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 we inscribe a regular n-gon with side b into a circle, then the perimeter of such a figure is 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, a 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.
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 fraction. 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 instrument 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
- This flat figure, which is a set of points equidistant from the center. They are all at the same distance and form a circle.
A 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 straight line connecting 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..
This is interesting : Number π. represents 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:
Let's look at an example of calculating the area of a circle using the radius. Let us be given a circle with radius R = 4 cm. Let us find the area of the figure.
The area of our circle will be 50.24 square meters. cm.
There is a formula area of a circle through diameter. It is also widely used to calculate the necessary parameters. These formulas can be used to find.
Let's consider an example of calculating the area of a circle through its diameter, knowing its radius. Let us be given a circle with radius R = 4 cm. First, let’s find the diameter, which, as we know, is twice the radius.
Now we use the data for an example of calculating the area of a circle using the above formula:
As you can see, the result is the same answer as in the first calculations.
Knowledge standard formulas calculating the area of a circle will help you easily determine in the future sector area and easily find missing values.
We already know that the formula for the area of a circle is calculated through the product constant valueπ per square of the radius of the circle. The radius can be expressed in terms of the circumference and substitute the expression in the formula for the area of a circle in terms of the circumference:
Now let’s substitute this equality into the formula for calculating the area of a circle and get a formula for finding the area of a circle using the circumference
Let's consider an example of calculating the area of a circle using the circumference. Let a circle with length l = 8 cm be given. Substitute the value into 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.
To do this, you only need 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’ll substitute everything into the basic formula for the area of a circle circumscribed around a square:
Many objects in the surrounding world have round shape. These are wheels, round window openings, pipes, various dishes and much more. You can calculate the length of a circle by knowing its diameter or radius.
There are several definitions of this geometric figure.
- This is a closed curve consisting of points that are located at the same distance from a given point.
- This is a curve consisting of points A and B, which are the ends of the segment, and all points from which A and B are visible at right angles. In this case, the segment AB is the diameter.
- For the same segment AB, this curve includes all points C such that the ratio AC/BC is constant and not equal to 1.
- This is a curve consisting of points for which the following is true: if you add the squares of the distances from one point to two given other points A and B, you get constant number, greater than 1/2 of the segment connecting A and B. This definition is derived from the Pythagorean theorem.
Note! There are other definitions. A circle is an area within a circle. The perimeter of a circle is its length. By different definitions the circle may or may not include the curve itself, which is its boundary.
Definition of a circle
Formulas
How to calculate the circumference of a circle using the radius? This is done using a simple formula:
where L is the desired value,
π is the number pi, approximately equal to 3.1413926.
Usually, to find the required value, it is enough to use π to the second digit, that is, 3.14, this will provide the required accuracy. On calculators, in particular engineering ones, there may be a button that automatically enters the value of the number π.
Designations
To find through the diameter there is the following formula:
If L is already known, the radius or diameter can be easily found out. To do this, L must be divided by 2π or π, respectively.
If a circle has already been given, you need to understand how to find the circumference from this data. The area of the circle is S = πR2. From here we find the radius: R = √(S/π). Then
L = 2πR = 2π√(S/π) = 2√(Sπ).
Calculating the area in terms of L is also easy: S = πR2 = π(L/(2π))2 = L2/(4π)
To summarize, we can say that there are three basic formulas:
- through the radius – L = 2πR;
- through diameter – L = πD;
- through the area of the circle – L = 2√(Sπ).
Pi
Without the number π it will not be possible to solve the problem under consideration. The number π was first found as the ratio of the circumference of a circle to its diameter. This was done by the ancient Babylonians, Egyptians and Indians. They found it quite accurately - their results differed from the currently known value of π by no more than 1%. The constant was approximated by such fractions as 25/8, 256/81, 339/108.
Further, the value of this constant was calculated not only from the point of view of geometry, but also from the point of view mathematical analysis through sums of series. The designation of this constant Greek letterπ was first used by William Jones in 1706 and became popular after the work of Euler.
It is now known that this constant is an infinite non-periodic decimal, it is irrational, that is, it cannot be represented as a ratio of two integers. Using supercomputer calculations, the 10-trillionth sign of the constant was discovered in 2011.
This is interesting! To remember the first few digits of the number π, various mnemonic rules. Some allow you to store in memory big number numbers, for example, one French poem will help you remember pi up to the 126th digit.
If you need the circumference, an online calculator will help you with this. There are many such calculators; you just need to enter the radius or diameter. Some of them have both of these options, others calculate the result only through R. Some calculators can calculate the desired value with different precision, you need to specify the number of decimal places. You can also calculate the area of a circle using online calculators.
Such calculators are easy to find with any search engine. There are also mobile applications, which will help solve the problem of how to find the circumference of a circle.
Useful video: circumference
Practical use
Solving such a problem is most often necessary for engineers and architects, but in everyday life knowledge necessary formulas may also come in handy. For example, you need to wrap a paper strip around a cake baked in a mold with a diameter of 20 cm. Then it will not be difficult to find the length of this strip:
L = πD = 3.14 * 20 = 62.8 cm.
Another example: you need to build a fence around a round pool at a certain distance. If the radius of the pool is 10 m, and the fence needs to be placed at a distance of 3 m, then R for the resulting circle will be 13 m. Then its length is:
L = 2πR = 2 * 3.14 * 13 = 81.68 m.
Useful video: circle - radius, diameter, circumference
Bottom line
The perimeter of a circle can be easily calculated by simple formulas, including diameter or radius. You can also find the desired quantity through the area of a circle. Online calculators or mobile applications in which you need to enter singular– diameter or radius.
Circle concept
Definition 1
Circle -- geometric figure, consisting of all points located at equal distances from a given point.
Definition 2
Within Definition 1, set point called the center of the circle.
Definition 3
The segment connecting the center of the circle with any of its points is called the radius of the circle $(r)$ (Fig. 1).
Figure 1. Circle with center at point $O$ and radius $r$
Equation of a circle
Let us derive the equation of a circle in Cartesian system coordinates $xOy$. Let the center of the circle $C$ have coordinates $(x_0,y_0)$, and the radius of the circle be equal to $r$. Let a point $M$ with coordinates $(x,y)$ -- arbitrary point this circle (Fig. 2).
Figure 2. Circle in Cartesian coordinate system
The distance from the center of the circle to the point $M$ is calculated as follows
But, since $M$ lies on the circle, then by definition 3, we get $CM=r$. Then we get the following
Equation (1) is the equation of a circle with center at point $(x_0,y_0)$ and radius $r$.
In particular, if the center of the circle coincides with the origin. That equation of a circle has the form
Circumference
Let us derive the formula for the circumference of a circle $C$ in terms of its radius. To do this, consider two circles with lengths $C$ and $C"$ and radii $R$ and $R"$. Let us inscribe in it regular $n-gons$ with perimeters $P$ and $P"$ and side lengths $a$ and $a"$, respectively. As we know, the side of an inscribed triangle is equal to
Then we get
Hence
Unlimitedly increasing the number of sides of regular polygons $n$ we get that
From here we get
We found that the ratio of the circumference of a circle to its diameter is a constant number for any circle. This constant is usually denoted by the number $\pi \approx 3.14$. Thus, we get
Formula (2) is the formula for calculating the circumference.
Area of a circle
Definition 4
Circle-- part of a plane bounded by a circle.
Let us derive a formula for calculating the area of a circle.
Consider the following situation. Let us be given a circle with radius $R$. Let's denote its area by $S$. A regular -gon with area $S_n$ is inscribed into it, into which, in turn, a circle with area $(S")_n$ is inscribed (Fig. 3).
Figure 3.
From the figure it is obvious that
We use the following well-known formula For regular polygon:
We will now increase the number of sides of a regular polygon without limit. Then, for $n\to \infty $, we get
According to the formula, the area of a regular polygon is equal to $S_n=\frac(1)(2)P_nr$, $P_n\to 2\pi R$, therefore
Formula (3) is the formula for calculating the area of a circle.
Example problem on the concept of a circle
Example 1
Find the equation of a circle with center at point $(1,\ 1)$. passing through the origin, find the length of the given circle and the area of the circle bounded by the given circle.
Solution.
Let's first find the equation of this circle. For this we will use formula (1). Since the center of the circle lies at the point $(1,\ 1)$, we get
\[((x-1))^2+((y-1))^2=r^2\]
Let's find the radius of the circle as the distance from the point $(1,\ 1)$ to the point $(0,0)$
We find that the equation of a circle has the form:
\[((x-1))^2+((y-1))^2=2\]
Let's find the circumference using formula (2). We get
Let's find the area using formula (3)
Answer:$((x-1))^2+((y-1))^2=2$, $C=2\sqrt(2)\pi $, $S=2\pi $