oblate/prolate A curved surface similar to a sphere
but lengthened or shortened in one direction is called a
spheroid. If the length of the diameter from pole to
pole is greater than the length of the diameter connecting two opposite points on the equator, then the
spheroid is said to be oblate. If, on the other hand, the
polar diameter is less than the equatorial diameter, then
the spheroid is called prolate. The Earth, for example,
is not a perfect sphere but is an oblate spheroid.
See also EARTH; ELLIPSOID; SPHERE.
oblique This term is used in a number of geometric
settings to mean either “not at right angles” or “does
not contain a right angle.” For example, two intersecting lines drawn in the plane are oblique if they meet at
an ANGLE different from 90°, or three lines meeting at a
point in three-dimensional space are oblique if they are
not mutually perpendicular. A single line drawn in the
plane is called an oblique line if it is neither horizontal
nor vertical, and an oblique coordinate system has axes
that are not at right angles.
Any angle that is not a multiple of 90° is called an
oblique angle. An oblique triangle is one that does not
contain a right angle.
An oblique cone is a cone with its vertex not
directly above the center of its base (and so the line
connecting the vertex to the center of the base is not at
right angles to the base). An oblique prism has lateral
edges that are not perpendicular to the base.
See also CARTESIAN COORDINATES; RIGHT ANGLE;
SKEW LINES.
obtuse angle An ANGLE between 90° and 180° is
called an obtuse angle. A TRIANGLE in which one of its
angles is obtuse is called an obtuse triangle. According
to the LAW OF COSINES, a triangle with side-lengths a, b,
and c, and corresponding angles A, B, C opposite those
sides, satisfies:
For the angle C to be obtuse, it must be the case that
cosC < 0, that is, a
2 + b
2 < c
2
. Thus a triangle a, b, c is
obtuse if one of the following inequalities holds: a
2 + b
2
< c
2
, c
2 + a
2 < b
2 or b
2 + c
2 < a
2
.
An angle between 180° and 360° is called a reflex
angle. An angle of 180° degrees is a straight angle, and
one of 360° is called a full turn or a PERIGON.
See also ACUTE ANGLE; PYTHAGORAS’S THEOREM;
TRIANGLE.
obverse Changing the predicate B of a statement of
the form “all A are B” from positive to negative, or
vice verse, and negating the statement as a whole produces the obverse of the statement: “no A is not B.”
For example, the obverse of the statement “all men are
mortal” is “no man is immortal.” Euler diagrams show
that any statement of the this type is logically equivalent to its obverse.
See also ARGUMENT.
cosC
a b c
ab
=
+ −
2
2
2
2
362
o
but lengthened or shortened in one direction is called a
spheroid. If the length of the diameter from pole to
pole is greater than the length of the diameter connecting two opposite points on the equator, then the
spheroid is said to be oblate. If, on the other hand, the
polar diameter is less than the equatorial diameter, then
the spheroid is called prolate. The Earth, for example,
is not a perfect sphere but is an oblate spheroid.
See also EARTH; ELLIPSOID; SPHERE.
oblique This term is used in a number of geometric
settings to mean either “not at right angles” or “does
not contain a right angle.” For example, two intersecting lines drawn in the plane are oblique if they meet at
an ANGLE different from 90°, or three lines meeting at a
point in three-dimensional space are oblique if they are
not mutually perpendicular. A single line drawn in the
plane is called an oblique line if it is neither horizontal
nor vertical, and an oblique coordinate system has axes
that are not at right angles.
Any angle that is not a multiple of 90° is called an
oblique angle. An oblique triangle is one that does not
contain a right angle.
An oblique cone is a cone with its vertex not
directly above the center of its base (and so the line
connecting the vertex to the center of the base is not at
right angles to the base). An oblique prism has lateral
edges that are not perpendicular to the base.
See also CARTESIAN COORDINATES; RIGHT ANGLE;
SKEW LINES.
obtuse angle An ANGLE between 90° and 180° is
called an obtuse angle. A TRIANGLE in which one of its
angles is obtuse is called an obtuse triangle. According
to the LAW OF COSINES, a triangle with side-lengths a, b,
and c, and corresponding angles A, B, C opposite those
sides, satisfies:
For the angle C to be obtuse, it must be the case that
cosC < 0, that is, a
2 + b
2 < c
2
. Thus a triangle a, b, c is
obtuse if one of the following inequalities holds: a
2 + b
2
< c
2
, c
2 + a
2 < b
2 or b
2 + c
2 < a
2
.
An angle between 180° and 360° is called a reflex
angle. An angle of 180° degrees is a straight angle, and
one of 360° is called a full turn or a PERIGON.
See also ACUTE ANGLE; PYTHAGORAS’S THEOREM;
TRIANGLE.
obverse Changing the predicate B of a statement of
the form “all A are B” from positive to negative, or
vice verse, and negating the statement as a whole produces the obverse of the statement: “no A is not B.”
For example, the obverse of the statement “all men are
mortal” is “no man is immortal.” Euler diagrams show
that any statement of the this type is logically equivalent to its obverse.
See also ARGUMENT.
cosC
a b c
ab
=
+ −
2
2
2
2
362
o