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Math & Conversions Circle Calculator Circle Calculator: Area, Circumference, Diameter, Radius This calculator can solve for the four main properties of a circle. Enter a value for **any one** of the properties below (Radius, Diameter, Circumference, or Area), and the tool will instantly derive all the others. Geometric Properties Radius ($r$) Diameter ($d$) Circumference ($C$) Area ($A$) Calculate Circle Results Radius ($r$) Diameter ($d$) Circumference ($C$) Area ($A$) Step-by-Step Derivation Circle Equation Calculator Find the standard form of the circle equation, $(x - h)^2 + (y - k)^2 = r^2$, given the center and radius. Standard Form: $(x - h)^2 + (y - k)^2 = r^2$ Center X ($h$) Center Y ($k$) Radius ($r$) Derive Equation Standard Equation General Form Geometric Formulas for Circle Properties The entire geometry of a circle can be calculated if the radius ($r$) is known. The relationships are defined by the constant $\pi \approx 3.14159$ . Property Formula (using $r$) Inverse Formula (to find $r$) Diameter ($d$) $d = 2r$ $r = d/2$ Circumference ($C$) $C = 2\pi r$ $$r = \frac{C}{2\pi}$$ Area ($A$) $A = \pi r^2$ $$r = \sqrt{\frac{A}{\pi}}$$ Circle Equations (Analytic Geometry) In a coordinate plane, the location and size of a circle are described by its equation. The circle is defined as the set of all points $(x, y)$ that are equidistant (distance $r$) from a fixed center point $(h, k)$. Standard Form (Center-Radius Form) This is the simplest form, directly showing the center $(h, k)$ and the radius $r$: $$(x - h)^2 + (y - k)^2 = r^2$$ General Form The general form is found by expanding the standard form: $$x^2 + y^2 + Dx + Ey + F = 0$$ Where the center coordinates are related by $h = -D/2$ and $k = -E/2$. Frequently Asked Questions (FAQ) What is the formula for the Area of a Circle? The area ($A$) of a circle is calculated by multiplying pi ($\pi \approx 3.14159$) by the square of the radius ($r$): $A = \pi r^2$. What is the formula for the Circumference of a Circle? The circumference ($C$), which is the distance around the circle, is calculated as pi ($\pi$) times the diameter ($d$), or two times pi times the radius ($r$): $C = \pi d = 2\pi r$. What is the standard form equation of a circle? The standard form equation of a circle with center $(h, k)$ and radius $r$ is: $(x - h)^2 + (y - k)^2 = r^2$. What is the relationship between the circle and the ellipse? A circle is a special case of an ellipse where the two semi-axes are equal ($a = b = r$) and the eccentricity ($e$) is zero. Geometrically, it is the only conic section with $A=C$ in the general form $A x^2 + C y^2 + D x + E y + F = 0$. Essential Circle Formulas Area $$A = \pi r^2$$ Circumference $$C = 2\pi r$$ Standard Equation $$(x-h)^2 + (y-k)^2 = r^2$$ Related Geometry Tools Ellipse Calculator Area Calculator (General) Arc Length Calculator Sector Area Calculator Distance Formula Calculator
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