Concepts about Geometry
abscissa
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.
altitude
in geometry, the perpendicular distance from a vertex (corner) of a figure, such as a triangle, to the base (the side opposite the vertex).
analytic geometry
another name for coordinate geometry.
annulus ("ring" in Latin)
in geometry, the plane area between two concentric circles, making a flat ring.
arc
| 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.
asymptote
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). |  |
base
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.
Cartesian coordinates
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).
chord
| 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. |  |
circumference
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.
concave
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.
cone
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).
congruent
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.
convex
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.
coordinate
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.
coplanar
in geometry, describing lines or points that all lie in the same plane.
cube
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
curve
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.
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.
cylinder
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.
decagon
in geometry, a ten-sided polygon.
determinant
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).
dimension
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.
eccentricity
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.
equation
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).
frustum
(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.
hyperbola
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).
median
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.
metageometry
non-Euclidean geometry.
oblate
dedicated (person); adj. Geometry, flattened at poles.
oblation, n. offering; sacrifice. oblational, adj.
ordinate
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.
point
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.
polygon
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.
Pythagorean theorem
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.
QED
abbreviation for quod erat demonstrandum (Latin “which was to be proved”), added
at the end of a geometry proof.
quadratic equation
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.
rhombus
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.
right triangle
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.
sector
in geometry, part of a circle enclosed by two radii and the arc that joins them.
segment
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.
square
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).
tangent
| 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.
trapezium
in geometry, a four-sided plane figure (quadrilateral) with no two sides parallel.
trapezoid
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.
triangle
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.
volume
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.