Slant Height of Truncated Cone Calculator

Slant Height Formula:

\[ h_{slant} = \sqrt{(r_{base} - r_{top})^2 + h^2} \]

Slant Height (h_slant):

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1. What is the Slant Height of a Truncated Cone?

The slant height of a truncated cone is the shortest distance along the lateral surface from the base circumference to the top circumference. It represents the length of the slanted side of the frustum.

2. How Does the Calculator Work?

The calculator uses the slant height formula:

\[ h_{slant} = \sqrt{(r_{base} - r_{top})^2 + h^2} \]

Where:

\( h_{slant} \) — Slant height of the truncated cone (m)
\( r_{base} \) — Base radius of the truncated cone (m)
\( r_{top} \) — Top radius of the truncated cone (m)
\( h \) — Height of the truncated cone (m)

Explanation: The formula is derived from the Pythagorean theorem, where the slant height forms the hypotenuse of a right triangle with legs equal to the height and the difference between the base and top radii.

3. Importance of Slant Height Calculation

Details: Calculating the slant height is essential for determining the lateral surface area of a truncated cone, which is crucial in various engineering, architectural, and manufacturing applications where material requirements need to be calculated.

4. Using the Calculator

Tips: Enter base radius, top radius, and height in meters. All values must be valid (radii ≥ 0, height > 0). The calculator will compute the slant height using the mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated cone?
A: A truncated cone (or frustum) is a cone with the apex cut off by a plane parallel to the base, resulting in two circular faces of different sizes.

Q2: How is slant height different from height?
A: Height is the vertical distance between the two bases, while slant height is the diagonal distance along the lateral surface between the two circumferences.

Q3: Can the slant height be smaller than the height?
A: No, the slant height is always greater than or equal to the height, as it forms the hypotenuse of a right triangle where the height is one of the legs.

Q4: What if the base and top radii are equal?
A: If the base and top radii are equal, the truncated cone becomes a cylinder, and the slant height equals the height.

Q5: How is slant height used in real-world applications?
A: Slant height is used to calculate lateral surface area for material estimation in construction, manufacturing of conical objects, and in various geometric calculations.