Radius in the context of Line segment

In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, which may be centered on Earth or the observer. If centered on the observer, half of the sphere would resemble a hemispherical screen over the observing location.

The celestial sphere is a conceptual tool used in spherical astronomy to specify the position of an object in the sky without consideration of its linear distance from the observer. The celestial equator divides the celestial sphere into northern and southern hemispheres.

In geometry, the area enclosed by a circle of radius r is πr. Here, the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.

One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons with an increasing number of sides. The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and because the sequence tends to a circle, the corresponding formula–that the area is half the circumference times the radius–namely, A = ⁠1/2⁠ × 2πr × r, holds for a circle.

A sphere (from Greek σφαῖρα, sphaîra) is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the center of the sphere, and the distance r is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.

The sphere is a fundamental surface in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings.

Earth's crust is its thick outer shell of rock, comprising less than one percent of the planet's radius and volume. It is the top component of the lithosphere, a solidified division of Earth's layers that includes the crust and the upper part of the mantle. The lithosphere is broken into tectonic plates whose motion allows heat to escape the interior of Earth into space.

The crust lies on top of the mantle, a configuration that is stable because the upper mantle is made of peridotite and is therefore significantly denser than the crust. The boundary between the crust and mantle is conventionally placed at the Mohorovičić discontinuity, a boundary defined by a contrast in seismic velocity.

In astronomy, the term compact object (or compact star) refers collectively to white dwarfs, neutron stars, and black holes. It could also include exotic stars if such hypothetical, dense bodies are confirmed to exist. All compact objects have a high mass relative to their radius, giving them a very high density compared to ordinary atomic matter. The term is used as a generalization for cases where the exact nature of a significant gravitational effect isolated to a small radius is not known.

Since most compact object types represent endpoints of stellar evolution, they are also called stellar remnants, and accordingly may be called dead stars in popular media reports. The state and type of a stellar remnant depends primarily on the mass of its progenitor star. A compact object that is not a black hole may be called a degenerate star.

The Port of New York and New Jersey is the port district of the New York–Newark metropolitan area, encompassing the region within approximately a 25-mile (40 km) radius of the Statue of Liberty National Monument.

It includes the system of navigable waterways in the New York–New Jersey Harbor Estuary, which runs along over 770 miles (1,240 km) of shoreline in the vicinity of New York City and northeastern New Jersey, and is considered one of the largest natural harbors in the world.

In mathematics, a surface is a mathematical model of the common concept of a surface. It is a generalization of a plane, but, unlike a plane, it may be curved (this is analogous to a curve generalizing a straight line). An example of a non-flat surface is the sphere.

There are several more precise definitions, depending on the context and the mathematical tools that are used for the study. The simplest mathematical surfaces are planes and spheres in the Euclidean 3-space. Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not.

In geometry, an angle subtended (from Latin for "stretched under") by a line segment at an arbitrary vertex is formed by the two rays between the vertex and each endpoint of the segment. For example, a side of a triangle subtends the opposite angle.

More generally, an angle subtended by an arc of a curve is the angle subtended by the corresponding chord of the arc.For example, a circular arc subtends the central angle formed by the two radii through the arc endpoints.

Angular distance or angular separation is the measure of the angle between the orientation of two straight lines, rays, or vectors in three-dimensional space, or the central angle subtended by the radii through two points on a sphere. When the rays are lines of sight from an observer to two points in space, it is known as the apparent distance or apparent separation.

Angular distance appears in mathematics (in particular geometry and trigonometry) and all natural sciences (e.g., kinematics, astronomy, and geophysics). In the classical mechanics of rotating objects, it appears alongside angular velocity, angular acceleration, angular momentum, moment of inertia and torque.

A fishing rod or fishing pole is a long, thin rod used by anglers to catch fish by manipulating a line ending in a hook (formerly known as an angle, hence the term "angling"). At its most basic form, a fishing rod is a straight rigid stick/pole with a line fastened to one end (as seen in traditional bamboo rod fishing such as Tenkara fishing); however, modern rods are usually more elastic and generally have the line stored in a reel mounted at the rod handle, which is hand-cranked and controls the line retrieval, as well as numerous line-restricting rings (also known as line guides) that distribute bending stress along the rod and help dampening down/prevent line whipping and entanglement. To better entice fish, baits or lures are dressed onto the hook attached to the line, and a bite indicator (e.g. a float) is typically used, some of which (e.g. quiver tip) might be incorporated as part of the rod itself.

Fishing rods act as an extended lever and allow the angler to amplify line movements while luring and pulling the fish. It also enhances casting distance by increasing the launch speed of the terminal tackles (the hook, bait/lure, and other accompanying attachments such as float and sinker/feeder), as a longer swing radius (compared to that of a human arm) corresponds to greater arc speed at the tip under the same angular velocity. The length of fishing rods usually vary between 0.6 m (2 ft) and 4.6 m (15 ft) depending on the style of angling, while the Guinness World Record is 22.45 m (73 ft 7.9 in).

A solar radius is a unit of distance, commonly understood as 695,700 km and expressed as ${\displaystyle R_{\odot }}$, used mostly to express the size of an astronomical objects relative to that of the Sun, or their distance from it. This length is also called the nominal solar radius. The sun's actual radius, from which the unit of measurement is derived, is usually calculated as the radius from the sun's center out to the layer in the Sun's photosphere where the optical depth equals 2/3. One solar radius can be described as follows:$${\displaystyle 1\,R_{\odot }=6.957\times 10^{8}{\hbox{ m}}}$$This is an approximation: both because such distance is difficult to measure and can be measured in various ways, and because the sun is not a perfectly spherical object itself, and thus the actual radius varies depending on the point(s) measured and modality of measurement employed.

695,700 kilometres (432,300 miles) is approximately 10 times the average radius of Jupiter; 109 times the 6378 km radius of the Earth at its equator; and ${\textstyle {1 \over 215}}$ or 0.0047 of an astronomical unit, the approximate average distance between Earth and the Sun. The solar radius to the sun's poles and that to the equator differ slightly due to the Sun's rotation, which induces an oblateness in the order of 10 parts per million.

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a disc.

The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus.

In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere.

A ball bearing is a type of rolling-element bearing that uses balls to maintain the separation between the bearing races.

The purpose of a ball bearing is to reduce rotational friction and support radial and axial loads. It achieves this by using at least two races to contain the balls and transmit the loads through the balls. In most applications, one race is stationary and the other is attached to the rotating assembly (e.g., a hub or shaft). As one of the bearing races rotates it causes the balls to rotate as well. Because the balls are rolling, they have a much lower coefficient of friction than if two flat surfaces were sliding against each other.

In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. Fractal geometry relates to the mathematical branch of measure theory by their Hausdorff dimension.

One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the conventional dimension of the filled sphere). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer and is in general greater than its conventional dimension. This power is called the fractal dimension of the geometric object, to distinguish it from the conventional dimension (which is formally called the topological dimension).

In geometry, the compass equivalence theorem is an important statement in compass and straightedge constructions. The tool advocated by Plato in these constructions is a divider or collapsing compass, that is, a compass that "collapses" whenever it is lifted from a page, so that it may not be directly used to transfer distances. The modern compass with its fixable aperture can be used to transfer distances directly and so appears to be a more powerful instrument. However, the compass equivalence theorem states that any construction via a "modern compass" may be attained with a collapsing compass. This can be shown by establishing that with a collapsing compass, given a circle in the plane, it is possible to construct another circle of equal radius, centered at any given point on the plane. This theorem is Proposition II of Book I of Euclid's Elements. The proof of this theorem has had a chequered history.

Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. Typically expressed as ratios that align with another system, these quantities do not necessitate explicitly defined units. For instance, alcohol by volume (ABV) represents a volumetric ratio; its value remains independent of the specific units of volume used, such as in milliliters per milliliter (mL/mL).

The number one is recognized as a dimensionless base quantity. Radians serve as dimensionless units for angular measurements, derived from the universal ratio of 2π times the radius of a circle being equal to its circumference.