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Circle Equation

A circle with center (h,k) and radius r has equation (x−h)²+(y−k)²=r². Area A=πr², circumference C=2πr. General form: x²+y²+Dx+Ey+F=0.

Concept Fundamentals
(x−h)²+(y−k)²=r²
Standard
A = πr²
Area
C = 2πr
Circumference
x²+y²+Dx+Ey+F=0
General

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Unit circle: x²+y²=1, center (0,0), radius 1. General form coefficients relate to center: h=−D/2, k=−E/2. A point (x,y) is on the circle iff it satisfies the equation.

Key quantities
(x−h)²+(y−k)²=r²
Standard
Key relation
A = πr²
Area
Key relation
C = 2πr
Circumference
Key relation
x²+y²+Dx+Ey+F=0
General
Key relation

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Why: Circle equations appear in geometry, trigonometry (unit circle), physics (circular motion), and computer graphics. Standard form makes center and radius explicit.

How: Standard form: (x−h)²+(y−k)²=r². Expand to get general form. Area = πr², circumference = 2πr. Completing the square converts general to standard.

Unit circle: x²+y²=1, center (0,0), radius 1.General form coefficients relate to center: h=−D/2, k=−E/2.
Sources:Khan Academy — Circle EquationWolfram MathWorld — Circle

Run the calculator when you are ready.

Circle EquationEnter center (h,k) and radius r

Circle Parameters

Visualization

Enter the circle parameters to see the visualization

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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(x−h)²+(y−k)²=r² for center (h,k), radius r.

— Coordinate Geometry

π

Area = πr², circumference = 2πr.

— Formulas

Key Takeaways

  • • The standard form (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2 directly shows the center (h,k)(h,k) and radius rr.
  • • The general form x2+y2+Dx+Ey+F=0x^2 + y^2 + Dx + Ey + F = 0 is useful for algebraic manipulation and solving systems.
  • • To convert general to standard form, use completing the square on both xx and yy terms.
  • • The area of a circle is A=pir2A = \\pi r^2; the circumference is C=2pirC = 2\\pi r.
  • • A circle is the locus of all points equidistant from a fixed center — a fundamental definition in coordinate geometry.

Did You Know?

Pi Through History

The symbol π for the circle constant was first used by William Jones in 1706. Ancient Babylonians approximated π as 3.125 around 1900 BCE.

Why Wheels Are Round

A circle has constant curvature — every point on the circumference is the same distance from the center. That's why wheels roll smoothly without bouncing.

Orbits Are Ellipses

Planetary orbits are ellipses, not perfect circles. A circle is a special ellipse where both foci coincide at the center.

Manhole Covers

Manhole covers are circular because a circle is the only shape that cannot fall through its own hole when rotated — the width is constant in all directions.

Circle of Willis

In anatomy, the Circle of Willis is a ring of arteries supplying blood to the brain. Its circular structure provides redundant blood flow.

Great Circles

On a sphere, a "great circle" (like the equator) is the largest possible circle. Airplane routes often follow great circles for the shortest path.

Understanding Circle Equations

A circle is the set of all points in a plane that are at a fixed distance (the radius) from a fixed point (the center).

Standard Form

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

where (h,k)(h,k) is the center and rr is the radius.

General Form

x2+y2+Dx+Ey+F=0x^2 + y^2 + Dx + Ey + F = 0

To find center and radius: complete the square on xx and yy.

Conversion Between Forms

Standard → General: expand the squares and move all terms to one side.

General → Standard: complete the square: x2+Dxx^2 + Dx becomes (x+D/2)2−(D/2)2(x + D/2)^2 - (D/2)^2, and similarly for yy.

Expert Tips

Completing the Square

To convert general to standard form, add (D/2)2(D/2)^2 and (E/2)2(E/2)^2 to both sides after grouping xx and yy terms.

Tangent Lines

A tangent to a circle at a point is perpendicular to the radius at that point. Use this to find tangent line equations.

Point Inside/Outside

Substitute a point into the left side of the standard form. If less than r2r^2, the point is inside; if greater, outside.

Intersection with Lines

Substitute the line equation into the circle equation to get a quadratic. The discriminant tells you 0, 1, or 2 intersection points.

Frequently Asked Questions

What is the standard form of a circle equation?

The standard form is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. It directly shows the circle's key properties.

How do I convert general form to standard form?

Use completing the square. Group x and y terms, add (coefficient/2)² to both sides for each variable, then factor into (x - h)² + (y - k)² = r².

Can the radius be negative?

No. The radius is a distance and must be positive. A negative value in the equation would not represent a real circle.

What is the unit circle?

The unit circle has center (0, 0) and radius 1. Its equation is x² + y² = 1. It is fundamental in trigonometry for defining sine and cosine.

How do I find if a point is on a circle?

Substitute the point (x, y) into the equation. If the left side equals the right side (or r² in standard form), the point lies on the circle.

What is the relationship between diameter and radius?

The diameter is twice the radius: d = 2r. The diameter is the longest chord of the circle, passing through the center.

How is the circle equation used in real life?

Circle equations appear in GPS (distance from a point), optics (lens curvature), engineering (circular motion), and computer graphics (rendering circles and arcs).

How to Use This Calculator

  1. Enter the center coordinates (h, k) and the radius (r), or click a sample example to auto-fill.
  2. Results update automatically as you type — standard form, general form, area, circumference, and diameter.
  3. View the visualization to see the circle on a coordinate grid with the center marked.
  4. Check the step-by-step solution to see how the standard form expands into general form.
  5. Copy the results to share or paste into assignments.

Note: This calculator uses standard floating-point arithmetic. Results are suitable for educational purposes, homework, and professional calculations. For very large radii, the visualization may not show the full circle.

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