Slant height of a right cone

Definition: The distance from the top of a cone, down the side to a point on the edge of the base.
Try this Drag the orange dots to adjust the radius and height of the cone and note how the slant height changes.

There are three dimensions of a cone.

  • The vertical height (or altitude) which is the perpendicular distance from the top down to the base.
  • The radius of the circular base
  • The slant height which is the distance from the top, down the side, to a point on the base circumference.

These three are related and we only need any two to define the cone. We can then find the third missing dimension. From the figure above, we can see that the three dimensions form a right triangle, with the slant height as the hypotenuse, so we can use the Pythagorean theorem to solve it*.

Drag either orange dot in the top figure and note how the slant height is calculated from the radius and altitude.

* We can actually use any method of solving this triangle we like. It just depends on what you are given and personal preference. See Solving the triangle.

Finding the slant height

By applying the Pythagorean Theorem, the slant height is given by the formula: where r is the base radius and h is the altitude.

If you are given the slant height

By rearranging the terms in the Pythagorean theorem, we can solve for other lengths:

  • The radius r can be found using the formula where s is the slant height h is the altitude.
  • The altitude h can be found using the formula where s is the slant height r is the base radius.

Things to try

  • In the top figure, click "hide details".
  • Drag the orange dots to set the radius and height of the cone.
  • Calculate the slant height of the cone using the formula
  • Click "show details" to check your answer.
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