1, circumference: C=2πr (r: radius)
2. The circumference of a semicircle: C=πr+2r
Second, the area of the circle.
1, area: S=πr?
2. semicircle area: S=πr? /2
Third, the arc length angle formula
1, arc length of sector: L= central angle (radian system) ×R= nπR/ 180(θ is central angle) (r is sector radius).
2. Sector area: S=nπ R? /360=LR/2(L is the arc length of the sector)
3. Radius of cone bottom surface: r=nR/360(r is the radius of bottom surface) (n is the central angle).
4. Sector area formula: S=nπr? /360=rl/2
R: radius, n: degree of central angle of arc, π: pi, l: arc length corresponding to sector.
You can also divide the area of the circle where the sector is located by 360 and multiply it by the angle n of the central angle of the sector.
Fourth, the equation of the circle:
The standard equation of 1. circle: In the plane rectangular coordinate system, the standard equation of a circle with a radius of r and a center of point O(a, b) is (x-a) 2+(y-b) 2 = r 2.
2. General equation of the circle: expand the standard equation of the circle, move the term and merge the similar terms, and the general equation of the circle can be obtained as X 2+Y 2+DX+EY+F = 0. Compared with the standard equation, in fact, D=-2a, E=-2b and f = a 2+b 2.
Five, the position relationship between the circle and the point:
Take point P and circle O as examples (if P is a point, PO is the distance from the point to the center of the circle), P is outside ⊙O, and PO > R；; P on ⊙O，po = r； P is within ⊙O, and po < r.
Six, there are three kinds of positional relationship between a straight line and a circle:
Do not separate public points;
There are two things in common;
Only one common point of a circle and a straight line is tangent. This straight line is called the tangent of the circle, and the only thing in common is called the tangent point. Take straight line AB and circle O as examples (let OP⊥AB be in P, then PO is the distance from AB to the center of the circle): AB is separated from O, and po > r；; AB is tangent to ⊙O, po = r；; AB and ⊙O intersect, po < r.
First of all, the nature of the circle
The (1) circle is an axisymmetric figure, and its symmetry axis is an arbitrary straight line passing through the center of the circle. A circle is also a central symmetric figure, and its symmetric center is the center of the circle.
Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord.
Inverse theorem of vertical diameter theorem: bisecting the diameter of a chord (not the diameter) is perpendicular to the chord and bisecting two arcs opposite to the chord.
(2) The properties and theorems of central angle and central angle.
(1) In the same circle or the same circle, if one of two central angles, two peripheral angles, two sets of arcs, two chords and the distance between two chords is equal, their corresponding other groups are equal respectively.
(2) In the same circle or equal circle, the circumferential angle of an equal arc is equal to half of the central angle it faces (the circumferential angle and the central angle are on the same side of the chord).
The circumferential angle of the diameter is a right angle. The chord subtended by a 90-degree circle angle is the diameter.
The formula for calculating the central angle is θ = (l/2π r) × 360 =180l/π r = l/r (radian).
That is, the degree of the central angle is equal to the degree of the arc it faces; The angle of a circle is equal to half the angle of the arc it faces.
(3) If the length of an arc is twice that of another arc, then the angle of circumference and center it subtends is also twice that of the other arc.
(3) The properties and theorems of circumscribed circle and inscribed circle.
① A triangle has a unique circumscribed circle and inscribed circle. The center of the circumscribed circle is the intersection of the perpendicular lines of each side of the triangle, and the distances to the three vertices of the triangle are equal;
(2) The center of the inscribed circle is the intersection of the bisectors of the inner angles of the triangle, and the distances to the three sides of the triangle are equal.
③R=2S△÷L(R: radius of inscribed circle, s: area of triangle, l: perimeter of triangle).
(4) The intersection of the connecting lines of two tangent circles. (line: a straight line with two centers connected)
⑤ The midpoint M of the chord PQ on the circle O, if the intersection point M is two chords AB and CD, and the chords AC and BD intersect PQ on X and Y respectively, then M is the midpoint of XY.
(4) If two circles intersect, the line segment (or straight line) connecting the centers of the two circles vertically bisects the common chord.
(5) The degree of the chord tangent angle is equal to half the degree of the arc it encloses.
(6) The degree of the angle inside a circle is equal to half of the sum of the degrees of the arcs subtended by the angle.
(7) The degree of the outer angle of a circle is equal to half of the difference between the degrees of two arcs cut by this angle.
(8) The perimeters are equal, and the area of a circle is larger than that of a square, rectangle or triangle.
Reference link: Circle _ Baidu Encyclopedia
Edited on 20 18-09-22.
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I feel that if you want to remember circles, you can remember plates first. After all, sectors make up a circle.
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All formulas for calculating circles
1065438+ It is the locus of the point where a circle with a fixed length and a half diameter 106 is equidistant from the two endpoints of a known line segment, the locus of the point where the middle perpendicular of the line segment 107 is equidistant from the two sides of a known angle, and the locus of the point where the bisector of this angle 108 is equidistant from two parallel lines. The vertical diameter theorem of straight line 65 1 10 bisects the chord perpendicular to the chord diameter and bisects the two arcs opposite to the chord.11inference 1 ① bisects the diameter (not the diameter) of the chord perpendicular to the chord, and the middle vertical lines of the two arcs opposite to the chord pass through the center of the circle. The perpendicular bisecting chord and bisecting another arc 1 12 Inference 2 The arcs sandwiched by two parallel chords of a circle are equal. 1 13 circle is a centrosymmetric figure with the center of the circle as the symmetry center. 1 14 Theorem In the same circle or an equal circle, equal central angles have equal arcs and equal chords. The distance between chords of a pair of chords is equal. 1 15 It is inferred that in the same circle or the same circle, if the distances between two central angles, two arcs, two chords or two chords are equal, the corresponding other components are equal. 1 16 Theorem: The circumferential angle of an arc is equal to half of its central angle. In the same circle or equal circle, the arc opposite to the equal circle angle is also equal. 1 18 infers that 2 semicircles (or diameters) are right angles; The chord subtended by the circumferential angle of 90 is 1 19 Inference 3 If the median line of one side of a triangle is equal to half of this side, then this triangle is the diagonal complement of the inscribed quadrilateral of the right triangle 120 theorem circle. And any outer angle is equal to the intersection point of the inner diagonal line 12 1① and ⊙O D < R2, and the tangent judgment theorem of ⊙O D = R3 and ⊙O D > R 122 passes through the outer end of the radius, and the straight line perpendicular to this radius is the tangent of the circle. 438+024 Inference 1 A straight line passing through the center and perpendicular to the tangent must pass through the tangent point 125 Inference 2 A straight line passing through the tangent point and perpendicular to the tangent must pass through the center 126 The tangent length theorem leads to two tangents from a point outside the circle, and their tangent lengths are equal. The line between the center of the circle and this point bisects the included angle of the two tangents. The sum of two opposite sides of the circumscribed quadrangle of a circle is equal. The tangent angle theorem is equal to the circumferential angle of the arc pair it clamps. It is deduced that if the arcs sandwiched by two chord tangent angles are equal, then the two chord tangent angles are equal to the two intersecting chords in the chord theorem circle. The product of the length of two lines divided by the intersection is equal to 13 1. It is deduced that if the chord intersects the diameter vertically, then half of the chord is the tangent and secant of the circle, which is drawn by the middle term 132 according to the ratio of two line segments formed by a point outside the circle. The tangent length is the ratio of the lengths of two lines from this point to the intersection of the secant and the circle. 133 This item infers that two secant lines are drawn from a point outside the circle, and the product of the lengths of the two lines from this point to the intersection of each secant line and the circle is equal to 134. If two circles are tangent, then the tangent point must be on the line 135① two circles are tangent to D > R+R ② two circles are tangent to d=R+r ③ two circles intersect R-R < D+R (R > R) ④ two circles are inscribed with D = R-R (R > R) ⑤ two circles contain D < R. Chord theorem 137 divides a circle into n (n ≥ 3): (1) The polygon obtained by connecting all points in turn is the inscribed regular n polygon of this circle; (2) A polygon whose vertex is the intersection of adjacent tangents is the circumscribed regular N polygon of the circle; (2) Theorem 138: Any regular polygon has a circumscribed circle and an inscribed circle, which are concentric circles.
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All formulas of a circle
1. perimeter formula 1. Circumference of a circle: C=2πr (r: radius) 2. The circumference of a semicircle: C=πr+2r 2. Area of circle: 1. Area: S=πr? 2. semicircle area: S=πr? /2 3. Arc length angle formula 1. Sector arc length: L= central angle (radian system) ×R= nπR/ 180(θ is central angle) (r is sector radius) 2. Sector area: S=nπ R? /360=LR/2(L is the arc length of the sector) 3. Radius of cone base: r=nR/360(r is the base radius) (n is the central angle) 4. Sector area formula: r: radius, n: degree of central angle of arc, π: π, L: arc length corresponding to sector. You can also divide the area of the circle where the sector is located by 360 and multiply it by the angle n of the central angle of the sector. Extended data:
There are three positional relationships between a straight line and a circle: there is no separated common point; There are two things in common; A circle and a straight line have a unique common tangent point. This straight line is called the tangent of the circle, and this unique common point is called the tangent point. There are five positional relationships between two circles: if there is nothing in common, one circle is called external separation and internal inclusion outside the other; If there is only one common point, a circle is called circumscribed by another circle and inscribed by another circle; There are two things in common called intersection. The distance between the centers of two circles is called the center distance. The radii of the two circles are R and R respectively, and R≥r, and the center distance is P: outward separation P > R+R; Circumscribed p = r+r; Intersection r-r < p < r+r; Inner cut p = r-r; Include P R；; P on ⊙O，PO = r； P is in the range of ≧O, and po < r. Extended data:
The ratio of the circumference of a circle to the length of its diameter is called pi. It is an infinite acyclic decimal, usually represented by the letter π, ≈3. 14 15926535 ... The approximate value is usually 3. 14. We can say that the circumference of a circle is π times the diameter, or about 3. 14 times the diameter, but we can't directly say that the circumference of a circle is 3. 14 times the diameter. Shape: 1. A figure surrounded by a chord and the arc it faces is called an arch. 2. The figure enclosed by two radii of the central angle and an arc corresponding to the central angle is called a sector. The POsitional relationship between po > and the circle ①P is outside the circle o, then po >; R .②P is on the circle o, then po = R. ③P is in the circle o, then po.
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All calculation formulas of circle (letter formula)
Circumference of a circle: c=2πr=πd Perimeter of a semi-circle: c=πr+2r Area of a circle: S=πr? Area of semicircle: S=(πr? ) ÷2 circular area: s great circle -s small circle =π(R? -r？ ) (r Great Circle Radius) Semicircle Perimeter =π×r+d Note: Radius of the circle: R Diameter: D pi (3. 14 15926 ...) Extended data circle property: 1, the circle is an axisymmetric figure, and its symmetry axis is an arbitrary straight line passing through the center. A circle is also a central symmetric figure, and its symmetric center is the center of the circle. 2. If one of two central angles, two peripheral angles, two sets of arcs, two chords and the distance between the two chords are equal in the same circle or equal circle, their corresponding other groups are equal respectively. 3. If the length of an arc is twice that of another arc, then the angle of circumference and center it subtends is also twice that of the other arc. A triangle has a unique circumscribed circle and inscribed circle. The center of the circumscribed circle is the intersection of the perpendicular lines of each side of the triangle, and the distances to the three vertices of the triangle are equal. 5. The center of the inscribed circle is the intersection point of the bisector of the inner angle of the triangle, and the distances to the three sides of the triangle are equal.
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All formulas about circles
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