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