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 value was named from this letter in Ancient Greek 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, 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 at 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.
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 of 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 singular– diameter or radius.
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 solving school problems in geometry or mathematics, as well as in working calculations in construction or in the repair and decoration of premises, when accurate calculations using this formula are required.
First, let's understand the difference between a circle and a circle. To see this difference, it is enough to consider what both figures are. These are an infinite number of points on 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 limits it (circle(r)), and an innumerable 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 segment that connects two points on a circle is its 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 is called that part of it that is located between its two points. These two points define two arcs of a circle. The chord CD subtends two arcs: CMD and CLD. Identical chords subtend equal arcs.
Central angle An angle that lies between two radii is called.
Arc length can be found using the formula:
- Using degree measure: CD = \frac(\pi R \alpha ^(\circ))(180^(\circ))
- Using radian measure: CD = \alpha R
The diameter, which is perpendicular to the chord, divides the chord and the arcs contracted by it in half.
If chords AB and CD of a circle intersect at point N, then the products of segments of chords separated by point N are equal to each other.
AN\cdot NB = CN\cdot ND
Tangent to a 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 common points, it is called secant.
If you draw the radius to the tangent point, 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 tangent segments 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 let’s draw a tangent and a secant to the circle from our point. We obtain that the square of the length of the tangent segment will be equal to the product of the entire secant segment and its outer part.
AC^(2) = CD \cdot BC
We can conclude: the product of an entire segment of the first secant and its external part is equal to the product of an entire segment of the second secant and its external 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 a diameter, inscribed angle, right angle.
\angle CBD = \angle CED = \angle CAD = 90^ (\circ)
Inscribed angles that subtend the same arc are identical.
Inscribed angles resting on one chord are identical or their sum is equal to 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 contained within 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 values of the arcs of the circle that are contained 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 a polygon.
At the point where the bisectors of the corners of a 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 semi-perimeter of the polygon,
r is the radius of the inscribed circle.
It follows that the radius of the inscribed circle is equal to:
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 fits into a convex quadrilateral if the sums of the lengths of opposite sides are identical.
AB + DC = AD + BC
It is possible to inscribe a circle in any of the triangles. Only one single one. At the point where the bisectors intersect internal corners figure, 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)
Circumcircle
If a circle passes through each vertex of a polygon, then such a circle is usually called described 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 the circle that is circumscribed about the triangle defined by any 3 vertices of the polygon.
There is the following condition: a circle can be described 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)
Around any triangle you can describe a circle, 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 using the formulas:
R = \frac(a)(2 \sin A) = \frac(b)(2 \sin B) = \frac(c)(2 \sin C)
R = \frac(abc)(4 S)
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 diagonals is identical to the sum of the products of opposite sides of a cyclic quadrilateral.
AC \cdot BD = AB \cdot CD + BC \cdot AD
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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 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.
