Circle Facts – Area, Circumference, Diameter, Radius


Circle Facts
A circle is a two-dimensional shape formed by all the points that are the same distance from the center.

Explore this collection of circle facts. Learn how to find the circumference, diameter, radius, and area of a circle and get definitions of circle terms used in geometry.

Circle Facts

  • A circle is a two-dimensional shape formed by all the points that are the same distance from a center point.
  • Technically, only the points equidistant from the center form the circle. The area enclosed within a circle is called a disc.
  • The word circle comes from the Greek word κρίκος (krikos), meaning “hoop” or “ring”.
  • A circle is the only one-sided shape containing an area. A straight line is a circle containing an infinite area.
  • Humans have recognized circles since ancient times. Natural circles include the shapes of the Sun and Moon, the human eye, tree cross-sections, some flowers, some shells, etc.
  • The distance around a circle is its circumference.
  • The distance from the center to the circle is its radius.
  • The longest distance between two points on a circle is the diameter, which is a line segment running through the center.
  • A circle is the shape with the shortest perimeter enclosing an area.
  • The circle is the most symmetric shape because every line through the center is a line of reflection symmetry. It has rotation symmetry for every angle around its center.
  • Pi (π) is an irrational number that is the ratio of a circle’s circumference to its diameter. It is approximately equal to 3.1415259.
  • Archimedes proved the area enclosed within a circle is the same as the area of a triangle with a base the length of the circle’s circumference and height equal to the circle’s radius.
  • The full arc of circle measures 360 degrees.
  • A circle is a special type of ellipse where the two foci are in the same location and the eccentricity is 0.
  • Written in 1700 BCE, the Rhind papyrus describes a method of finding the area of a circle. The result comes out as 256/81, which is about 3.16 (close to pi).
  • You can draw a special circle inside every triangle, called the incircle, where each of the three triangle sides are tangent to the circle.

How to Find the Circumference of a Circle

The circumference (C) is the distance around a circle. There are a few ways to find the circumference. You can calculate it from either the radius (r) or diameter (d) or you can measure it.

  • C = 2πr
  • C = πd
  • It’s easiest to measure a circle’s circumference using a string. Shape the string around the circle, mark the length, and then use a ruler or meter stick to measure the length of the string.

How to Find the Diameter of a Circle

The diameter (d) is the length of the line segment with end points on the circle that passes through its center. It is the longest distance across a circle. The diameter is twice the length of the radius.

  • d = 2r
  • d = C/π
  • Measure the diameter by finding the longest line segment across a circle.

How to Find the Radius of a Circle

The radius (r) is the distance from the center of a circle to its border. It is half the length of the diameter.

  • r = d/2
  • r = C/2π
  • If you draw a circle using a compass, the radius is the distance between its two points. Measuring the radius of a circle is a bit tricky unless you know its center. Sometimes its easier to measure the circumference or diameter and calculate the radius.

How to Find the Area of a Circle

The area (A) of a circle is the region enclosed by a circle or the area of its disc.

  • A = πr2
  • A = π(d/2)2
  • A = Cr/2 – You can use Archimedes’ proof to find the circle area using its circumference and radius. Set the base of the triangle equal to circumference C and height equal to radius r. The triangle area formula 1/2 bh becomes A = Cr/2

Circle Vocabulary Terms

Here are key circle vocabulary terms to know:

Circle Slices
  • Annulus: An annulus is a ring-shape formed between two concentric circles.
  • Arc: An arc is any segment of a circle formed by connected points.
  • Center (Centre): The center is the point that is equidistant from all points on a circle. It is also called the origin.
  • Chord: A chord is a line segment with endpoints on the circle. The diameter is the longest chord.
  • Circumference: The circumference is the distance around a circle.
  • Closed: A region that includes its boundaries.
  • Diameter: The diameter is the line segment with endpoints on the circle and midpoint at its center. It is the largest distance between any two points on a circle.
  • Disc: A disc is the area inside a circle.
  • Lens: A lens is a region shared by two overlapping discs.
  • Open: Any region, excluding its boundaries.
  • Passant: A passant is a coplanar line that has no points in common with a circle.
  • Radius: A radius is a line segment running from the center to the circle.
  • Sector: A sector is an area within a circle bounded by two radii.
  • Segment: A segment is an area bounded by an arc and a chord.
  • Secant: A secant is a chord that extends beyond the circle. In other words, it is a coplanar line that intersects a circle at two points.
  • Semicircle: A semicircle is an arc which has the diameter as endpoints and center as midpoint. The interior of a semicircle is a half-disc.
  • Tangent: A tangent is a coplanar line sharing one single point in common with a circle.

References

  • Gamelin, Theodore (1999). Introduction to Topology. Mineola, N.Y: Dover Publications. ISBN 0486406806.
  • Harkness, James (1898). “Introduction to the theory of analytic functions”. Nature. 59 (1530): 30. doi:10.1038/059386a0
  • Katz, Victor J. (1998). A History of Mathematics / An Introduction (2nd ed.). Addison Wesley Longman. ISBN 978-0-321-01618-8.
  • Ogilvy, C. Stanley (1969). Excursions in Geometry. Dover.