Overview

An obtuse angle is any angle whose measure is strictly greater than 90 degrees and strictly less than 180 degrees. In radian measure this range is between π/2 and π. The term describes a broad class of angles that are larger than an acute angle but smaller than a straight angle (180°). For a concise definition, one can state: 90° < θ < 180°.

Characteristics and basic properties

Several simple facts distinguish obtuse angles from other types:

  • Trigonometry: cosine of an obtuse angle is negative, sine is positive, and tangent is negative (between π/2 and π).
  • Complements and supplements: an obtuse angle cannot have a complement (sum 90°) but it does have a unique supplement that is acute (sums to 180°).
  • Triangles: a triangle can have at most one obtuse interior angle. The side opposite the obtuse angle is the longest in such a triangle.

Occurrence in polygons and examples

Interior angles of many common polygons are obtuse. For example, each interior angle of a regular pentagon measures 108°, and each interior angle of a regular hexagon measures 120°; both are obtuse. The general formula for an interior angle of a regular n-gon is ((n-2)·180°)/n, which produces obtuse values for moderately small n (e.g., n=5,6).

Measurement and construction

Obtuse angles are measured with standard angle tools such as a protractor. On a coordinate plane an obtuse angle between two vectors corresponds to a dot product that is negative. When drawing geometric figures, obtuse angles change the locations of important triangle centers: in an obtuse triangle the circumcenter and orthocenter lie outside the triangle, while the centroid remains inside.

History and terminology

The English word "obtuse" comes from Latin roots meaning "blunted" or "dull," reflecting the idea of an angle that is more open than a right angle. Classical geometry—dating back to Euclid and earlier traditions—routinely distinguishes acute, right and obtuse angles, and modern elementary geometry continues to use the same categories.

Practical importance and distinctions

Recognizing obtuse angles is useful in design, architecture and computer graphics because they affect visibility, required clearances and structural behavior. Distinguish an obtuse angle from a reflex angle (greater than 180°) and from straight or right angles; this classification controls many geometric constructions and the sign of trigonometric functions.