Name: circle equations in standard form
Uploaded: 2025-11-26T05:44:19.000Z
Duration: 1 min 18 s
Channel: JoVE

Um den Radius des größten Wertungsrings einer Dartscheibe zu bestimmen, kann er als Kreis modelliert werden.Ein Kreis besteht aus allen Punkten, die den gleichen festen Abstand haben, den sogenannten Radius, von einem zentralen Punkt, der als Mittelpunkt bezeichnet wird.Die Gleichung eines Kreises stammt aus der Entfernungsformel, die auf dem Satz des Pythagoras basiert. Er misst den Abstand zwischen dem Mittelpunkt und einem beliebigen Punkt auf dem Kreis. Wenn sich Punkt P entlang der Kante bewegt, bleibt sein Abstand vom Mittelpunkt konstant. Wenn Sie beide Seiten quadrieren, wird die Quadratwurzel entfernt und die Standardform der Gleichung erzeugt.Mit dem Mittelpunkt am Ursprung reduziert sich die Gleichung auf die Summe der Quadrate der x- und y-Koordinaten.Um einen Kreis aus seiner Gleichung grafisch darzustellen, werden die Koordinaten des Mittelpunkts aus der Standardform entnommen und dann der Radius identifiziert.Umgekehrt ermöglicht das Lokalisieren des Mittelpunkts und das Messen des Radius bei einem Graphen eines Kreises eine direkte Substitution in die Standardform, um die Gleichung zu erhalten.Wenn die Gleichung nicht in der Standardform vorliegt, werden durch das Gruppieren von Termen und das Vervollständigen des Quadrats der Mittelpunkt und der Radius des Kreises angezeigt.

kreise

Ein Kreis in der Koordinatenebene ist die Menge aller Punkte, die von einem festen Punkt, dem Mittelpunkt, in konstantem Abstand (dem Radius) liegen. Diese Beziehung wird mithilfe der Abstandsformel beschrieben. F&#252;r einen Punkt (x, y) auf dem Kreis und den Mittelpunkt (h, k) ist der Abstand gleich dem Radius r. Durch Quadrieren beider Seiten der Abstandsformel erh&#228;lt man die Standardform der Kreisgleichung:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40041/40041_Equation_1_1__editor_40041.svg' alt='40041_Equation_1_1_.svg' /></figure>Aufstellen der Gleichung aus geometrischen AngabenSind Mittelpunkt und Radius eines Kreises bekannt, l&#228;sst sich die Gleichung durch Einsetzen unmittelbar bestimmen. Betrachten Sie einen Kreis mit dem Mittelpunkt (1, 2) und dem Radius 4. Einsetzen in die Standardform ergibt:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40041/40041_Equation_2_1__editor_40041.svg' alt='40041_Equation_2_1_.svg' /></figure>Diese Gleichung beschreibt alle Punkte (x, y), die 4 Einheiten vom Punkt (1, 2) entfernt liegen. Dieser einfache Ansatz gen&#252;gt h&#228;ufig, wenn die geometrischen Eigenschaften des Kreises explizit vorgegeben sind.Identifizierung des Kreises aus einer allgemeinen GleichungLiegt die Kreisgleichung nicht in Standardform vor, hilft die Quadratvervollst&#228;ndigung, Mittelpunkt und Radius zu bestimmen. Betrachten Sie die Gleichung<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40041/40041_Equation_3_1__editor_40041.svg' alt='40041_Equation_3_1_.svg' /></figure>Fassen Sie geeignete Terme zusammen und f&#252;hren Sie eine Quadratvervollst&#228;ndigung durch:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40041/40041_Equation_4_1__editor_40041.svg' alt='40041_Equation_4_1_.svg' /></figure><figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40041/40041_Equation_5_1__editor_40041.svg' alt='40041_Equation_5_1_.svg' /></figure>Daraus ergeben sich der Mittelpunkt (3, 2) und der Radius 4. Das Umschreiben in die Standardform ist f&#252;r eine effektive grafische Darstellung und Analyse von Kreisen unerl&#228;sslich.

Kreise

Ein Kreis in der Koordinatenebene ist die Menge aller Punkte, die von einem festen Punkt, dem Mittelpunkt, in konstantem Abstand (dem Radius) liegen. ...

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JoVE Core

Math Fundamentals

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Coordinates and Graphs

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KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

Circles

circle equations in standard form

Circle equations in the coordinate plane use a center and a radius to describe every point on the curve. A circle is the set of all points that stay the same distance from a fixed point called the center. That constant distance is the radius.

The distance formula connects a point x, y on the circle to the center h, k. The distance between those two points equals the radius r. When both sides of the distance formula are squared, the equation of the circle can be written in standard form.

If the center and radius are known, the equation can be built directly. For a circle with center 1, 2 and radius 4, substitute those values into the standard form. The result describes every point that is 4 units from 1, 2.

Sometimes a circle is given in a more general equation instead of standard form. In that case, completing the square helps reveal the center and radius. For example, grouping the terms and completing the square can show a center at 3, 2 and a radius of 4. Rewriting the equation in standard form makes circles easier to graph and analyze.

Circle Equations in Standard Form

coordinate plane

Finding Points on the Coordinate Plane

The Cartesian coordinate plane is used to find and plot points in two dimensions. It is made of two number lines that cross at right angles. The horizontal line is the x-axis, and the vertical line is the y-axis. Their meeting point is the origin, where both values are zero. The axes divide the plane into four quadrants. These quadrants are labeled in a counterclockwise direction, starting from the upper right. Every point on the plane is written as an ordered pair. The first number is the...

the distance formula

Finding Distance on a Coordinate Plane

Finding distance on a coordinate plane uses the distance formula and the Pythagorean Theorem. In geometry, this method helps measure the direct distance between two points on a flat plane. It is useful in many practical and theoretical settings. The formula comes from a right triangle. On a coordinate plane, the horizontal and vertical differences between two points become the legs of that triangle. The line connecting the points is the hypotenuse, and that is the shortest path between them. <br...

the midpoint formula

Finding the Center of a Line Segment

Finding the center of a line segment is a key idea in coordinate geometry. The midpoint is the point that lies exactly halfway between two points on a coordinate plane. It is used in mathematics, physics, engineering, and planning work where a central location matters. To work with a midpoint, start with two points, A(x1, y1) and B(x2, y2). A straight line segment can be drawn between them on the coordinate plane. The midpoint, labeled M, divides that segment into two equal parts. The...

graphs of equations in two variables

Plotting Two-Variable Equations with Tables

Plotting two-variable equations with tables helps show how x and y are related. An equation written as y = f(x) or Ax + By = C gives ordered pairs that satisfy the rule. Each pair can be checked by substituting the x and y values into the equation. A value table is a common way to build the graph. Start by choosing several values for the independent variable x. Then substitute each x value into the equation to find the matching y value. The results become ordered pairs on the Cartesian...

circles

Circle equations in the coordinate plane use a center and a radius to describe every point on the curve. A circle is the set of all points that stay the same distance from a fixed point called the center. That constant distance is the radius. The distance formula connects a point x, y on the circle to the center h, k. The distance between those two points equals the radius r. When both sides of the distance formula are squared, the equation of the circle can be written in standard form. <br...

symmetry

Ellipse Symmetry at the Origin

An ellipse centered at the origin shows clear symmetry in its graph. Its equation describes a smooth, closed curve whose distance from the center keeps a constant ratio between the horizontal and vertical axes. Because the ellipse stretches farther along the x-axis than the y-axis, it has a horizontal shape. This ellipse has symmetry across the x-axis. That means the top half and bottom half are mirror images across the horizontal axis. Algebraically, the equation stays the same when y is...

solving equations graphically

Graphing Equations to Find Intercepts

Graphing equations helps students find intercepts and solutions on the coordinate plane. It gives a visual way to solve equations and understand how a function behaves. To use this method, write the equation in the form y = f(x). The solution to the original equation is the x-value where the graph crosses the x-axis. At that point, f(x) = 0. For example, the linear equation 2x - 4 = 0 can be written as y = 2x - 4. When this function is graphed, it has one x-intercept at x = 2, and that is the...

solving inequalities graphically

Graphing Inequality Solution Ranges

Graphing inequality solution ranges gives a visual way to find the x-values that make an inequality true. Students look at where a graph sits compared with the x-axis or with another graph. This shows the interval where the condition is satisfied. For a quadratic inequality, the first step is to plot the related function. For example, with x² − 4x + 2 &lt; 0, graph y = x² − 4x + 2. The solution is the set of x-values where the curve lies below the x-axis. Inequalities can also compare two...

key terms and concepts

Learn how to write a circle equation from a center and radius, and how completing the square reveals the center and radius from a general equation.

Circle Equations in Standard Form | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Circle Equations in Standard Form

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