Mathematics · Euclidean Geometry

Triangle Perimeter (Geometry)

Reference entry · last updated September 13, 2026

Triangle perimeter denotes the total linear length of the three line segments that bound a triangle in the Euclidean plane [1].

1. First principles and basic formula

In Euclidean geometry, a triangle is a polygon with three vertices and three edges. Let the lengths of the three sides be \(a\), \(b\), and \(c\). The perimeter \(P\) is the scalar sum of these lengths [1]:

$$P = a + b + c$$

Perimeter is a one-dimensional measure. Its physical unit matches the unit of edge length (such as meters, centimeters, or units in a Cartesian coordinate system) [1].

On a Cartesian coordinate plane with vertices \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\), side lengths derive from the Euclidean distance formula:

$$P = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} + \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2} + \sqrt{(x_1 - x_3)^2 + (y_1 - y_3)^2}$$

2. Special triangles

Symmetries reduce the number of independent measurements needed to determine \(P\) [1, 2]:

• Equilateral triangle: All three sides have equal length \(a\).

  $$P = 3a$$

• Isosceles triangle: Two legs have equal length \(a\) and the base has length \(b\).

  $$P = 2a + b$$

• Right triangle: Legs \(a\) and \(b\) form a right angle, with hypotenuse \(c = \sqrt{a^2 + b^2}\) by the Pythagorean theorem [2].

  $$P = a + b + \sqrt{a^2 + b^2}$$

• Isosceles right triangle (45-45-90): Legs have length \(a\), yielding hypotenuse \(a\sqrt{2}\).

  $$P = 2a + a\sqrt{2} = a(2 + \sqrt{2})$$

3. Semi-perimeter and geometric applications

The semi-perimeter \(s\) is defined as half of the total perimeter [1, 3]:

$$s = \frac{P}{2} = \frac{a + b + c}{2}$$

The semi-perimeter appears in fundamental geometric identities:

• Heron's formula: Computes the area \(K\) of a triangle from its side lengths without requiring an explicit height [3]:

  $$K = \sqrt{s(s - a)(s - b)(s - c)}$$

• Inradius: The radius \(r\) of the incircle tangent to all three sides satisfies \(K = r \cdot s\), so [1, 3]:

  $$r = \frac{K}{s} = \frac{2K}{P}$$

4. Inequality constraints and existence bounds

A set of three positive real numbers \(\{a, b, c\}\) forms a non-degenerate triangle if and only if each side is strictly shorter than the sum of the remaining two sides. This is the triangle inequality theorem [2]:

$$a + b > c, \quad a + c > b, \quad b + c > a$$

Consequently, no single edge may exceed or equal the semi-perimeter:

$$\max(a, b, c) < s = \frac{P}{2}$$

Degenerate triangles where \(a + b = c\) collapse into a line segment of length \(c\), yielding perimeter \(P = 2c\) with zero interior area.

See also

• Triangle Inequality Theorem
• Polygon Perimeter

References

1. ^H.S.M. Coxeter, Introduction to Geometry, 2nd ed., John Wiley & Sons, 1969.
2. ^Euclid, Elements, Book I, Definitions 20–21 (Classification of Triangles), Proposition 20 (Triangle Inequality), and Proposition 47 (Pythagorean Theorem).
3. ^Heron of Alexandria, Metrica, Book I (Semi-perimeter, Heron's Formula, and Incircle Radius).