Slant height

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The slant height of a right circular cone is the distance from any point on the circle to the apex of the cone.

The slant height of a cone is given by the formula $\sqrt{r^2+h^2}$ , where $r$ is the radius of the circle and $h$ 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 is inscribed inside the cone, with one leg of the triangle being 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 $h$ , another leg will have length $r$ , and by the Pythagorean theorem, $r 2 + h 2 = d 2$ , and $d=\sqrt{r^2+h^2}$ gives the length of the hypotenuse, or distance from the edge of the circle to the apex of the cone. Since the cone is symmetric and the point on the edge of the circle was chosen arbitrarily, we can say with no loss of generality that the same formula holds for the distance from any point on the circle to the apex of the cone.