How to calculate the circumference of a circle knowing the diameter. How to find and what will be the circumference of a circle
Very often, when solving school assignments in or 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 understand what formulas, concepts and definitions are required for this.
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Basic concepts and definitions
- The radius is the line connecting the center of the circle and its arbitrary point. It is denoted by the Latin letter r.
- A chord is a line connecting two arbitrary points 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.
- - this is a line consisting of all points that are at an equal distance from one chosen point, called its center. Its length will be denoted by the Latin letter l.
The area of a circle is the entire area enclosed within a circle. It's measured in square units and is denoted by the Latin letter s.
Using our definitions, we conclude 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!
Circle diameter.
Finding the circumference of a circle and its area
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, is l \u003d 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 mathematics, it is considered to be a known tabular value equal to 3.14!
Now let's rewrite the previous formula to find the circumference of a circle in terms of its diameter, remembering what its difference is in relation to the radius. Get: l \u003d 2 * P * r \u003d 2 * r * P \u003d P * d.
From the course of mathematics it is known that the formula describing the area of a circle has the form: s \u003d P * r ^ 2.
Now let's rewrite the previous formula to find the area of a circle in terms of its diameter. We get
s = P*r^2 = P*d^2/4.
One of the most difficult tasks in this topic is determining the area of a circle in terms of the circumference and vice versa. We use the fact that s = P*r^2 and l = 2*P*r. From here we get r = l/(2*П). We substitute the resulting expression for the radius into the formula for the area, we get: s = l^2/(4P). The circumference of a circle is determined in exactly the same way in terms of the area of a circle.
Determining Radius Length and Diameter
Important! First of all, we will learn how to measure the diameter. It's very simple - we draw any radius, extend it in the opposite direction until it intersects with the arc. We measure the resulting distance with a compass and with the help of any metric tool we find out what we are looking for!
Let's 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 \u003d P * d. We get d = l/P.
We already know how to find its diameter from the circumference of a circle, and we will find the radius in the same way.
l \u003d 2 * P * r, hence r \u003d l / 2 * P. In general, to find out the radius, it must be expressed in terms of the diameter and vice versa.
Let now it is required to determine the diameter, knowing the area of the circle. We use the fact that s \u003d P * d ^ 2/4. We express from here d. It turns out d^2 = 4*s/P. To determine the diameter itself, you need to extract square root of the right side. It turns out d \u003d 2 * sqrt (s / P).
Solution of typical tasks
- Learn how to find the diameter given the circumference of a circle. Let it be equal to 778.72 kilometers. Need to find d. d \u003d 778.72 / 3.14 \u003d 248 kilometers. Let's remember what the diameter is and immediately determine the radius, for this we divide the value d defined above in half. It turns out r=248/2=124 kilometers.
- 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 translate 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 will be equal to l=2*3.14*87=546.36cm. Let's translate our obtained value into integers of metric values l \u003d 546.36 cm \u003d 5 m 4 dm 6 cm 3.6 mm.
- Suppose we need to determine the area of a given circle using the formula in terms of its known diameter. Let d = 815 meters. Recall the formula for finding the area of a circle. Substituting the given values here, we get s \u003d 3.14 * 815 ^ 2/4 \u003d 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 we know. Substitute here the value given to us by condition. You get the following: s \u003d 3.14 * 38 ^ 2 \u003d 4534.16 square meters. cm.
- The last task is to determine the area of the circle from the known circumference. Let l = 47 meters. s \u003d 47 ^ 2 / (4P) \u003d 2209 / 12.56 \u003d 175.87 sq. m.
Circumference
§ 117. Circumference and area of a circle.1. Circumference. A circle is a closed flat curved line, all points of which are at an equal distance from one point (O), called the center of the circle (Fig. 27).
The circle is drawn with a compass. To do this, the sharp leg of the compass is placed in the center, and the other (with a pencil) is rotated around the first until the end of the pencil draws a complete circle. The distance from the center to any point on the circle is called its radius. It follows from the definition that all radii of one circle are equal to each other.
A straight line segment (AB) connecting any two points of the circle and passing through its center is called diameter. All diameters of one circle are equal to each other; the diameter is equal to two radii.
How to find the circumference of a circle? In practice, in some cases, the circumference can be found by direct measurement. This can be done, for example, when measuring the circumference of relatively small objects (bucket, glass, etc.). To do this, you can use a tape measure, braid or cord.
In mathematics, the method of indirectly determining the circumference of a circle is used. It consists in the calculation according to the ready-made formula, which we will now derive.
If we take several large and small round objects (coin, glass, bucket, barrel, etc.) and measure the circumference and diameter of each of them, we will get two numbers for each object (one measuring the circumference, and the other is the length of the diameter). Naturally, for small objects, these numbers will be small, and for large objects, they will be large.
However, if in each of these cases we take the ratio of the two numbers obtained (circumference and diameter), then with careful measurement we will find almost the same number. Denote the circumference by the letter With, the length of the diameter by the letter D, then their relation will look like C:D. Actual measurements are always accompanied by inevitable inaccuracies. But, having performed the indicated experiment and having made the necessary calculations, we will obtain for the relation C:D approximately the following numbers: 3.13; 3.14; 3.15. These numbers differ very little from each other.
In mathematics, by theoretical considerations, it is established that the desired ratio C:D never changes and it is equal to an infinite non-periodic fraction, the approximate value of which, with an accuracy of ten thousandths, is equal to 3,1416 . This means that any circle is longer than its diameter by the same number of times. This number is usually denoted by the Greek letter π (pi). Then the ratio of the circumference to the diameter is written as: C:D = π . We will limit this number only to hundredths, i.e., take π = 3,14.
Let's write a formula for determining the circumference of a circle.
As C:D= π , then
C = πD
i.e. the circumference is equal to the product of the number π for diameter.
Task 1. Find the circumference ( With) of a round room if its diameter D= 5.5 m.
Taking into account the above, we must increase the diameter by 3.14 times to solve this problem:
5.5 3.14 = 17.27 (m).
Task 2. Find the radius of a wheel whose circumference is 125.6 cm.
This problem is the reverse of the previous one. Find the wheel diameter:
125.6: 3.14 = 40 (cm).
Now let's find the radius of the wheel:
40:2 = 20 (cm).
2. Area of a circle. To determine the area of a circle, one could draw a circle of a given radius on paper, cover it with transparent checkered paper, and then count the cells inside the circle (Fig. 28).
But this method is inconvenient for many reasons. First, near the contour of the circle, a number of incomplete cells are obtained, the size of which is difficult to judge. Secondly, you cannot cover a large object with a sheet of paper (a round flower bed, a pool, a fountain, etc.). Thirdly, having counted the cells, we still do not get any rule that allows us to solve another similar problem. Because of this, let's do it differently. Let's compare the circle with some figure familiar to us and do it as follows: cut out a circle from paper, cut it first in diameter in half, then cut each half in half again, each quarter in half again, etc., until we cut the circle, for example, into 32 parts having the shape of teeth (Fig. 29).
Then we fold them as shown in Figure 30, i.e., first we place 16 teeth in the form of a saw, and then we put 15 teeth into the holes formed, and finally, cut the last remaining tooth along the radius in half and attach one part to the left, the other - on right. Then you get a figure resembling a rectangle.
The length of this figure (the base) is approximately equal to the length of the semicircle, and the height is approximately equal to the radius. Then the area of such a figure can be found by multiplying the numbers expressing the length of the semicircle and the length of the radius. If we denote the area of a circle by the letter S, the circumference of the letter With, radius letter r, then we can write a formula for determining the area of a circle:
which reads like this: The area of a circle is equal to the length of the semicircle times the radius.
Task. Find the area of a circle whose radius is 4 cm. First find the circumference, then the length of the semicircle, and then multiply it by the radius.
1) Circumference With = π D= 3.14 8 = 25.12 (cm).
2) Half circle length C / 2 \u003d 25.12: 2 \u003d 12.56 (cm).
3) Circle area S = C / 2 r\u003d 12.56 4 \u003d 50.24 (sq. cm).
§ 118. Surface and volume of a cylinder.
Task 1. Find the total surface area of a cylinder with a base diameter of 20.6 cm and a height of 30.5 cm.
The shape of a cylinder (Fig. 31) is: a bucket, a glass (not faceted), a saucepan and many other items.
The full surface of a cylinder (like the full surface of a rectangular parallelepiped) consists of the side surface and the areas of the two bases (Fig. 32).
To visualize what we are talking about, you need to carefully make a model of a cylinder out of paper. If we subtract two bases from this model, that is, two circles, and cut the lateral surface lengthwise and unfold it, then it will be quite clear how the full surface of the cylinder should be calculated. The side surface will unfold into a rectangle, the base of which is equal to the circumference of the circle. Therefore, the solution to the problem will look like:
1) Circumference: 20.6 3.14 = 64.684 (cm).
2) Side surface area: 64.684 30.5= 1972.862(sq.cm).
3) The area of one base: 32.342 10.3 \u003d 333.1226 (sq. cm).
4) Full surface of the cylinder:
1972.862 + 333.1226 + 333.1226 = 2639.1072 (sq cm) ≈ 2639 (sq cm).
Task 2. Find the volume of an iron barrel shaped like a cylinder with dimensions: base diameter 60 cm and height 110 cm.
To calculate the volume of a cylinder, you need to remember how we calculated the volume of a rectangular parallelepiped (it is useful to read § 61).
The unit of measure for volume is the cubic centimeter. First you need to find out how many cubic centimeters can be placed on the base area, and then multiply the found number by the height.
To find out how many cubic centimeters can be placed on the base area, you need to calculate the base area of \u200b\u200bthe cylinder. Since the base is a circle, you need to find the area of the circle. Then, to determine the volume, multiply it by the height. The solution to the problem looks like:
1) Circumference: 60 3.14 = 188.4 (cm).
2) Area of a circle: 94.230 = 2826 (sq. cm).
3) Cylinder volume: 2826 110 \u003d 310 860 (cc).
Answer. The volume of the barrel is 310.86 cubic meters. dm.
If we denote the volume of a cylinder by the letter V, base area S, cylinder height H, then you can write a formula for determining the volume of a cylinder:
V = S H
which reads like this: The volume of a cylinder is equal to the area of the base times the height.
§ 119. Tables for calculating the circumference of a circle by diameter.
When solving various production problems, it is often necessary to calculate the circumference. Imagine a worker who manufactures round parts according to the diameters indicated to him. He must each time, knowing the diameter, calculate the circumference. To save time and insure himself against mistakes, he turns to ready-made tables that indicate the diameters and the corresponding circumferences.
Here is a small part of these tables and tell you how to use them.
Let it be known that the diameter of the circle is 5 m. We are looking for in the table in the vertical column under the letter D number 5. This is the length of the diameter. Next to this number (to the right, in the column called "Circumference") we will see the number 15.708 (m). In exactly the same way, we find that if D\u003d 10 cm, then the circumference is 31.416 cm.
The same tables can be used to perform reverse calculations. If the circumference is known, then you can find the corresponding diameter in the table. Let the circumference be approximately 34.56 cm. Let's find in the table the number closest to the given one. This will be 34.558 (0.002 difference). The diameter corresponding to such a circumference is approximately 11 cm.
The tables mentioned here are available in various reference books. In particular, they can be found in the book "Four-digit mathematical tables" by V. M. Bradis. and in the problem book on arithmetic by S. A. Ponomarev and N. I. Syrnev.
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So the circumference ( C) can be calculated by multiplying the constant π per diameter ( D), or by 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 is the 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.
Problems for the circumference
Task 1. Find the circumference of a circle if its diameter is 5 cm.
Since the circumference is π multiplied by the diameter, then the circumference of a circle with a diameter of 5 cm will be equal to:
C≈ 3.14 5 = 15.7 (cm)
Task 2. Find the circumference 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 find the circumference of the circle 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 given its length, divide the circumference by 2. π
Area of a circle
The area of a circle is equal to the product of the number π to the square of the radius. The formula for finding the area of a circle:
S = pr 2
where S is the area of the circle, and r is the radius of the circle.
Since the diameter of a circle is twice the radius, the radius is equal to the diameter divided by 2:
Problems for 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 \u003d 3.14 4 \u003d 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 we calculate the area of the circle using the formula:
S = pr 2 ≈ 3.14 3.5 2 \u003d 3.14 12.25 \u003d 38.465 (cm 2)
This problem can be solved in another way. Instead of first finding the radius, you can use the formula for finding the area of a circle in terms of the diameter:
| S = π | D 2 | ≈ 3,14 | 7 2 | = 3,14 | 49 | = | 153,86 | \u003d 38.465 (cm 2) |
| 4 | 4 | 4 | 4 |
Task 3. Find the radius of the circle if its area is 12.56 m 2.
To find the radius of a circle given its area, divide the area of the circle π , and then take the square root of the result:
r = √S : π
so the radius will be:
r≈ √12.56: 3.14 = √4 = 2 (m)
Number π
The circumference of objects surrounding us can be measured using a centimeter tape or a rope (thread), the length of which can then be measured separately. But in some cases it is difficult or almost impossible to measure the circumference, for example, the inner circumference of a bottle or just the circumference 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 a few round objects, from which you can measure both the circumference and the diameter. We calculate the ratio of length to diameter, as a result we get the following series of numbers:
From this we can conclude that the ratio of the circumference 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 circumference of a circle with a radius of 3 cm, you need to multiply the radius by 2 (so we get the diameter), and multiply the resulting diameter by π . Finally, with the number π we learned that the circumference of a circle with a radius of 3 cm is 18.84 cm.
Let's first understand the difference between a circle and a circle. To see this difference, it is enough to consider what both figures are. This is an infinite number of points in the plane, located at an equal distance from a single central point. But, if the circle also consists of internal space, then it does not belong to the circle. It turns out that a circle is both a circle that bounds it (o-circle (g)ness), and an uncountable number of points that are inside the circle.
For any point L lying on the circle, the equality OL=R applies. (The length of the segment OL is equal to the radius of the circle).
A line segment that connects two points on a circle is chord.
A chord passing directly through the center of a circle is diameter this circle (D) . The diameter can be calculated using the formula: D=2R
Circumference calculated by the formula: C=2\pi R
Area of a circle: S=\pi R^(2)
arc of a circle called that part of it, which is located between two of its points. These two points define two arcs of a circle. The chord CD subtends two arcs: CMD and CLD. The same chords subtend the same arcs.
Central corner is the angle between two radii.
arc length can be found using the formula:
- Using degrees: CD = \frac(\pi R \alpha ^(\circ))(180^(\circ))
- Using a radian measure: CD = \alpha R
The diameter that is perpendicular to the chord bisects the chord and the arcs it spans.
If the chords AB and CD of the circle intersect at the point N, then the products of the segments of the chords separated by the point N are equal to each other.
AN\cdot NB = CN \cdot ND
Tangent to circle
Tangent to a circle It is customary to call a straight line that has one common point with a circle.
If a line has two points in common, it is called secant.
If you draw a radius at the point of contact, it will be perpendicular to the tangent to the circle.
Let's draw two tangents from this point to our circle. It turns out that the segments of the tangents will be equal to one another, and the center of the circle will be located on the bisector of the angle with the vertex at this point.
AC=CB
Now we draw a tangent and a secant to the circle from our point. We get that the square of the length of the tangent segment will be equal to the product of the entire secant segment by its outer part.
AC^(2) = CD \cdot BC
We can conclude: the product of an integer segment of the first secant by its outer part is equal to the product of an integer segment of the second secant by its outer part.
AC \cdot BC = EC \cdot DC
Angles in a circle
The degree measures of the central angle and the arc on which it rests are equal.
\angle COD = \cup CD = \alpha ^(\circ)
Inscribed angle is an angle whose vertex is on a circle and whose sides contain chords.
You can calculate it by knowing the size of the arc, since it is equal to half of this arc.
\angle AOB = 2 \angle ADB
Based on diameter, inscribed angle, straight.
\angle CBD = \angle CED = \angle CAD = 90^ (\circ)
Inscribed angles that lean on the same arc are identical.
The inscribed angles based on the same chord are identical or their sum equals 180^ (\circ) .
\angle ADB + \angle AKB = 180^ (\circ)
\angle ADB = \angle AEB = \angle AFB
On the same circle are the vertices of triangles with identical angles and a given base.
An angle with a vertex inside the circle and located between two chords is identical to half the sum of the angular values of the arcs of the circle that are inside the given and vertical angles.
\angle DMC = \angle ADM + \angle DAM = \frac(1)(2) \left (\cup DmC + \cup AlB \right)
An angle with a vertex outside the circle and located between two secants is identical to half the difference in the angular magnitudes of the arcs of a circle that are inside the angle.
\angle M = \angle CBD - \angle ACB = \frac(1)(2) \left (\cup DmC - \cup AlB \right)
Inscribed circle
Inscribed circle is a circle tangent to the sides of the polygon.
At the point where the bisectors of the angles of the polygon intersect, its center is located.
A circle may not be inscribed in every polygon.
The area of a polygon with an inscribed circle is found by the formula:
S=pr,
p is the semiperimeter of the polygon,
r is the radius of the inscribed circle.
It follows that the radius of the inscribed circle is:
r = \frac(S)(p)
The sums of the lengths of opposite sides will be identical if the circle is inscribed in a convex quadrilateral. And vice versa: a circle is inscribed in a convex quadrilateral if the sums of the lengths of opposite sides in it are identical.
AB+DC=AD+BC
It is possible to inscribe a circle in any of the triangles. Only one single. At the point where the bisectors of the inner angles of the figure intersect, the center of this inscribed circle will lie.
The radius of the inscribed circle is calculated by the formula:
r = \frac(S)(p) ,
where p = \frac(a + b + c)(2)
Circumscribed circle
If a circle passes through every vertex of a polygon, then such a circle is called circumscribed about a polygon.
At the point of intersection of the perpendicular bisectors of the sides of this figure will be the center of the circumscribed circle.
The radius can be found by calculating it as the radius of a circle that is circumscribed about a triangle defined by any 3 vertices of the polygon.
There is the following condition: a circle can be circumscribed around a quadrilateral only if the sum of its opposite angles is equal to 180^( \circ) .
\angle A + \angle C = \angle B + \angle D = 180^ (\circ)
Near any triangle it is possible to describe a circle, and one and only one. The center of such a circle will be located at the point where the perpendicular bisectors of the sides of the triangle intersect.
The radius of the circumscribed circle can be calculated by the formulas:
R = \frac(a)(2 \sin A) = \frac(b)(2 \sin B) = \frac(c)(2 \sin C)
R = \frac(abc)(4S)
a, b, c are the lengths of the sides of the triangle,
S is the area of the triangle.
Ptolemy's theorem
Finally, consider Ptolemy's theorem.
Ptolemy's theorem states that the product of the diagonals is identical to the sum of the products of the opposite sides of an inscribed quadrilateral.
AC \cdot BD = AB \cdot CD + BC \cdot AD
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