# Height (geometry)

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.${\ displaystyle h_ {a}}$${\ displaystyle h_ {c}}$${\ displaystyle h_ {b}}$

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 . ${\ displaystyle h}$

## 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: ${\ displaystyle h_ {a}}$${\ displaystyle h_ {b}}$${\ displaystyle h_ {c}}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle c}$${\ displaystyle s = {\ frac {a + b + c} {2}}}$${\ displaystyle r, \, R}$${\ displaystyle \ alpha, \, \ beta, \, \ gamma}$${\ displaystyle A, \, B, \, C}$

${\ displaystyle h_ {a}: h_ {b}: h_ {c} = {\ frac {1} {a}}: {\ frac {1} {b}}: {\ frac {1} {c}} }$

Beyond this relationship equation, the following applies more precisely:

{\ displaystyle {\ begin {aligned} h_ {a} & = {\ frac {bc} {2R}} = c \ sin (\ beta) = b \ sin (\ gamma) \\ h_ {b} & = { \ frac {ac} {2R}} = c \ sin (\ alpha) = a \ sin (\ gamma) \\ h_ {c} & = {\ frac {ab} {2R}} = b \ sin (\ alpha ) = a \ sin (\ beta) \\\ end {aligned}}}

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

${\ displaystyle h_ {a} h_ {b} h_ {c} = {\ frac {(abc) ^ {2}} {8R ^ {3}}}}$

The relation to the radius of the inscribed circle is:

${\ displaystyle {\ frac {1} {h_ {a}}} + {\ frac {1} {h_ {b}}} + {\ frac {1} {h_ {c}}} = {\ frac {1 } {r}}}$

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.

## literature

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