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圆的方程和面积公式都是什么?

时间:2019年08月06日

 
 
圆是一种几何图形,指的是平面中到一个定点距离为定值的所有点的集合。以下就是关于圆的方程和面积的公式和定义
 
圆的方程和面积公式
 
这个给定的点称为圆的圆心。作为定值的距离称为圆的半径。当一条线段绕着它的一个端点在平面内旋转一周时,它的另一个端点的轨迹就是一个圆。
 
圆的概念
 
1.到定点的距离等于定长的点的集合叫做圆。这个定点叫做圆的圆心,通常用字母“o”表示。
 
2.连接圆心和圆周上任意一点之间的连线叫做半径,通常用字母“r”表示。
 
3.通过圆心并且两个端点都在圆周上的线段叫做直径,通常用字母“d”表示。
 
4.连接圆上任意两点的线段叫做弦。在同圆或等圆中,最长的弦是直径。
 
5.圆上任意两点间的部分叫做圆弧,简称弧。大于半圆的弧称为优弧,优弧是用三个字母表示。小于半圆的弧称为劣弧,劣弧用两个字母表示。半圆既不是优弧,也不是劣弧。
 
圆的方程
 
1、圆的标准方程:在平面直角坐标系中,以点O(a,b)为圆心,以r为半径的圆的标准方程是(x-a)^2+(y-b)^2=r^2。
 
特别地,以原点为圆心,半径为r(r>0)的圆的标准方程为x^2+y^2=r^2。
 
2、圆的一般方程:方程x^2+y^2+Dx+Ey+F=0可变形为(x+D/2)^2+(y+E/2)^2=(D^2+E^2-4F)/4.故有:
 
(1)、当D^2+E^2-4F>0时,方程表示以(-D/2,-E/2)为圆心,以(√D^2+E^2-4F)/2为半径的圆;
 
(2)、当D^2+E^2-4F=0时,方程表示一个点(-D/2,-E/2);
 
(3)、当D^2+E^2-4F<0时,方程不表示任何图形。
 
3、圆的参数方程:以点O(a,b)为圆心,以r为半径的圆的参数方程是 x=a+r*cosθ,y=b+r*sinθ,(其中θ为参数)
 
圆的端点式:若已知两点A(a1,b1),B(a2,b2),则以线段AB为直径的圆的方程为 (x-a1)(x-a2)+(y-b1)(y-b2)=0
 
圆的离心率e=0,在圆上任意一点的半径都是r。
 
经过圆x^2+y^2=r^2上一点M(a0,b0)的切线方程为a0*x+b0*y=r^2
 
在圆(x^2+y^2=r^2)外一点M(a0,b0)引该圆的两条切线,且两切点为A,B,则A,B两点所在直线的方程也为a0*x+b0*y=r^2
 
面积公式
 
圆的面积:S=πr2=πd2/4
 
扇形弧长:L=圆心角(弧度制) * r = n°πr/180°(n为圆心角)
 
扇形面积:S=nπ r2/360=Lr/2(L为扇形的弧长)
 
圆的直径: d=2r
 
圆锥侧面积: S=πrl(l为母线长)
 
圆锥底面半径: r=n°/360°L(L为母线长)(r为底面半径)
 
 
A circle is a geometric figure, which refers to the set of all points in a plane whose distance from a fixed point is a fixed value. Here are the formulas and definitions of the equation and area of a circle.
 
Equations and Area Formulas of Circles
 
This given point is called the center of a circle. The distance as a fixed value is called the radius of the circle. When a line segment rotates around one of its endpoints in a plane, the trajectory of the other endpoint is a circle.
 
The concept of circle
 
1. The set of points whose distance to a fixed point equals a fixed length is called a circle. This fixed point is called the center of a circle and is usually represented by the letter "o".
 
2. The line connecting the center of the circle and any point on the circle is called radius, which is usually expressed by the letter "r".
 
3. Lines passing through the center of a circle and having both ends on the circumference are called diameters, which are usually represented by the letter "d".
 
4. A line segment connecting any two points on a circle is called a string. In the same circle or equal circle, the longest string is the diameter.
 
5. The part between any two points on a circle is called an arc, or arc for short. An arc larger than a semicircle is called a superior arc, which is expressed in three letters.  An arc smaller than a semicircle is called a bad arc, which is expressed in two letters.  A semicircle is neither a good arc nor a bad arc.
 
The equation of circle
 
1. The standard equation of a circle: In the plane rectangular coordinate system, the standard equation of a circle with point O (a, b) as its center and radius r as its radius is (x-a) ^ 2+ (y-b) ^ 2 = R ^ 2.
 
In particular, the standard equation of a circle with origin as its center and radius R (r > 0) is x ^ 2 + y ^ 2 = R ^ 2.
 
2. General equation of circle: equation x ^ 2 + y ^ 2 + Dx + Ey + F = 0 can be changed into (x + D / 2) ^ 2 + (y + E / 2) ^ 2 = (D ^ 2 + E ^ 2 - 4F) / 4.
 
(1) When D ^ 2 + E ^ 2 - 4F > 0, the equation represents a circle with (- D/2, - E/2) as its center and (D ^ 2 + E ^ 2 - 4F) / 2 as its radius.
 
(2) When D ^ 2 + E ^ 2 - 4F = 0, the equation represents a point (- D/2, - E/2);
 
(3) When D ^ 2 + E ^ 2-4F < 0, the equation does not represent any graphics.
 
3. Parametric equation of a circle: The parametric equation of a circle with point O (a, b) as its center and radius r as its radius is x=a+r*cos theta and y=b+r*sin theta (where theta is its parameter).
 
The Endpoint Formula of a Circle: If two points A (a1, b1), B (a2, b2) are known, then the equation of the circle with segment AB as its diameter is (x-a1) (x-a2) + (y-b1) (y-b2) = 0.
 
The eccentricity e of a circle is equal to 0, and the radius of any point on the circle is r.
 
The tangent equation of a point M (a0, b0) passing through a circle x ^ 2 + y ^ 2 = R ^ 2 is A0 * x + B0 * y = R ^ 2
 
At the point M (a0, b0) outside the circle (x ^ 2 + y ^ 2 = R ^ 2), two tangents of the circle are introduced. If the two tangents are A, B, then the equation of the straight line between A and B is also A0 * x + B0 * y = R ^ 2.
 
Area formula
 
Area of a circle: S = Pi r_ = Pi d_/4
 
Sector arc length: L = center angle (radian system)* r = n PI R / 180 (n is center angle)
 
Sector area: S = n PI r/360 = Lr/2 (L is the arc length of the sector)
 
Diameter of a circle: d = 2R
 
Conical side area: S = Pirl (l is bus length)
 
The bottom radius of cone: r = n / 360 L is bus length (r is bottom radius)
 




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