Slant Height Calculator Cone

Cone Slant Height Formula:

\[ l = \sqrt{r^2 + h^2} \]

Slant Height (l):

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1. What is the Slant Height of a Cone?

The slant height (l) of a cone is the distance from the apex (top point) to any point on the edge of the circular base. It forms the hypotenuse of a right triangle with the radius (r) and height (h) of the cone.

2. How Does the Calculator Work?

The calculator uses the slant height formula:

\[ l = \sqrt{r^2 + h^2} \]

Where:

\( l \) — Slant height (units)
\( r \) — Radius of the base (units)
\( h \) — Height of the cone (units)

Explanation: The formula is derived from the Pythagorean theorem, as the slant height forms the hypotenuse of a right triangle with the radius and height.

3. Importance of Slant Height Calculation

Details: The slant height is crucial for calculating the lateral surface area of a cone, which is important in construction, manufacturing, and geometry problems.

4. Using the Calculator

Tips: Enter the radius and height of the cone in the same units. Both values must be positive numbers. The result will be in the same units as the input.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between height and slant height?
A: Height is the perpendicular distance from base to apex, while slant height is the diagonal distance along the cone's side.

Q2: Can I use this for truncated cones?
A: No, this formula is only for regular cones. Truncated cones (frustums) have a different formula for slant height.

Q3: What units should I use?
A: Any consistent length units (cm, m, inches, etc.) can be used as long as both radius and height are in the same units.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact for perfect right circular cones.

Q5: What's the relationship between slant height and surface area?
A: Lateral surface area = π × r × l, where l is the slant height.