Equation of a Circle Calculator – Find the Equation from Center and Radius

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How to Write the Equation of a Circle: Formula, Steps & Examples

Equation of a Circle Calculator – Free Online Tool with Formula, Steps & Examples

Equation of a Circle Calculator

Center (h, k):
Radius (r):

What is Equation of a Circle Calculator?

An Equation of a Circle Calculator is an interactive online tool that helps you instantly find the algebraic equation of a circle given its center point (h, k) and radius (r) in the coordinate plane. By simply entering the values, you get the equation in standard form, which is widely used in mathematics, physics, engineering, and exams. This calculator is ideal for students preparing for school, board exams, Olympiads, or anyone needing a quick, error-free way to express a circle mathematically.

Formula or Logic Behind Equation of a Circle Calculator

The core formula used is the standard equation of a circle in coordinate geometry. If a circle has its center at (h, k) and radius r, every point (x, y) on the circle satisfies:
(x − h)² + (y − k)² = r²
Here,

h = x-coordinate of the center
k = y-coordinate of the center
r = radius of the circle

Expanding or rearranging this form can also produce the general or expanded equations of a circle, but calculators for exams almost always use the standard center-radius form for clarity and simplicity.

Examples of the Equation of a Circle (Standard Form)

Center (h, k)	Radius (r)	Equation (Standard Form)
(0, 0)	7	(x)² + (y)² = 49
(3, −2)	5	(x−3)² + (y+2)² = 25
(–1, 4)	2	(x+1)² + (y−4)² = 4
(6, 6)	10	(x−6)² + (y−6)² = 100

Steps to Use the Equation of a Circle Calculator

Enter the x- and y-coordinates (h, k) of the circle's center.
Enter the circle's radius (r).
Click on the 'Calculate Equation' button.
Read your equation of the circle in standard form instantly below.

Why Use Vedantu’s Equation of a Circle Calculator?

Our calculator is easy to use, mobile-friendly, and trusted by lakhs of students across India for fast, accurate results. It minimizes calculation mistakes, saves exam time, and is ideal for CBSE, ICSE, IIT JEE, Olympiad, and school homework preparations. Vedantu’s platform ensures reliability and up-to-date methods for score improvement.

Real-life Applications of Equation of a Circle Calculator

The equation of a circle is used in a variety of fields. Some practical applications include:

Solving geometry problems in competitive exams and school tests (CBSE, ICSE, JEE)
Engineering and Design – modeling wheels, holes, gears, or parts using CAD software
Physics – representing wavefronts, sound propagation, or circular motions
Geography and GPS – mapping regions that are equidistant from a point (e.g., coverage zones)
Computer Graphics – rendering round objects, arcs, and collision detection

Whether in academics or practical design tasks, quick and correct circle equations are essential.

For more math tools and concepts, check out these helpful resources from Vedantu:
Area of a Circle, Prime Numbers, Conic Sections, HCF Calculator, Algebra Topics

FAQs on Equation of a Circle Calculator – Find the Equation from Center and Radius

1. What is the equation of a circle?

The equation of a circle represents all points equidistant from a central point. The standard form is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

2. How do you find the equation of a circle given its center and radius?

Substitute the coordinates of the center (h, k) and the radius (r) directly into the standard form equation: (x - h)² + (y - k)² = r². For example, a circle with center (2, 3) and radius 5 has the equation (x - 2)² + (y - 3)² = 25.

3. What is the general form of a circle equation?

The general form is x² + y² + Dx + Ey + F = 0, where D, E, and F are constants. This form is less intuitive but can be useful in certain situations. You can convert it to standard form to find the center and radius.

4. How do I convert the general form of a circle equation to standard form?

Complete the square for both the x and y terms. This involves manipulating the equation to resemble (x - h)² + (y - k)² = r². The process involves adding and subtracting specific constants to create perfect squares.

5. How can I find the center and radius of a circle from its equation?

If the equation is in standard form [(x - h)² + (y - k)² = r²], the center is (h, k) and the radius is r. If it's in general form, convert it to standard form first to identify the center and radius.

6. What is the equation of a circle with center (0, 0) and radius 4?

Substituting into the standard form, we get x² + y² = 16.

7. What is the radius of a circle with equation (x + 1)² + (y - 2)² = 9?

The radius is the square root of the constant term, which is 3. The center is (-1, 2).

8. How do you find the equation of a circle given three points on the circle?

This involves solving a system of three equations with three unknowns (h, k, and r). Substitute the coordinates of each point into the general form equation and then solve the resulting system to find the circle's equation.

9. What are some real-world applications of the circle equation?

Circle equations are used in many fields including engineering (designing circular components), GPS systems (defining areas), and physics (modeling circular motion). They are fundamental in geometry and coordinate geometry.

10. Can a circle have a negative radius?

No, a radius must always be a non-negative value (greater than or equal to zero). A negative radius is not physically or mathematically meaningful.

11. What happens if the radius of a circle is 0?

If the radius is 0, the circle becomes a single point, specifically the center point (h, k).

12. What's the difference between the standard and general forms of the circle equation?

The standard form, (x - h)² + (y - k)² = r², directly shows the center (h, k) and radius r. The general form, x² + y² + Dx + Ey + F = 0, is a less intuitive representation but is useful for solving certain problems. Conversion between the two forms is often necessary.