Height (geometry)

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Heights in a triangle ABC:
The heights and run outside the triangle because there is an obtuse angle at B. If these heights are extended beyond the corresponding plumb points L a and L c and the height beyond corner point B, all three straight lines intersect at the height intersection point H.

In geometry, a height is understood as a special perpendicular (perpendicular) to a line or a surface and its length . Heights play an important role in the calculation of areas and volumes ( volumes ). They can also lie outside of figures and bodies , e.g. B. with obtuse triangles .

Heights for triangles

If the perpendicular falls from one corner to the opposite side of the triangle, it cuts this side at the base of the perpendicular . The distance between the corner and the plumb line is called the height and the triangle formed by the three plumb points is also known as the height base triangle . Each triangle has exactly three heights. These intersect at a common point, the height intersection . It is inside the triangle for acute triangles and outside the triangle for obtuse triangles. In the case of a right triangle, it coincides with the right-angled corner. The heights of a triangle are also the bisectors of its height base triangle. For the heights , and in a triangle with sides , and denote in the following half the circumference of the triangle, the radii of the inner and circumference and the inner angles in the corner points . The relationship between the sides and heights of the triangle is as follows:

Beyond this relationship equation, the following applies more precisely:

This gives the following representation of the product of the three heights:

The relation to the radius of the inscribed circle is:

With right triangles, Euclid's theorem of heights plays a major role.

The distance between the two parallels is called the height in the trapezoid
Heights in the parallelogram

Height of trapezoid and parallelogram

  • A trapezoid has two opposite sides that are parallel to each other. The distance between these two parallels is called the height of the trapezoid.
  • The height of a parallelogram is the vertical distance between the opposite sides.

Heights of other geometric objects

  • With prisms and cylinders , the height is the vertical distance between the base and top surface.
  • For pyramids and cones , the height is the perpendicular distance between the tip and the base.
  • Also in higher-dimensional geometric objects, such as the hyperpyramid , the (vertical) distance of a corner point in n-dimensional space from a hypersurface lying in a hyperplane is called the height.

cylinder pyramid cone


  • Student dudes: Mathematik I, Dudenverlag, 8th edition, Mannheim 2008, pp. 192–193

Web links

Wiktionary: Height  - explanations of meanings, word origins, synonyms, translations
Wikiquote: Height  - Quotes