Triangles — Mathematics Knowledge System

Overview

A triangle is a polygon with three vertices and three sides, the simplest possible polygon and one of the basic shapes of geometry. Its three vertices are zero-dimensional points, and its three sides are line segments joining them in pairs. A triangle has three interior angles, and a foundational fact of Euclidean geometry is that these three angles always sum to 180° (a straight angle). Triangles are classified by side lengths (scalene, isosceles, equilateral) and by angles (acute, right, obtuse), and the area of any triangle equals one-half the product of a base and the corresponding height.

Intuition

A triangle is the most rigid polygon: given three fixed side lengths, there is only one possible triangle shape (up to reflection) — unlike a quadrilateral, which can flex like a hinge even with fixed side lengths. This rigidity is why triangles are used throughout engineering (trusses, bridges, scaffolding): they cannot deform without one of their sides changing length. The angle-sum property (always 180°) is a direct consequence of Euclid's parallel postulate — draw a line through one vertex parallel to the opposite side, and alternate interior angles show the three angles fit together to exactly form a straight line.

Interactive Graph

Formal Definition

Definition

For a triangle with vertices A, B, C, side lengths a, b, c (opposite the respectively-named vertices), interior angles α, β, γ, base b and corresponding height h, and semiperimeter s:

\alpha + \beta + \gamma = 180^\circ

Angle-sum theorem

T = \tfrac{1}{2}bh

Area from base and height

T = \tfrac{1}{2}ab\sin\gamma

Area from two sides and the included angle

s = \tfrac{1}{2}(a+b+c)

Semiperimeter

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

Heron's formula for area from side lengths alone

Notation

Notation	Meaning
$\triangle ABC$	Triangle with vertices A, B, C
$a, b, c$	Side lengths, each named opposite its corresponding vertex (a is opposite A, etc.)
$\alpha, \beta, \gamma$	Interior angles at vertices A, B, C respectively
$s$	Semiperimeter, half the perimeter

Properties

Triangle angle-sum theorem

\alpha + \beta + \gamma = 180^\circ

Condition: Holds for every triangle in Euclidean (flat) geometry

Example: A triangle with angles 90°, 60°, 30° sums to exactly 180°.

Triangle inequality

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

Condition: Necessary and sufficient for three positive lengths a, b, c to form a valid (possibly degenerate if equality holds) triangle: the sum of any two sides must exceed the third.

Example: Sides 2, 3, 4 form a valid triangle since 2+3=5>4, 2+4=6>3, 3+4=7>2. Sides 1, 2, 5 do NOT (1+2=3 < 5).

Exterior angle theorem

\theta_{\text{ext}} = \alpha + \beta

Condition: An exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.

Classification by sides

\text{Scalene: all sides different. Isosceles: at least two sides equal. Equilateral: all three sides equal.}

Classification by angles

\text{Acute: all angles} < 90^\circ.\ \text{Right: one angle} = 90^\circ.\ \text{Obtuse: one angle} > 90^\circ.

Applications

Truss and frame structures use triangular substructures because a triangle is rigid — it cannot change shape without changing a side length, unlike quadrilaterals.

Worked Examples

Use the angle-sum theorem: all three angles sum to 180°.
$\gamma = 180^\circ - 50^\circ - 70^\circ = 60^\circ$

Answer: 60°

Practice Problems

Difficulty 2/10

A triangle has sides 7 and 10. What is the range of possible lengths for the third side x?

Difficulty 3/10

Find the area of a triangle with sides 5, 6, and 7 using Heron's formula.

Common Mistakes

Common Mistake

Thinking any three lengths can form a triangle as long as they're positive.

The triangle inequality must hold for all three pairs: the sum of any two sides must strictly exceed the third. If the sum equals the third side, the 'triangle' degenerates into a straight line segment.

Common Mistake

Believing the angle-sum theorem depends on the triangle's size or shape.

Every Euclidean triangle — no matter how large, small, or skewed — has interior angles summing to exactly 180°. This fails only in non-Euclidean geometries (e.g. on a sphere, angles sum to more than 180°).

Summary

A triangle's three interior angles always sum to exactly 180° in Euclidean geometry.
The triangle inequality (a+b>c for every pair) is necessary and sufficient for three lengths to form a triangle.
Triangles are classified by sides (scalene/isosceles/equilateral) and by angles (acute/right/obtuse).
Area = ½·base·height, or ½ab·sin(γ), or via Heron's formula √(s(s-a)(s-b)(s-c)) from side lengths alone.
Triangles are the only rigid polygon, which is why they're used structurally in trusses and frames.

References

WebsiteWikipedia — Triangle