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. Diameter of this figure equal to length 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.
- Consists of points, for each of which the following equality holds: the sum of the squares of the distances to the other two is set value, which is always more than half the length of the segment between them.
Terminology
Not everyone at school had good teacher mathematics. 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 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
From the definitions it follows directly geometric formulas, which 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 in the following way: C = π*D.
- Radius equal to half 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 only have 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 required amount 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 It is not at all necessary to know each other's language. 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 the right 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.
Examples of 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 real math problems which are contained in the textbook. After all, the teacher gives points for them! So let's look at the problem increased complexity. 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 similar tasks. 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!
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 approximate value numbers π. In everyday life, to find the length of a circle, they use special device– 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.
Instructions
First you need the initial data for the task. The fact is that its condition cannot explicitly say what the radius is circle. Instead, the problem may give the length of the diameter circle. Diameter circle- a segment that connects two opposite points circle, passing through its center. Having analyzed the definitions circle, we can say that the length of the diameter is twice the length of the radius.
Now we can accept the radius circle equal to R. Then for the length circle you need to use the formula:
L = 2πR = πD, where L is the length circle, D - diameter circle, which is always 2 times the radius.
note
A circle can be inscribed in a polygon or described around it. Moreover, if the circle is inscribed, then at the points of contact with the sides of the polygon it will divide them in half. To find out the radius of the inscribed circle, you need to divide the area of the polygon by half its perimeter:
R = S/p.
If a circle is circumscribed around a triangle, then its radius is found using the following formula:
R = a*b*c/4S, where a, b, c are the sides given triangle, S is the area of the triangle around which the circle is circumscribed.
If you want to describe a circle around a quadrilateral, this can be done if two conditions are met:
The quadrilateral must be convex.
In total opposite angles quadrilaterals must be 180°
In addition to the traditional caliper, stencils can also be used to draw a circle. Modern stencils include circles of different diameters. These stencils can be purchased at any office supply store.
Sources:
- How to find the circumference of a circle?
A circle is a closed curved line, all points of which are on equal distance from one point. This point is the center of the circle, and the segment between the point on the curve and its center is called the radius of the circle.
Instructions
If a straight line is drawn through the center of a circle, then its segment between two points of intersection of this line with the circle is called the diameter of the given circle. Half the diameter, from the center to the point where the diameter intersects the circle is the radius
circles. If a circle is cut at an arbitrary point, straightened and measured, then the resulting value is the length of the given circle.
Draw several circles with different compass solutions. Visual comparison allows us to conclude that the larger diameter outlines larger circle, circumscribed with longer length. Therefore, between the diameter of a circle and its length there is a direct relationship proportional dependence.
By physical meaning the “circumference length” parameter corresponds to , bounded by a broken line. If you fit it into a circle regular n-gon with side b, then the perimeter of such a figure P equal to the product sides b by the number of sides n: P=b*n. Side b can be determined by the formula: b=2R*Sin (π/n), where R is the radius of the circle into which the n-gon is inscribed.
As the number of sides increases, the perimeter of the inscribed polygon will increasingly approach L. Р= b*n=2n*R*Sin (π/n)=n*D*Sin (π/n). The relationship between the circumference L and its diameter D is constant. The ratio L/D=n*Sin (π/n) as the number of sides of an inscribed polygon tends to infinity tends to the number π, constant value, called "pi" and expressed as an infinite decimal fraction. For calculations without application computer technology the value π=3.14 is accepted. The circumference of a circle and its diameter are related by the formula: L= πD. For a circle, divide its length by π=3.14.
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 the solution school tasks in geometry or mathematics, as well as for working calculations in construction or in the repair and decoration of premises, when accurate calculations using this formula are required.
Very often when deciding school assignments in physics, 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.
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Basic concepts and definitions
- Radius is the line connecting the center of the circle and its arbitrary point. It is designated 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 V 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. IN school math 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/(4P). 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 by the opposite side 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