Slant height

# Slant height

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Encyclopedia
The slant height of a right circular cone is the distance from any point on the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

to the apex of the cone.

The slant height of a cone is given by the formula , where is the radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

of the circle and is the height from the center of the circle to the apex of the cone.

It is trivial to see why this formula holds true. If a right triangle
Right triangle
A right triangle or right-angled triangle is a triangle in which one angle is a right angle . The relation between the sides and angles of a right triangle is the basis for trigonometry.-Terminology:The side opposite the right angle is called the hypotenuse...

is inscribed inside the cone, with one leg of the triangle being the line segment
Line segment
In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...

from the center of the circle to its radius, and the second leg of the triangle being from the apex of the cone to the center of the circle, then one leg will have length , another leg will have length , and by the Pythagorean theorem
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...

, , and gives the length of the circle to the apex of the cone.