A triangle is the simplest polygon in geometry, formed by three straight line segments that connect three non-collinear points. Every triangle has three sides, three vertices, and three interior angles. Despite its simplicity, the triangle is the foundation of trigonometry, structural engineering, and countless real-world applications, from roof trusses to satellite navigation.

Types of Triangles by Sides

Triangles are classified by their side lengths into three categories. An equilateral triangle has all three sides of equal length, which also means all three interior angles are exactly 60 degrees. An isosceles triangle has exactly two sides of the same length, and the angles opposite those equal sides are also equal. A scalene triangle has no sides of equal length, so all three interior angles differ as well. Recognizing the type of triangle you are working with often simplifies the calculation because equilateral and isosceles triangles have built-in symmetry that reduces the number of unknowns.

Types of Triangles by Angles

Triangles can also be classified by their largest interior angle. An acute triangle has all three angles less than 90 degrees. A right triangle has one angle that is exactly 90 degrees, making it the cornerstone of the Pythagorean theorem and trigonometric ratios. An obtuse triangle has one angle greater than 90 degrees. Every triangle falls into exactly one of these angle-based categories while simultaneously belonging to one of the side-based categories, so you could have, for instance, an isosceles right triangle.

Area Formulas

The most familiar area formula is A = 1/2 × base × height, where the base is any side of the triangle and the height is the perpendicular distance from that base to the opposite vertex. When the height is not directly measurable, Heron's formula provides an elegant alternative. First compute the semi-perimeter s = (a + b + c) / 2, then calculate A = √(s(s − a)(s − b)(s − c)). For problems involving two sides and the included angle, the SAS area formula is A = 1/2 × a × b × sin(C). Each formula serves a different set of known values, so choosing the right one is the first step in any triangle area problem.

The Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, a² + b² = c², where c is the hypotenuse (the side opposite the 90-degree angle) and a and b are the two legs. This relationship allows you to find any missing side when the other two are known. The converse is equally useful: if the side lengths of a triangle satisfy the equation, the triangle is a right triangle. Well-known Pythagorean triples include 3-4-5, 5-12-13, and 8-15-17.

Angle Sum Property and Trigonometric Relationships

In Euclidean geometry, the interior angles of any triangle always sum to exactly 180 degrees. This means that knowing two angles immediately gives you the third. The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C), connecting each side to its opposite angle. The Law of Cosines generalizes the Pythagorean theorem to any triangle: c² = a² + b² − 2ab·cos(C). Together, these laws allow you to solve any triangle when at least three independent elements (sides or angles) are known.

Real-World Applications

• Surveying and Navigation: Surveyors use triangulation to measure distances to remote points by forming triangles and applying trigonometric laws.
• Architecture and Engineering: Triangular trusses distribute weight efficiently, making them essential in bridge and roof design.
• Computer Graphics: 3D models are rendered as meshes of triangles because any polygon can be decomposed into triangles for efficient computation.
• Astronomy: Stellar parallax uses triangle geometry to measure distances to nearby stars from different positions in Earth's orbit.

Work through these three examples to practice the most common triangle calculation techniques: base-height area, Heron's formula, and finding a missing angle.