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 denoted by the 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 applies 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 used: 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 is the area of the 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.
Instructions
If only the diameter is known, the formula will look like “R = D/2”.
If length circle is unknown, but there is data on the length of a certain , then the formula will look like “R = (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). A chord is a segment that connects two points circle.
note
It is necessary to distinguish between the concepts of “circle” 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 the circle. In this case, the equation will be “R = (S/π)^1/2”, where S is the area. To calculate the area, in turn, you need to 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 figures. This follows from the fact that the diameters of circles inscribed or circumscribed around such figures coincide with the lengths of their sides or diagonals.
Instructions
If you need to find square(S) according to its known length diameter(D), multiply pi (π) by its length diameter, and divide the result by four: S=π ²*D²/4. For example, a circle is twenty centimeters, then its square can be calculated as follows: 3.14² * 20² / 4 = 9.86 * 400 / 4 = 986 centimeters.
If you need to find square square (S) along the diameter of the circle (D) around it, construct 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 = 400 / 2 = 200 square centimeters.
If square square (S) must be found by the diameter of the circle inscribed in it (D), it is enough to construct the length diameter squared: S=D². For example, if the diameter of the inscribed circle is twenty centimeters, then square square can be calculated as follows: 20² = 400 square centimeters.
If you need to find square(S) according to known diameter m inscribed (d) and circumscribed (D) circles around it, then construct the length diameter inscribed circle into a square and divide by four, and to the result add half the product of the lengths of the inscribed and circumscribed circles: 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 triangle can be calculated as follows: 10² / 4 + 20 * 10/2 = 25 + 100 = 125 square centimeters.
Use Google's built-in search engine to perform the necessary calculations. For example, so that using this search engine square a right triangle according to the example from the fourth step, you need to enter the following search query: “10^2 / 4 + 20*10/2” and press the Enter key.
Sources:
- how to find the area of a circle by diameter
A circle is a flat geometric figure, all points of which are at the same and non-zero distance from a 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 diameter. The total length of all the boundaries of a two-dimensional figure, which is usually called the perimeter, is more often referred to as the “circumference” of a circle. Knowing the circumference of a circle, you can calculate its diameter.
Instructions
To find the diameter, use one of the main properties of a circle, 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 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 length 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 obtain.
Use some to calculate the length of the diameter if you can’t do it in your head. For example, you can use the one that is built into the Nigma or Google search engine - it is mathematical operations entered in “human” language. 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” into the search query field, then the search engine will understand this formulation of the problem. In any case, the answer will be “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 “globe” is very arbitrary. A real ball will always have the same diameter, no matter where a segment is drawn connecting two points on the surface of the sphere and passing through the center.
In relation to the Earth, it does not seem possible, since its spherical shape is far from ideal (in nature there are no ideal geometric figures and bodies at all; they are abstract geometric concepts). To accurately designate the Earth, scientists even had to introduce a special concept - “geoid”.
Official diameter of the Earth
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 at 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 slightly less than 43 km.
These numbers do not make much of an impression; they are even inferior to the distance between Moscow and Krasnodar - two cities located in the same country. However, it was not easy to figure them out.
Calculating the diameter of the Earth
The diameter of the planet is calculated using the same geometric formula as any other diameter.
To find the perimeter of a circle, you need to multiply its diameter by the number pi. Consequently, to find the diameter of the Earth, you need to measure its circumference in the appropriate section (along the equator or in the plane of the poles) and divide it by the number pi.
The first person to try 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 was at its zenith, illuminating the bottom of a deep well. In Alexandria on that day it was 1/50 of the circle away from the zenith. From this, the scientist concluded that the distance from Alexandria to Syene is 1/50 of the circumference of the Earth. The distance between these cities is 5,000 Greek stadia (approximately 787.5 km), therefore 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 basis corresponds to the idea of Eratosthenes. At two points located several hundred kilometers from each other, the position of the Sun or certain stars in the sky is recorded and the difference between the results of the two measurements is calculated in degrees. 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 at the equator is 40,075.017 km, and at the equator – 40,007.86. Eratosthenes was only slightly mistaken.
The size of both the circumference and diameter of the Earth is increasing due to meteorite matter that constantly falls on the Earth, but this process is very slow.
Sources:
- How the Earth was measured in 2019
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.
Thus, the circumference ( C) can be calculated by multiplying the constant π per diameter ( D), or multiplying π by twice the radius, since the diameter is equal to two radii. Hence, circumference formula will look like this:
C = πD = 2πR
Where C- circumference, π - constant, D- circle diameter, R- radius of the circle.
Since a circle is the boundary of a circle, the circumference of a circle can also be called the length of a circle or the perimeter of a circle.
Circumference problems
Task 1. Find the circumference of a circle if its diameter is 5 cm.
Since the circumference is equal to π multiplied by the diameter, then the length of a circle with a diameter of 5 cm will be equal to:
C≈ 3.14 5 = 15.7 (cm)
Task 2. Find the length of a circle whose radius is 3.5 m.
First, find the diameter of the circle by multiplying the length of the radius by 2:
D= 3.5 2 = 7 (m)
Now let's find the circumference by multiplying π per diameter:
C≈ 3.14 7 = 21.98 (m)
Task 3. Find the radius of a circle whose length is 7.85 m.
To find the radius of a circle based on its length, you need to divide the circumference by 2 π
Area of a circle
The area of a circle is equal to the product of the number π per square radius. Formula for finding the area of a circle:
S = πr 2
Where S is the area of the circle, and r- radius of the circle.
Since the diameter of a circle is equal to twice the radius, the radius is equal to the diameter divided by 2:
Problems involving the area of a circle
Task 1. Find the area of a circle if its radius is 2 cm.
Since the area of a circle is π multiplied by the radius squared, then the area of a circle with a radius of 2 cm will be equal to:
S≈ 3.14 2 2 = 3.14 4 = 12.56 (cm 2)
Task 2. Find the area of a circle if its diameter is 7 cm.
First, find the radius of the circle by dividing its diameter by 2:
7:2=3.5(cm)
Now let's calculate the area of the circle using the formula:
S = πr 2 ≈ 3.14 3.5 2 = 3.14 12.25 = 38.465 (cm 2)
This problem can be solved in another way. Instead of finding the radius first, you can use the formula for finding the area of a circle using the diameter:
| S = π | D 2 | ≈ 3,14 | 7 2 | = 3,14 | 49 | = | 153,86 | = 38.465 (cm 2) |
| 4 | 4 | 4 | 4 |
Task 3. Find the radius of the circle if its area is 12.56 m2.
To find the radius of a circle from its area, you need to divide the area of the circle π , and then take the square root of the result:
r = √S : π
therefore the radius will be equal to:
r≈ √12.56: 3.14 = √4 = 2 (m)
Number π
The circumference of objects surrounding us can be measured using a measuring tape or rope (thread), the length of which can then be measured separately. But in some cases, measuring the circumference is difficult or practically impossible, for example, the inner circumference of a bottle or simply the circumference of a circle drawn on paper. In such cases, you can calculate the circumference of a circle if you know the length of its diameter or radius.
To understand how this can be done, let’s take several round objects whose circumference and diameter can be measured. Let's calculate the ratio of length to diameter, and as a result we get the following series of numbers:
From this we can conclude that the ratio of the length of a circle to its diameter is a constant value for each individual circle and for all circles as a whole. This relationship is denoted by the letter π .
Using this knowledge, you can use the radius or diameter of a circle to find its length. For example, to calculate the length of a circle with a radius of 3 cm, you need to multiply the radius by 2 (this is how we get the diameter), and multiply the resulting diameter by π . As a result, using the number π We learned that the length of a circle with a radius of 3 cm is 18.84 cm.
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 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 the knowledge gained 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 receive 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!