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.
- 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:
- 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.
Practice finding the circumference and area of circles with these math worksheets.
- 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.