A circle is a curved line that encloses a circle. In geometry, figures are flat, so the definition refers to a two-dimensional image. It is assumed that all points of this curve are at an equal distance from the center of the circle.
The circle has several characteristics, on the basis of which the calculations associated with this geometric figure are made. These include: diameter, radius, area and circumference. These characteristics are interrelated, that is, information about at least one of the components is sufficient to calculate them. For example, knowing only the radius geometric figure using the formula you can find the circumference, diameter, and its area.
- The radius of a circle is a segment inside the circle connected to its center.
- Diameter is a line segment inside a circle that connects its points and passes through the center. In fact, the diameter is two radii. This is exactly what the formula for calculating it looks like: D=2r.
- There is another component of the circle - the 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 the circumference of a circle? Now let's find out.
Circumference: formula
The Latin letter p has been chosen to designate this characteristic. Archimedes also proved that the ratio of the circumference of a circle to its diameter is the same number for all circles: it is the number π, which is approximately equal to 3.14159. The formula for calculating π looks like this: 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 circumference formula can be written as p=2πr. Consider the application of the formula using simple problems as an example:
Task 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
- We substitute the existing value in the formula: p \u003d 3.14 * 6.6 \u003d 20.724
Answer: The circumference of the base of the bell is 20.7 meters.
Task 2
An 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 circular orbit of the satellite using the formula: P=2πr
- 3.P=2*3.14*6690=42013.2
Answer: the length of the circular orbit of the Earth's satellite is 42013.2 km.
Methods for measuring the circumference
The calculation of the circumference of a circle is not often used in practice. The reason for this is the approximate value of the number π. In everyday life, a special device is used to find the length of a circle - a curvimeter. An arbitrary reference point is marked on the circle and the device is guided from it strictly along the line until they again reach this point.
How to find the circumference of a circle? You just need to keep in mind simple formulas for calculations.
Instruction
At first it is necessary the initial data to the task. The fact is that its condition cannot be explicitly said what is the radius circles. Instead, the problem can be given the length of the diameter circles. Diameter circles a line segment that connects two opposite points circles passing through its center. Having analyzed the definitions circles, we can say that the length of the diameter is twice the length of the radius.
Now we can accept the radius circles equal to R. Then for the length circles you need to use the formula:
L = 2πR = πD, where L is the length circles, D - diameter circles, 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 it will divide them in half at the points of contact with the sides of the polygon. To find the radius of an 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 by the following formula:
R \u003d a * b * c / 4S, where a, b, c are the sides of the given triangle, S is the area of \u200b\u200bthe triangle around which the circle is described.
If it is required to describe a circle around a quadrilateral, then this can be done subject to two conditions:
The quadrilateral must be convex.
The sum of the opposite angles of the quadrilateral should be 180°
In addition to the traditional caliper, stencils can also be used to draw a circle. Modern stencils include a circle different diameters. These stencils can be purchased at any stationery store.
Sources:
- How to find the circumference of a circle?
Circle - 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.
Instruction
If a straight line is drawn through the center of a circle, then its segment between the two points of intersection of this line with the circle is called the diameter of this circle. Half the diameter, from the center to the point where the diameter intersects with the circle, is the radius
circles. If the 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 larger diameter draws a larger circle bounded by a larger circle. Therefore, between the diameter of a circle and its length, there is a direct proportional dependence.
According to the physical meaning, the parameter "circumference" corresponds to, limited by a broken line. If a regular n-gon with side b is inscribed in a circle, then the perimeter of such a figure P is equal to the product of side b by the number of sides n: P \u003d 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 \u003d n * Sin (π / n) as the number of sides of the inscribed polygon tends to infinity tends to the number π, a constant value called “pi number” and expressed as infinite decimal. For calculations without using computer science the value π=3.14 is taken. 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.
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 consists of inner 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 magnitudes 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)
Length sums 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 intersect internal corners figure, will lie the center of this inscribed circle.
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.
The center of the circumscribed circle will be at the point of intersection of the perpendicular bisectors of the sides of this figure.
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 next condition: a circle can only be circumscribed around a quadrilateral if the sum of its opposite angles is 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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