Concepts about Geometry
in coordinate geometry, the x-coordinate of a point - that is, the horizontal distance of that point from the vertical or y-axis. For example, a point with the coordinates (4, 3) has an abscissa of 4. The y-coordinate of a point is known as the ordinate.
in geometry, the perpendicular distance from a vertex (corner) of a figure, such as a triangle, to the base (the side opposite the vertex).
another name for coordinate geometry.
annulus ("ring" in Latin)
in geometry, the plane area between two concentric circles, making a flat ring.
|in geometry, a section
of a curved line or circle. A circle has three types of arc: a semicircle,
which is exactly half of the circle; minor arcs, which are less than the
semicircle; and major arcs, which are greater than the semicircle.
An arc of a circle is measured in degrees, according to the angle formed by joining its two ends to the center of that circle. A semicircle is therefore 180°, whereas a minor arc will always be less than 180° (acute or obtuse) and a major arc will always be greater than 180° but less than 360° (reflex).
arc minute, arc second
units for measuring small angles, used in geometry, surveying, map-making, and astronomy. An arc minute (symbol ´) is one-sixtieth of a degree, and an arc second (symbol ") is one-sixtieth of an arc minute. Small distances in the sky, as between two close stars or the apparent width of a planet's disk, are expressed in minutes and seconds of arc.
in coordinate geometry, a straight line that a curve approaches more and more closely but never reaches. The x and y axes are asymptotes to the graph of xy = constant (a rectangular hyperbola).
If a point on a curve approaches a straight line such that its distance from the straight line is d, then the line is an asymptote to the curve if limit d tends to zero as the point moves toward infinity. Among conic sections (curves obtained by the intersection of a plane and a double cone), a hyperbola has two asymptotes, which in the case of a rectangular hyperbola are at right angles to each other.
axis (plural axes)
|in geometry, one of the reference lines by which a point on a graph may be located. The horizontal axis is usually referred to as the x-axis, and the vertical axis as the y-axis. The term is also used to refer to the imaginary line about which an object may be said to be symmetrical (axis of symmetry) - for example, the diagonal of a square - or the line about which an object may revolve (axis of rotation).|
in mathematics, the number of different single-digit symbols used in a particular number system. In our usual (decimal) counting system of numbers (with symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) the base is 10. In the binary number system, which has only the symbols 1 and 0, the base is two. A base is also a number that, when raised to a particular power (that is, when multiplied by itself a particular number of times as in 102 = 10 × 10 = 100), has a logarithm equal to the power. For example, the logarithm of 100 to the base ten is 2.
In geometry, the term is used to denote the line or area on which a polygon or solid stands.
in coordinate geometry, components used to define the position of a point by its perpendicular distance from a set of two or more axes, or reference lines. For a two-dimensional area defined by two axes at right angles (a horizontal x-axis and a vertical y- axis), the coordinates of a point are given by its perpendicular distances from the y-axis and x-axis, written in the form (x,y). For example, a point P that lies three units from the y-axis and four units from the x-axis has Cartesian coordinates (3,4) (see abscissa and ordinate). In three-dimensional coordinate geometry, points are located with reference to a third, z- axis, mutually at right angles to the x and y axes.
The Cartesian coordinate system can be extended to any finite number of dimensions (axes), and is used thus in theoretical mathematics. It is named for the French mathematician, René Descartes. The system is useful in creating technical drawings of machines or buildings, and in computer-aided design (CAD).
|in geometry, a straight line joining any two points on a curve. The chord that passes through the center of a circle (its longest chord) is the diameter. The longest and shortest chords of an ellipse (a regular oval) are called the major and minor axes respectively.|
in geometry, the curved line that encloses a curved plane figure, for example a circle or an ellipse. Its length varies according to the nature of the curve, and may be ascertained by the appropriate formula. The circumference of a circle is pd or 2pr, where d is the diameter of the circle, r is its radius, and p is the constant pi, approximately equal to 3.1416.
of a surface, curving inward, or away from the eye. For example, a bowl appears concave when viewed from above. In geometry, a concave polygon is one that has an interior angle greater than 180°. Concave is the opposite of convex.
in geometry, a solid or surface consisting of the set of all straight lines passing through a fixed point (the vertex) and the points of a circle or ellipse whose plane does not contain the vertex.
A circular cone of perpendicular height, with its apex above the center of the circle, is known as a right circular cone; it is generated by rotating an isosceles triangle or framework about its line of symmetry. A right circular cone of perpendicular height h and base of radius r has a volume V = 1/3 pr2h.
The distance from the edge of the base of a cone to the vertex is called the slant height. In a right circular cone of slant height l, the curved surface area is prl, and the area of the base is pr2. Therefore the total surface area A = prl + pr2 = pr(l + r).
in geometry, having the same shape and size, as applied to two-dimensional or solid figures. With plane congruent figures, one figure will fit on top of the other exactly, though this may first require rotation and/or rotation of one of the figures.
of a surface, curving outward, or toward the eye. For example, the outer surface of a ball appears convex. In geometry, the term is used to describe any polygon possessing no interior angle greater than 180°. Convex is the opposite of concave.
in geometry, a number that defines the position of a point relative to a point or axis (reference line). Cartesian coordinates define a point by its perpendicular distances from two or more axes drawn through a fixed point mutually at right angles to each other. Polar coordinates define a point in a plane by its distance from a fixed point and direction from a fixed line.
coordinate geometry or analytical geometry
system of geometry in which points, lines, shapes, and surfaces are represented by algebraic expressions. In plane (two-dimensional) coordinate geometry, the plane is usually defined by two axes at right angles to each other, the horizontal x-axis and the vertical y-axis, meeting at O, the origin. A point on the plane can be represented by a pair of Cartesian coordinates, which define its position in terms of its distance along the x-axis and along the y-axis from O. These distances are respectively the x and y coordinates of the point.
Lines are represented as equations; for example, y = 2x + 1 gives a straight line, and y = 3x2 + 2x gives a parabola (a curve). The graphs of varying equations can be drawn by plotting the coordinates of points that satisfy their equations, and joining up the points. One of the advantages of coordinate geometry is that geometrical solutions can be obtained without drawing but by manipulating algebraic expressions. For example, the coordinates of the point of intersection of two straight lines can be determined by finding the unique values of x and y that satisfy both of the equations for the lines, that is, by solving them as a pair of simultaneous equations. The curves studied in simple coordinate geometry are the conic sections (circle, ellipse, parabola, and hyperbola), each of which has a characteristic equation.
in geometry, describing lines or points that all lie in the same plane.
in geometry, a regular solid figure whose faces are all squares. It has six equal-area faces and 12 equal-length edges.
If the length of one edge is l, the volume V of the cube is given by:
V = l3
and its surface area A by:
A = 6l2
in geometry, the locus of a point moving according to specified conditions. The circle is the locus of all points equidistant from a given point (the center). Other common geometrical curves are the ellipse, parabola, and hyperbola, which are also produced when a cone is cut by a plane at different angles.
Many curves have been invented for the solution of special problems in geometry and mechanics - for example, the cissoid (the inverse of a parabola) and the cycloid.
in geometry, a curve resembling a series of arches traced out by a point on the circumference of a circle that rolls along a straight line. Its applications include the study of the motion of wheeled vehicles along roads and tracks.
in geometry, a tubular solid figure with a circular base. In everyday use, the term applies to a right cylinder, the curved surface of which is at right angles to the base.
The volume V of a cylinder is given by the formula V = pr2h, where r is the radius of the base and h is the height of the cylinder. Its total surface area A has the formula A = 2pr(h + r), where 2prh is the curved surface area, and 2pr2 is the area of both circular ends.
in geometry, a ten-sided polygon.
in mathematics, an array of elements written as a square, and denoted by two vertical lines enclosing the array. For a 2 × 2 matrix, the determinant is given by the difference between the products of the diagonal terms. Determinants are used to solve sets of simultaneous equations by matrix methods.
When applied to transformational geometry, the determinant of a 2 × 2 matrix signifies the ratio of the area of the transformed shape to the original and its sign (plus or minus) denotes whether the image is direct (the same way round) or indirect (a mirror image).
in science, any directly measurable physical quantity such as mass (M), length (L), and time (T), and the derived units obtainable by multiplication or division from such quantities.
For example, acceleration (the rate of change of velocity) has dimensions (LT-2), and is expressed in such units as km s-2. A quantity that is a ratio, such as relative density or humidity, is dimensionless.
In geometry, the dimensions of a figure are the number of measures needed to specify its size. A point is considered to have zero dimension, a line to have one dimension, a plane figure to have two, and a solid body to have three.
in geometry, a property of a conic section (circle, ellipse, parabola, or hyperbola). It is the distance of any point on the curve from a fixed point (the focus) divided by the distance of that point from a fixed line (the directrix). A circle has an eccentricity of zero; for an ellipse it is less than one; for a parabola it is equal to one; and for a hyperbola it is greater than one.
in mathematics, expression that represents the equality of two expressions involving constants and/or variables, and thus usually includes an equals sign (=). For example, the equation A = pr2 equates the area A of a circle of radius r to the product pr2.
The algebraic equation y = mx + c is the general one in coordinate geometry for a straight line.
If a mathematical equation is true for all variables in a given domain, it is sometimes called an identity and denoted by º.
Thus (x + y)2 º x2 + 2xy + y2 for all x, y Î R.
An indeterminate equation is an equation for which there is an infinite set of solutions - for example, 2x = y. A diophantine equation is an indeterminate equation in which the solution and terms must be whole numbers (after Diophantus of Alexandria, c. AD 250).
(from Latin for “a piece cut off”)
in geometry, a “slice” taken out of a solid figure by a pair of parallel planes. A conical frustum, for example, resembles a cone with the top cut off. The volume and area of a frustum are calculated by subtracting the volume or area of the “missing” piece from those of the whole figure.
in geometry, a curve formed by cutting a right circular cone with a plane so that the angle between the plane and the base is greater than the angle between the base and the side of the cone. All hyperbolae are bounded by two asymptotes (straight lines which the hyperbola moves closer and closer to but never reaches).
A hyperbola is a member of the family of curves known as conic sections.
A hyperbola can also be defined as a path traced by a point that moves such that the ratio of its distance from a fixed point (focus) and a fixed straight line (directrix) is a constant and greater than 1; that is, it has an eccentricity greater than 1.
maximum and minimum
in coordinate geometry, points at which the slope of a curve representing a function changes from positive to negative (maximum), or from negative to positive (minimum). A tangent to the curve at a maximum or minimum has zero gradient.
Maxima and minima can be found by differentiating the function for the curve and setting the differential to zero (the value of the slope at the turning point). For example, differentiating the function for the parabola y = 2x2 - 8x gives dy/dx = 4x - 8. Setting this equal to zero gives x = 2, so that y = -8 (found by substituting x = 2 into the parabola equation). Thus the function has a minimum at the point (2, -8).
in mathematics and statistics, the middle number of an ordered group of numbers. If there is no middle number (because there is an even number of terms), the median is the mean (average) of the two middle numbers. For example, the median of the group 2, 3, 7, 11, 12 is 7; that of 3, 4, 7, 9, 11, 13 is 8 (the average of 7 and 9).
In geometry, the term refers to a line from the vertex of a triangle to the midpoint of the opposite side.
dedicated (person); adj. Geometry, flattened at poles.
oblation, n. offering; sacrifice. oblational, adj.
in coordinate geometry, the y coordinate of a point; that is, the vertical distance of the point from the horizontal or x-axis. For example, a point with the coordinates (3,4) has an ordinate of 4. See abscissa.
parallel lines and parallel planes
in mathematics, straight lines or planes that always remain a constant distance from one another no matter how far they are extended. This is a principle of Euclidean geometry. Some non-Euclidean geometries, such as elliptical and hyperbolic geometry, however, reject Euclid's parallel axiom.
in geometry, a basic element, whose position in the Cartesian system may be determined by its coordinates.
Mathematicians have had great difficulty in defining the point, as it has no size, and is only the place where two lines meet. According to the Greek mathematician Euclid, (i) a point is that which has no part; (ii) the straight line is the shortest distance between two points.
in geometry, a plane (two-dimensional) figure with three or more straight-line sides. Common polygons have names which define the number of sides (for example, triangle, quadrilateral, pentagon).
These are all convex polygons, having no interior angle greater than 180°. The sum of the internal angles of a polygon having n sides is given by the formula (2n - 4) × 90°; therefore, the more sides a polygon has, the larger the sum of its internal angles and, in the case of a convex polygon, the more closely it approximates to a circle.
in geometry, a theorem stating that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides (legs). If the hypotenuse is c units long and the lengths of the legs are a and b, then c2 = a2 + b2.
The theorem provides a way of calculating the length of any side of a right triangle if the lengths of the other to sides are known. It is also used to determine certain trigonometrical relationships such as sin2 q + cos2 q = 1.
furthur elaboration ....
abbreviation for quod erat demonstrandum (Latin “which was to be proved”), added at the end of a geometry proof.
in mathematics, a polynomial equation of second degree (that is, an equation containing as its highest power the square of a variable, such as x2). The general formula of such equations is ax2 + bx + c = 0, in which a, b, and c are real numbers, and only the coefficient a cannot equal 0.
In coordinate geometry, a quadratic function represents a parabola.
Some quadratic equations can be solved by factorization, or the values of x can be found by using the formula for the general solution x = [-b ± Ö(b2 -4ac)]/2a. Depending on the value of the discriminant b2 -4ac, a quadratic equation has two real, two equal, or two complex roots (solutions). When b2 -4ac > 0, there are two distinct real roots. When b2 - 4ac = 0, there are two equal real roots. When b2 - 4ac < 0, there are two distinct complex roots.
in geometry, an equilateral (all sides equal) parallelogram. Its diagonals bisect each other at right angles, and its area is half the product of the lengths of the two diagonals. A rhombus whose internal angles are 90° is called a square.
triangle in which one of the angles is a right angle (90°). It is the basic form of triangle for defining trigonometrical ratios (for example, sine, cosine, and tangent) and for which the Pythagorean theorem holds true. The longest side of a right triangle is called the hypotenuse.
Its area is equal to half the product of the lengths of the two shorter sides. A triangle constructed with its hypotenuse as the diameter of a circle with its opposite vertex on the circumference is a right triangle. This is a fundamental theorem in geometry, first credited to the Greek mathematician Thales about 580 BC.
in geometry, part of a circle enclosed by two radii and the arc that joins them.
in geometry, part of a circle cut off by a straight line or chord, running from one point on the circumference to another. All angles in the same segment are equal.
in geometry, a quadrilateral (four-sided) plane figure with all sides equal and each angle a right angle. Its diagonals bisect each other at right angles. The area A of a square is the length l of one side multiplied by itself (A = l × l).
Also, any quantity multiplied by itself is termed a square, represented by an exponent of power 2; for example, 4 × 4 = 42 = 16 and 6.8 × 6.8 = 6.82 = 46.24.
An algebraic term is squared by doubling its exponent and squaring its coefficient if it has one; for example, (x2)2 = x4 and (6y3)2 = 36y6. A number that has a whole number as its square root is known as a perfect square; for example, 25, 144 and 54,756 are perfect squares (with roots of 5, 12 and 234, respectively).
|in geometry, a straight line that touches a curve and gives the gradient of the curve at the point of contact. At a maximum, minimum, or point of inflection, the tangent to a curve has zero gradient. Also, in trigonometry, a function of an acute angle in a right triangle, defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to it; a way of expressing the gradient of a line.|
tetrahedron (plural tetrahedra)
in geometry, a solid figure (polyhedron) with four triangular faces; that is, a pyramid on a triangular base. A regular tetrahedron has equilateral triangles as its faces.
In chemistry and crystallography, tetrahedra describe the shapes of some molecules and crystals; for example, the carbon atoms in a crystal of diamond are arranged in space as a set of interconnected regular tetrahedra.
in geometry, a four-sided plane figure (quadrilateral) with no two sides parallel.
in geometry, a four-sided plane figure (quadrilateral) with only two sides parallel. If the parallel sides have lengths a and b and the perpendicular distance between them is h (the height of the trapezoid), its area A= 1/2 h(a + b).
An isosceles trapezoid has its sloping sides (legs) equal, is symmetrical about a line drawn through the midpoints of its parallel sides, and has equal base angles.
in geometry, a three-sided plane figure, the sum of whose interior angles is 180°. Triangles can be classified by the relative lengths of their sides. A scalene triangle has three sides of unequal length; an isosceles triangle has at least two equal sides; an equilateral triangle has three equal sides (and three equal angles of 60°).
Triangles can also be classified by their angle measures: a right triangle has one right (90°) angle; an acute triangle has three acute (less than 90°) angles; an obtuse triangle has one obtuse (greater than 90°) angle; an equiangular triangle has three equal angles. (All equilateral triangles are equiangular, and vice versa.) If the length of one side of a triangle is l and the perpendicular distance from that side to the opposite corner is h (the height or altitude of the triangle), its area A = 1/2 (lh).
vertex (plural vertices)
in geometry, a point shared by three or more sides of a solid figure; the point farthest from a figure's base; or the point of intersection of two sides of a plane figure or the two rays of an angle.
in geometry, the space occupied by a three-dimensional solid object. A prism (such as a cube) or a cylinder has a volume equal to the area of the base multiplied by the height. For a pyramid or cone, the volume is equal to one-third of the area of the base multiplied by the perpendicular height. The volume of a sphere is equal to 4/3 × pr3, where r is the radius. Volumes of irregular solids may be calculated by the technique of integration.