Dimension of the Circle in Standard Form

The circle is one of the main shapes in geometry. We need to learn about the circle as we encounter the circle in our daily life many times. The clock we use is a circle, plates are circular, and many other things.

Circle, just like any other mathematical term, properties, and equations. There are many ways in which the equation of a circle is represented. There is the standard form, polar form, general form, and graphical form. Let us learn about the standard form of the equation of the circle in detail and also take up a brief look at other forms.

Circle

The circle is a shape in geometry. A circle is an ellipse in which the eccentricity of the lips is zero, and focus 1 and focus 2 of the ellipse are also coincident. The circle is a locus of points, where all points are at an equal distance from the center. 

The distance between the center of the circle to its outer line is recorded as its radius. The diameter of a circle is twice the radius of the circle. The circle is a two-dimensional shape. And the measuring parameter of the circle is its radius.

The Equation of the Circle

All geometrical shapes have equations that define them. The equation of the circle is the algebraic representation of the circle with the help of the center and the length of the circle.

However, the equation of a circle is not the same as the formulas used to calculate the circumference and area of a circle.

The equation of a circle is used in coordinate geometry. 

Significance of the Equation of the Circle

 A circle is represented on the Cartesian plane with the help of its equation. Once we know the equation of the circle, we can construct the circle in the cartesian plane. The graph can also be used to represent the equation of the circle.

The equation of the circle is a representation of each and every point which lies on the circumference of that circle. All these points are at equal distances from the center of the circle.

The equation of a circle in a standard form whose center lies at (x1, y1) and the radius r of the circle is given below:

(x – x1)² + (y – y1)² = r2

Various Representations of the Equation of the Circle

As we discussed, the equation of the circle in standard form is (x – x1)² + (y – y1)² = r2.

However, there are many other forms in which the equation of a circle can be represented.

Those forms are given below:

• The standard form

• The parametric form

• The general form

• The polar form

Let’s take a look at all these forms of the equation of circle one by one.

• The General Form:

The equation of the circle in its  general form is:

x2 + y2 + 2gx + 2fy + c = 0

The above mission is used to find out the coordinates of the circle r and the length of its radius.

The g, f, and c here are constants.

• The Parametric Form:

As we know, the general form of the equation of the circle is:

x2 + y2 + 2gx + 2fy + c = 0

Suppose we consider a point on the boundary of a circle with (x, y) as its coordinates. The center of the circle is (-h, -k), and θ is the angle made with the center of the circle.

The equation of circle becomes

x2 + y2 + 2hx + 2ky + C = 0

where x = -h + rcosθ and y = -k + rsinθ.

• The Standard Form:

The standard form of the equation of the circle is

(x – x1)2 + (y – y1)2 = r2

Here the x1 and y1 are the coordinates of the center of the circle. The r is the radius of the circle, while (x,y ) represents any point at the circumference of the circle.

The distance formula is used to form the standard form of the equation of the circle.

The standard form is preferred over other forms.

Conclusion

The geometric shape or the representation of the shapes on a cartesian plane. With the help of these equations, we can find out about the radius and the center of the circle. The equation of circle helps construct the circle in a cartesian plane. 

The equation of a circle represents any point that lies on the circumference of the circle.

The standard form is the most preferred form for representing the equation of a circle. It is easy to read. Other forms to represent the equation of the circle are 

polar form, general form, and parametric form.