How to find out the circumference of a circle knowing the diameter formula. How to calculate the circumference of a circle if the diameter and radius of the circle are not specified
A ruler alone is not enough; you need to know special formulas. The only thing we need to do is determine the diameter or radius of the circle. In some problems these quantities are indicated. But what if we have nothing but a drawing? No problem. The diameter and radius can be calculated using a regular ruler. Now let's get down to the basics.
Formulas everyone should know
Almost 4,000 years ago, scientists discovered an amazing relationship: if the circumference of a circle is divided by its diameter, the result is the same number, which is approximately 3.14. This meaning was named with this letter in the ancient Greek language, the words “perimeter” and “circumference” began. Based on the discovery made by ancient scientists, you can calculate the length of any circle:
Where P means the length (perimeter) of the circle,
D - diameter, P - number "Pi".
The circumference of a circle can also be calculated through its radius (r), which is equal to half the length of the diameter. Here is the second formula you need to remember:
How to find out the diameter of a circle?
It is a chord that passes through the center of the figure. At the same time, it connects the two most distant points in the circle. Based on this, you can independently draw the diameter (radius) and measure its length using a ruler.
Method 1: enter right triangle in a circle
Calculating the circumference of a circle will be easy if we find its diameter. It is necessary to draw in a circle where the hypotenuse will be equal to the diameter of the circle. To do this, you need to have a ruler and a square on hand, otherwise nothing will work.
Method 2: fit any triangle
On the side of the circle we mark any three points, connect them - we get a triangle. It is important that the center of the circle lies in the area of the triangle; this can be done by eye. We draw medians to each side of the triangle, the point of their intersection coincides with the center of the circle. And when we know the center, we can easily draw the diameter using a ruler.
This method is very similar to the first, but can be used in the absence of a square or in cases where it is not possible to draw on a figure, for example on a plate. You need to take a sheet of paper with right angles. We apply the sheet to the circle so that one vertex of its corner touches the edge of the circle. Next, we mark with dots the places where the sides of the paper intersect with the circle line. Connect these points using a pencil and ruler. If you don't have anything on hand, just fold the paper. This line will be equal to the length of the diameter.
Sample task
- We look for the diameter using a square, ruler and pencil according to method No. 1. Let's assume it turns out to be 5 cm.
- Knowing the diameter, we can easily insert it into our formula: P = d P = 5 * 3.14 = 15.7 In our case, it turned out to be about 15.7. Now you can easily explain how to calculate the circumference of a circle.
A circle consists of many points that are at equal distances from the center. It's 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 have round 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. Interior 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 also 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. According to these parameters there is required size circle: 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:
- At small sizes of a round object, 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 Romanesque architectural style, portholes in ships. Architects, engineers, scientists, mechanics and designers every day in their field professional activity are faced with the need to calculate the size of a circle.
Many objects in the world around us are round in shape. These are wheels, round window openings, pipes, various dishes and much more. Calculate what it is equal to circumference, you can, 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 a 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. According to different definitions, a 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 by the Greek letter π was first used by William Jones in 1706, and it 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! Various mnemonic rules have been invented to remember the first few digits of the number π. 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 of the necessary formulas can also be useful. 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 a single number - diameter or radius, will help you solve this problem.
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 image 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 is used in in this case, looks like that:
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 of 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.
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, then the radius equal to 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 task 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 by its area, you need to divide the area of the circle π , and then extract from the obtained result Square root:
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.