Circle Calculator

How does Circle Calculator work?

Circle Calculator is a simple calculator for calculating circumference, diameter, area and other attributes of a circle. With Circle Calculator, you can easily enter known information about a circle (radius, diameter, perimeter, or area) and then click the Calculate button to derive additional information about the circle.

What is a Circle?

A circle is a geometric shape defined by two elements: the center and the radius:

• Center: The center is the central point of the circle, and all points on the circle are equidistant from this point.
• Radius: The radius is the distance from the center to any point on the circle. All radii of a circle are of equal length.

Based on these elements, a circle can be defined as:

• Geometric Definition: The set of all points that are at a constant distance (the radius) from a given point (the center).
• Algebraic Definition: In the Cartesian coordinate system, a circle can be defined by the equation:
  { \color{Orange} (x - h)^2 + (y - k)^2 = r^2}
  Here, { \color{Orange} (h, k)} are the coordinates of the center, { \color{Orange} r} is the radius, and { \color{Orange} (x, y)} are the coordinates of any point on the circle.
• Topological Definition: In topology, a circle can be considered as a closed curve formed by a line segment rotating continuously through 360 degrees.
• Parametric Equations: The parametric equations of a circle are:
  { \color{Orange} x = h + r \cos(\theta)}
{ \color{Orange} y = k + r \sin(\theta)}
  Here, { \color{Orange} \theta} is the parameter representing the angle from the center to a point on the circle (usually measured in radians).
• Polar Coordinate Definition: In polar coordinates, a circle can be defined by the equation:
  { \color{Orange} r = \text{constant}}
  Here, { \color{Orange} r} is the distance from the origin to any point on the circle, which is the radius.

These definitions describe the circle from different perspectives, but all are based on the circle’s central symmetry and the equidistance of all points from the center.

Circle Calculations

Given the Radius of a Circle

Given the radius { \color{Orange} r }, you can calculate the diameter { \color{Orange} d }, circumference { \color{Orange} C }, and area { \color{Orange} A } of the circle as follows:

• Diameter { \color{Orange} d }:
  { \color{Orange} d = 2r}
• Circumference { \color{Orange} C }:
  { \color{Orange} C = 2\pi r}
• Area { \color{Orange} A }:
  { \color{Orange} A = \pi r^2}

Here, { \color{Orange} \pi} is the mathematical constant pi, approximately equal to 3.14159.

For example, if the radius of a circle is 5 units, then:

• The diameter { \color{Orange} d } would be { \color{Orange} 2 \times 5 = 10} units.
• The circumference { \color{Orange} C } would be approximately { \color{Orange} 2 \times 3.14159 \times 5 \approx 31.4159} units.
• The area { \color{Orange} A } would be approximately { \color{Orange} 3.14159 \times 25 \approx 78.5398} square units.

Given the Diameter of a Circle

Given the diameter { \color{Orange} d }, you can calculate the radius { \color{Orange} r }, circumference { \color{Orange} C }, and area { \color{Orange} A } of the circle as follows:

• Radius { \color{Orange} r }:
  { \color{Orange} r = \frac{d}{2}}
• Circumference { \color{Orange} C }:
  { \color{Orange} C = \pi d}
• Area { \color{Orange} A }:
  { \color{Orange} A = \frac{\pi d^2}{4}}

For example, if the diameter of a circle is 10 units, then:

• The radius { \color{Orange} r } would be { \color{Orange} \frac{10}{2} = 5} units.
• The circumference { \color{Orange} C } would be approximately { \color{Orange} \pi \times 10 \approx 31.4159} units.
• The area { \color{Orange} A } would be approximately { \color{Orange} \frac{\pi \times 10^2}{4} \approx \frac{3.14159 \times 100}{4} \approx 78.5398} square units.

Given the Circumference of a Circle

Given the circumference { \color{Orange} C }, you can calculate the radius { \color{Orange} r }, diameter { \color{Orange} d }, and area { \color{Orange} A } of the circle using the following formulas:

• Radius { \color{Orange} r }:
  From the circumference formula { \color{Orange} C = 2\pi r}, we get:
  { \color{Orange} r = \frac{C}{2\pi}}
• Diameter { \color{Orange} d }:
  The diameter is twice the radius, so:
  { \color{Orange} d = 2r}
{ \color{Orange} d = 2 \times \frac{C}{2\pi}}
{ \color{Orange} d = \frac{C}{\pi}}
• Area { \color{Orange} A }:
  The area formula is:
  { \color{Orange} A = \pi r^2}
  Substituting the expression for radius { \color{Orange} r }:
  { \color{Orange} A = \pi \left(\frac{C}{2\pi}\right)^2}
{ \color{Orange} A = \frac{C^2}{4\pi}}

For example, if the circumference of a circle is 31.4159 units (approximately equal to { \color{Orange} 2\pi \times 5}), then:

• The radius { \color{Orange} r } would be approximately { \color{Orange} \frac{31.4159}{2\pi} \approx 5} units.
• The diameter { \color{Orange} d } would be approximately { \color{Orange} \frac{31.4159}{\pi} \approx 10} units.
• The area { \color{Orange} A } would be approximately { \color{Orange} \frac{31.4159^2}{4\pi} \approx 78.5398} square units.

Given the Area of a Circle

Given the area { \color{Orange} A }, you can calculate the radius { \color{Orange} r }, diameter { \color{Orange} d }, and circumference { \color{Orange} C } of the circle using the following formulas:

• Radius { \color{Orange} r }:
  From the area formula { \color{Orange} A = \pi r^2}, we get:
  { \color{Orange} r = \sqrt{\frac{A}{\pi}}}
• Diameter { \color{Orange} d }:
  The diameter is twice the radius, so:
  { \color{Orange} d = 2r}
{ \color{Orange} d = 2 \times \sqrt{\frac{A}{\pi}}}
• Circumference { \color{Orange} C }:
  The circumference formula is:
  { \color{Orange} C = 2\pi r}
  Substituting the expression for radius { \color{Orange} r }:
  { \color{Orange} C = 2\pi \times \sqrt{\frac{A}{\pi}}}
{ \color{Orange} C = 2\sqrt{\pi A}}

For example, if the area of a circle is 78.5398 square units (approximately equal to { \color{Orange} \pi \times 5^2}), then:

• The radius { \color{Orange} r } would be approximately { \color{Orange} \sqrt{\frac{78.5398}{\pi}} \approx 5} units.
• The diameter { \color{Orange} d } would be approximately { \color{Orange} 2 \times \sqrt{\frac{78.5398}{\pi}} \approx 10} units.
• The circumference { \color{Orange} C } would be approximately { \color{Orange} 2\sqrt{\pi \times 78.5398} \approx 31.4159} units.

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