Triangle area

Triangle area is the measure of the plane region enclosed by a triangle. It equals half the product of a base and its corresponding perpendicular height.

Base and height: A base is a chosen side of a triangle; its corresponding height is the perpendicular distance to the opposite vertex. Their product, divided by two, gives the triangle’s area.
Area: A measure of the size of a two-dimensional region, expressed in square units. Triangle area is one specific case of this general geometric measure.
Surveying: The measurement and mapping of land, boundaries, and spatial features. Surveyors divide irregular plots into triangles to calculate their areas.
Spherical triangle: A triangle drawn on a sphere whose sides are arcs of great circles. Its area depends on spherical excess, not the flat half-base-times-height rule.
Heron's formula: A formula for a triangle’s area using its three side lengths and semiperimeter. It computes area when side lengths are known but no height is given.
Perpendicularity: The relation between lines or directions that meet at a right angle. A triangle’s height is measured perpendicular to its chosen base.
Triangulation: A method for determining positions or distances by forming triangles from measured baselines and angles. Once a triangle’s dimensions are known, its area helps quantify mapped regions.
Hyperbolic geometry: A non-Euclidean geometry in which parallel lines diverge and triangle angle sums are less than 180 degrees. Triangle areas follow curvature-dependent relations rather than the Euclidean formula alone.
Trigonometric area formula: A triangle-area formula stating that two sides and their included angle give area as one-half their product times the angle’s sine. It replaces an explicitly measured height with the included angle.
Parallelogram: A quadrilateral with both pairs of opposite sides parallel. Two congruent copies of a triangle can form a parallelogram with twice its area.

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