
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the triangle inequality states that for any
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
[
] This statement permits the inclusion of
degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If , , and are the lengths of the sides of a triangle then the triangle inequality states that
:
with equality only in the degenerate case of a triangle with zero area.
In
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
and some other geometries, the triangle inequality is a theorem about vectors and vector lengths (
norms):
:
where the length of the third side has been replaced by the length of the vector sum . When and are
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, they can be viewed as vectors in
, and the triangle inequality expresses a relationship between
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
s.
In Euclidean geometry, for
right triangle
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 ( turn or 90 degrees).
The side opposite to the right angle i ...
s the triangle inequality is a consequence of the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
, and for general triangles, a consequence of the
law of cosines
In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides , , and , opposite respective angles , , and (see ...
, although it may be proved without these theorems. The inequality can be viewed intuitively in either
or
. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a angle and two angles, making the three
vertices collinear
In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...
, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.
In
spherical geometry
300px, A sphere with a spherical triangle on it.
Spherical geometry or spherics () is the geometry of the two-dimensional surface of a sphere or the -dimensional surface of higher dimensional spheres.
Long studied for its practical applicati ...
, the shortest distance between two points is an arc of a
great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Discussion
Any arc of a great circle is a geodesic of the sphere, so that great circles in spher ...
, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in ) with those endpoints.
[
][
]
The triangle inequality is a ''defining property'' of
norms and measures of
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...
. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s,
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
s, the
Lp spaces (), and
inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
s.
Euclidean geometry

The triangle inequality theorem is stated in
Euclid's Elements
The ''Elements'' ( ) is a mathematics, mathematical treatise written 300 BC by the Ancient Greek mathematics, Ancient Greek mathematician Euclid.
''Elements'' is the oldest extant large-scale deductive treatment of mathematics. Drawing on the w ...
, Book I, Proposition 20:
''
��in the triangle ABC the sum of any two sides is greater than the remaining one, that is, the sum of BA and AC is greater than BC, the sum of AB and BC is greater than AC, and the sum of BC and CA is greater than AB.''
[
Euclid proved the triangle inequality for distances in ]plane geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
using the construction in the figure.[
] Beginning with triangle , an isosceles triangle is constructed with one side taken as and the other equal leg along the extension of side . It then is argued that angle has larger measure than angle , so side is longer than side . However:
:
so the sum of the lengths of sides and is larger than the length of . This proof appears in Euclid's Elements
The ''Elements'' ( ) is a mathematics, mathematical treatise written 300 BC by the Ancient Greek mathematics, Ancient Greek mathematician Euclid.
''Elements'' is the oldest extant large-scale deductive treatment of mathematics. Drawing on the w ...
, Book I, Proposition 20.[
]
Mathematical expression of the constraint on the sides of a triangle
For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths that are all positive and excludes the degenerate case of zero area):
:
A more succinct form of this inequality system can be shown to be
:
Another way to state it is
:
implying
:
and thus that the longest side length is less than the semiperimeter.
A mathematically equivalent formulation is that the area of a triangle with sides must be a real number greater than zero. Heron's formula for the area is
:
In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero).
The triangle inequality provides two more interesting constraints for triangles whose sides are , where and is the golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if
\fr ...
, as
:
:
Right triangle
In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum.[
]
The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle . An isosceles triangle is constructed with equal sides . From the triangle postulate, the angles in the right triangle satisfy:
:
Likewise, in the isosceles triangle , the angles satisfy:
:
Therefore,
:
and so, in particular,
:
That means side , which is opposite to angle , is shorter than side , which is opposite to the larger angle . But . Hence:
:
A similar construction shows , establishing the theorem.
An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point :[
] (i) as depicted (which is to be proved), or (ii) coincident with (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle , which would violate the triangle postulate), or lastly, (iii) interior to the right triangle between points and (in which case angle is an exterior angle of a right triangle and therefore larger than , meaning the other base angle of the isosceles triangle also is greater than and their sum exceeds in violation of the triangle postulate).
This theorem establishing inequalities is sharpened by Pythagoras' theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides.
Examples of use
Consider a triangle whose sides are in an arithmetic progression
An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that ...
and let the sides be . Then the triangle inequality requires that
:
To satisfy all these inequalities requires
: