radius

Examples of radius in the following topics:

• Radians

  • If we divide both sides of this equation by $r$, we create the ratio of the circumference, which is always $2\pi$ to the radius, regardless of the length of the radius.
  • So the circumference of any circle is $2\pi \approx 6.28$ times the length of the radius.
  • Just as the full circumference of a circle always has a constant ratio to the radius, the arc length produced by any given angle also has a constant relation to the radius, regardless of the length of the radius.
  • (a) In an angle of 1 radian; the arc lengths equals the radius $r$.
  • The circumference of a circle is a little more than 6 times the length of the radius.

• Circles as Conic Sections

  • You probably know how to find the area and the circumference of a circle, given its radius.
  • The point is known as the center of the circle, and the distance is known as the radius.
  • The length of the circumference, C, is related to the radius, r, and diameter, d, by:
  • As proved by Archimedes, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference, and whose height equals the circle's radius, which comes to π multiplied by the radius squared:
  • Radius: a line segment joining the center of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.

• Introduction to Circles

  • This definition is what gives us the concept of the radius of a circle, which is equal to that certain distance.
  • Remember that the distance between the center $\left(a,b\right)$ and any point $\left(x,y\right)$ on the circle is that fixed distance, which is called the radius.
  • This is the general formula for a circle with center $\left(a,b\right)$ and radius $r$.
  • It is equal to twice the radius, so:
  • Notice that the radius is the only defining parameter for the size of any particular circle, and so it is the only variable that the area depends on.

• Introduction to Ellipses

  • In this equation, $r$ is the radius of the circle.
  • A circle has only one radius—the distance from the center to any point is the same.
  • First, let's start with a specific circle that's easy to work with, the circle centered at the origin with radius $1$.

• Applications of Circles and Ellipses

  • The water reaches 6 feet out from the sprinkler, so the circle radius is 6 feet.
  • The sprinkler is at coordinate $\left(6,7\right)$, and the radius of the sprinkler is 6 feet.
  • The radius of the circle is $r$.
  • The radius of the circle is $r$.

• Graphing Equations

  • Now you can begin seeing that we're drawing a circle with a radius of 10:
  • This is a graph of a circle with radius 10 and center at the origin.

• Types of Conic Sections

  • A radius, which the distance from any point on the circle to the center point
  • where $(h,k)$ are the coordinates of the center of the circle, and $r$ is the radius.
  • This is a single point intersection, or equivalently a circle of zero radius.
  • The degenerate form of an ellipse is a point, or circle of zero radius, just as it was for the circle.

• Parabolas As Conic Sections

  • The vertex is therefore also a point on the cone, and the distance between that point and the cone's central axis is the radius of a circle.
  • Using the definition of sine as opposite over hypotenuse, we can find a formula for the focal length "$f$" in terms of the radius and the angle:
  • The vertex of the parabola here is point $P$, and the diagram shows the radius $r$ between that point and the cone's central axis, as well as the angle $\theta$ between the parabola's axis of symmetry and the cone's central axis.

• Ellipses

  • These are sometimes called (especially in technical fields) the major and minor semi-axes, or major radius and minor radius.

• Pythagorean Identities

  • For a triangle drawn inside a unit circle, the length of the hypotenuse of the triangle is equal to the radius of the circle, which is $1$.
  • For a triangle drawn inside a unit circle, the length of the hypotenuse is equal to the radius of the circle.