GEOMETRYCircleMathematics Calculator
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Equation of a Circle

The standard form (x-h)²+(y-k)²=r² defines a circle with center (h,k) and radius r. The general form x²+y²+Dx+Ey+F=0 encodes the same with D=-2h, E=-2k, F=h²+k²-r².

Concept Fundamentals
(x-h)²+(y-k)²=r²
Standard form
x²+y²+Dx+Ey+F=0
General form
D=-2h, E=-2k, F=h²+k²-r²
D, E, F
x²+y²=1
Unit circle

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The unit circle (center 0,0, r=1) is fundamental in trigonometry: x²+y²=1. D and E in general form give the center: h=-D/2, k=-E/2. When F=0, the circle passes through the origin (0,0).

Key quantities
(x-h)²+(y-k)²=r²
Standard form
Key relation
x²+y²+Dx+Ey+F=0
General form
Key relation
D=-2h, E=-2k, F=h²+k²-r²
D, E, F
Key relation
x²+y²=1
Unit circle
Key relation

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Why: Circle equations appear in coordinate geometry, signal coverage (radar range), and computer graphics. Converting between forms is needed for algebra and geometry problems.

How: Standard form comes from the distance formula: every point (x,y) on the circle is distance r from (h,k). Expand to get general form; complete the square to reverse.

The unit circle (center 0,0, r=1) is fundamental in trigonometry: x²+y²=1.D and E in general form give the center: h=-D/2, k=-E/2.
Sources:Khan Academy - GeometryWolfram MathWorld - Circle

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Start CalculatingEnter center (h,k) and radius r to get both equation forms

Circle Equation — Standard & General Form

Enter center (h,k) and radius r. Get both equation forms.

⭕ Sample Examples — Click to Load

Circle Parameters

(0,0)
circle_equation.shCALCULATED
$ circle_eq --center=(0,0) --radius=1
Standard Form
(x )² + (y )² = 1²
General Form
x² + y² x y - 1 = 0
D
0
E
0
F
-1
Area
3.1416 ²
Circumference
6.2832
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Circle Equation
(x )² + (y )² = 1²
D=0, E=0, F=-1
numbervibe.com/calculators/mathematics/circle/equation
Step 1:Standard form: (x-h)²+(y-k)²=r² with h=0, k=0, r=1
Step 2:Expand: D=-2h=0, E=-2k=0, F=h²+k²-r²=-1

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🧮 Fascinating Math Facts

○

(x-h)²+(y-k)²=r² — distance from (h,k) to (x,y) equals r.

— Standard form

📐

Complete the square on x and y to convert general to standard.

— Conversion

📋 Key Takeaways

  • • Standard form: (x-h)² + (y-k)² = r² — center (h,k), radius r
  • • General form: x² + y² + Dx + Ey + F = 0 where D=-2h, E=-2k, F=h²+k²-r²
  • • Converting: expand standard form and collect terms to get general form
  • • Area A = πr² and Circumference C = 2πr derive from the radius

💡 Did You Know?

⭕The unit circle (center 0,0, r=1) is fundamental in trigonometry — x²+y²=1Source: Khan Academy
📐D and E in general form directly give the center: h=-D/2, k=-E/2Source: Purplemath
📏When F=0 in general form, the circle passes through the origin (0,0)Source: Wolfram MathWorld
📐The distance formula √[(x-h)²+(y-k)²]=r squared gives the standard formSource: Purplemath

📖 How It Works

Standard form (x-h)²+(y-k)²=r² comes from the distance formula: every point (x,y) on the circle is distance r from (h,k).

(x−h)2+(y−k)2=r⇒(x−h)2+(y−k)2=r2\sqrt{(x-h)^2 + (y-k)^2} = r \quad \Rightarrow \quad (x-h)^2 + (y-k)^2 = r^2

Expanding: x²-2hx+h²+y²-2ky+k²=r² → x²+y²-2hx-2ky+h²+k²-r²=0. So D=-2h, E=-2k, F=h²+k²-r².

D=−2h,E=−2k,F=h2+k2−r2D = -2h, \quad E = -2k, \quad F = h^2 + k^2 - r^2

🎯 Expert Tips

💡 Use Standard Form

When you know center and radius, standard form is most direct.

💡 General Form for Algebra

Use general form for solving systems of equations or intersections.

💡 Complete the Square

To convert general to standard: complete the square for x and y terms.

💡 Real-World Uses

Signal coverage, radar range, circular structures — all use circle equations.

⚖️ Comparison Table

FormEquationUse Case
Standard(x-h)²+(y-k)²=r²Direct center, radius
Generalx²+y²+Dx+Ey+F=0Algebraic manipulation
CoefficientsD=-2h, E=-2k, F=h²+k²-r²Conversion

❓ Frequently Asked Questions

What is D, E, F in general form?

D=-2h, E=-2k, F=h²+k²-r². They encode center and radius. From h=-D/2 and k=-E/2 you recover the center.

Is x²+y²=1 a circle?

Yes — unit circle centered at origin with radius 1. Fundamental in trigonometry.

How do I convert general to standard form?

Complete the square for x and y: x²+Dx → (x+D/2)²-(D/2)². Same for y. Then rearrange to get (x-h)²+(y-k)²=r².

What if F is negative?

F can be positive, negative, or zero. When F=0, the circle passes through the origin (0,0).

Can I find the equation from three points?

Yes! Substitute each point into x²+y²+Dx+Ey+F=0 to get a system of 3 equations in D, E, F. Solve and convert to standard form.

📊 Infographic Stats

(x-h)²+(y-k)²
Standard Form
x²+y²+Dx+Ey+F
General Form
D=-2h
D Coefficient
F=h²+k²-r²
F Coefficient

⚠️ Disclaimer: This calculator provides educational estimates based on standard circle formulas. For precision engineering or scientific applications, verify results with authoritative sources. All formulas assume a perfect circle.

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