Triangle in the context of Straight line

A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle (1⁄4 turn or 90 degrees).

The side opposite to the right angle is called the hypotenuse (side ${\displaystyle c}$ in the figure). The sides adjacent to the right angle are called legs (or catheti, singular: cathetus). Side ${\displaystyle a}$ may be identified as the side adjacent to angle ${\displaystyle B}$ and opposite (or opposed to) angle ${\displaystyle A,}$ while side ${\displaystyle b}$ is the side adjacent to angle ${\displaystyle A}$ and opposite angle ${\displaystyle B.}$

In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or ⁠${\displaystyle \pi }$/2⁠ radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. The term is a calque of Latin angulus rectus; here rectus means "upright", referring to the vertical perpendicular to a horizontal base line.

Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to vectors. The presence of a right angle in a triangle is the defining factor for right triangles, making the right angle basic to trigonometry.

A spear is a polearm consisting of a shaft, usually of wood, with a pointed head. The head may be simply the sharpened end of the shaft itself, as is the case with fire hardened spears, or it may be made of a more durable material fastened to the shaft, such as bone, flint, obsidian, copper, bronze, iron, or steel. The most common design for hunting and/or warfare, since modern times has incorporated a metal spearhead shaped like a triangle, diamond, or leaf. The heads of fishing spears usually feature multiple sharp points, with or without barbs.

Spears can be divided into two broad categories: those designed for thrusting as a melee weapon (including weapons such as lances and pikes) and those designed for throwing as a ranged weapon (usually referred to as javelins).

A pyramid is a polyhedron (a geometric figure) formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. A pyramid is a conic solid with a polygonal base. Many types of pyramids can be found by determining the shape of bases, either by based on a regular polygon (regular pyramids) or by cutting off the apex (truncated pyramid). It can be generalized into higher dimensions, known as hyperpyramid. All pyramids are self-dual.

In geometry, a polygon (/ˈpɒlɪɡɒn/) is a plane figure made up of line segments connected to form a closed polygonal chain.

The segments of a closed polygonal chain are called its edges or sides. The points where two edges meet are the polygon's vertices or corners. An n-gon is a polygon with n sides; for example, a triangle is a 3-gon.

A fractal landscape or fractal surface is generated using a stochastic algorithm designed to produce fractal behavior that mimics the appearance of natural terrain. In other words, the surface resulting from the procedure is not a deterministic, but rather a random surface that exhibits fractal behavior.

Many natural phenomena exhibit some form of statistical self-similarity that can be modeled by fractal surfaces. Moreover, variations in surface texture provide important visual cues to the orientation and slopes of surfaces, and the use of almost self-similar fractal patterns can help create natural looking visual effects.The modeling of the Earth's rough surfaces via fractional Brownian motion was first proposed by Benoit Mandelbrot.

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.

In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about figure F". A circle or ellipse inscribed in a convex polygon (or a sphere or ellipsoid inscribed in a convex polyhedron) is tangent to every side or face of the outer figure (but see Inscribed sphere for semantic variants). A polygon inscribed in a circle, ellipse, or polygon (or a polyhedron inscribed in a sphere, ellipsoid, or polyhedron) has each vertex on the outer figure; if the outer figure is a polygon or polyhedron, there must be a vertex of the inscribed polygon or polyhedron on each side of the outer figure. An inscribed figure is not necessarily unique in orientation; this can easily be seen, for example, when the given outer figure is a circle, in which case a rotation of an inscribed figure gives another inscribed figure that is congruent to the original one.

Familiar examples of inscribed figures include circles inscribed in triangles or regular polygons, and triangles or regular polygons inscribed in circles. A circle inscribed in any polygon is called its incircle, in which case the polygon is said to be a tangential polygon. A polygon inscribed in a circle is said to be a cyclic polygon, and the circle is said to be its circumscribed circle or circumcircle.

In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle ${\displaystyle \theta }$, the sine and cosine functions are denoted as ${\displaystyle \sin(\theta )}$ and ${\displaystyle \cos(\theta )}$.

The definitions of sine and cosine have been extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.

In geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also be equiangular (have all angles equal), but if it does then it is a regular polygon. If the number of sides is at least four, an equilateral polygon does not need to be a convex polygon: it could be concave or even self-intersecting.

In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points.

A geodetic control network is a network, often of triangles, that are measured precisely by techniques of control surveying, such as terrestrial surveying or satellite geodesy. It is also known as a geodetic network, reference network, control point network, or simply control network.

A geodetic control network consists of geodetic markers, which are stable, identifiable points or vertices with published coordinate values derived from observations that tie the points together.

A bicycle frame is the main component of a bicycle, onto which wheels and other components are fitted. The modern and most common frame design for an upright bicycle is based on the safety bicycle, and consists of two triangles: a main triangle and a paired rear triangle. This is known as the diamond frame. Frames are required to be strong, stiff and light, which they do by combining different materials and shapes.

A frameset consists of the frame and fork of a bicycle and sometimes includes the headset and seat post. Frame builders will often produce the frame and fork together as a paired set.

Futura is a geometric sans-serif typeface designed by Paul Renner and released in 1927. Designed as a contribution on the New Frankfurt-project, it is based on geometric shapes, especially the circle, similar in spirit to the Bauhaus design style of the period. It was developed as a typeface by Bauersche Gießerei, in competition with Ludwig & Mayer's seminal Erbar typeface.

Although Renner was not associated with the Bauhaus, he shared many of its idioms and believed that a modern typeface should express modern models, rather than be a revival of a previous design. Renner's design rejected the approach of most previous sans-serif designs (now often called grotesques), which were based on the models of sign painting, condensed lettering, and nineteenth-century serif typefaces, in favour of simple geometric forms: near-perfect circles, triangles and squares. It is based on strokes of near-even weight, which are low in contrast. The lowercase has tall ascenders, which rise above the cap line, and uses nearly-circular, single-storey forms for the "a" and "g", the former previously more common in handwriting than in printed text. The uppercase characters present proportions similar to those of classical Roman capitals. The original metal type showed extensive adaptation of the design to individual sizes, and several divergent digitisations have been released by different companies.

The triangle, or musical triangle, is a musical instrument in the percussion family, classified as an idiophone in the Hornbostel-Sachs classification system. Triangles are made from a variety of metals including aluminum, beryllium copper, brass, bronze, iron, and steel. The metal is bent into a triangular shape with one open end. The instrument is usually held by a loop of some form of thread or wire at the top curve to enable the triangle to vibrate, and it is struck with a metal rod called a "beater". The triangle theoretically has indefinite pitch, and produces a plurality of overtones when struck with an appropriate beater.

In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle. A polygon whose vertices are concyclic is called a cyclic polygon, and the circle is called its circumscribing circle or circumcircle. All concyclic points are equidistant from the center of the circle.

Three points in the plane that do not all fall on a straight line are concyclic, so every triangle is a cyclic polygon, with a well-defined circumcircle. However, four or more points in the plane are not necessarily concyclic. After triangles, the special case of cyclic quadrilaterals has been most extensively studied.

In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class.

Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important step in the process of classifying mathematical objects.