the second is 12,714, the difference between them is a little less than 43 km.
These numbers do not make much impression, they are even inferior to the distance between Moscow and Krasnodar - two cities located on the territory of one country. However, it was not easy to calculate them.
Calculating the Earth's Diameter
The diameter of the planet is calculated using the same geometric formula like any other diameter.
To find the perimeter of a circle, multiply its diameter by pi. Therefore, to find the diameter of the Earth, it is necessary to measure its circumference in the corresponding section (along the equator or in the plane of the poles) and divide it by the number pi.
The first person to attempt to measure the circumference of the Earth was the ancient Greek scientist Eratosthenes of Cyrene. He noticed that in Siena (now Aswan) on the day of the summer solstice, the Sun is at its zenith, illuminating the bottom of a deep well. In Alexandria, on that day, it was 1/50 of the circle from the zenith. From this, the scientist concluded that the distance from Alexandria to Siena is 1/50 of the circumference of the Earth. The distance between these cities is 5,000 Greek stadia (approximately 787.5 km), hence the circumference of the Earth is 250,000 stadia (approximately 39,375 km).
Modern scientists have more advanced means of measurement at their disposal, but their theoretical background corresponds to the idea of Eratosthenes. At two points located several hundred kilometers apart, the position of the Sun or certain stars in the sky is fixed and the difference between the results of the two measurements in degrees is calculated. Knowing the distance in kilometers, it is easy to calculate the length of one degree, and then multiply it by 360.
To clarify the size of the Earth, both laser ranging and satellite observation systems are used.
Today it is believed that the circumference of the Earth along the equator is 40,075.017 km, and along - 40,007.86. Eratosthenes was only slightly wrong.
The magnitude of both the circumference and the diameter of the Earth is increasing due to the meteorite substance that is constantly falling on the Earth, but this process is very slow.
Sources:
- How the Earth was measured in 2019
- This is a flat figure, which is a set of points equidistant from the center. All of them are at the same distance and form a circle.
A line segment that connects the center of a circle with points on its circumference is called radius. In each circle, all radii are equal to each other. A line joining two points on a circle and passing through the center is called diameter. The formula for the area of a circle is calculated using a mathematical constant - the number π ..
It is interesting : The number pi. is the ratio of the circumference of a circle to the length of its diameter and is a constant value. The value π = 3.1415926 was used after the work of L. Euler in 1737.
The area of a circle can be calculated using the constant π. and the radius of the circle. The formula for the area of a circle in terms of radius looks like this:
Consider an example of calculating the area of a circle using the radius. Let a circle with radius R = 4 cm be given. Let's find the area of the figure.
The area of our circle will be equal to 50.24 square meters. cm.
There is a formula the area of a circle in terms of the diameter. It is also widely used to calculate the required parameters. These formulas can be used to find .
Consider an example of calculating the area of a circle through the diameter, knowing its radius. Let a circle be given with a radius R = 4 cm. First, let's find the diameter, which, as you know, is twice the radius.
Now we use the data for the example of calculating the area of a circle using the above formula:
As you can see, as a result we get the same answer as in the first calculations.
Knowledge of the standard formulas for calculating the area of a circle will help in the future to easily determine sector area and it is easy to find the missing quantities.
We already know that the formula for the area of a circle is calculated through the product of the constant value π and the square of the radius of the circle. The radius can be expressed in terms of the circumference of a circle and substitute the expression in the formula for the area of a circle in terms of the circumference:
Now we substitute this equality into the formula for calculating the area of a circle and get the formula for finding the area of \u200b\u200bthe circle, through the circumference
Consider an example of calculating the area of a circle through the circumference. Let a circle be given with length l = 8 cm. Let's substitute the value in the derived formula: 
The total area of the circle will be 5 square meters. cm.
Area of a circle circumscribed around a square
It is very easy to find the area of a circle circumscribed around a square.
This will require only the side of the square and knowledge of simple formulas. The diagonal of the square will be equal to the diagonal of the circumscribed circle. Knowing the side a, it can be found using the Pythagorean theorem: from here.
After we find the diagonal, we can calculate the radius: .
And then we substitute everything into the basic formula for the area of a circle circumscribed around a square:
