Slant Height of Truncated Cone given Total Surface Area Calculator

Formula Used:

\[ Slant Height = \frac{Total Surface Area - \pi \times (Base Radius^2 + Top Radius^2)}{\pi \times (Base Radius + Top Radius)} \]

Slant Height (hSlant):

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

The slant height of a truncated cone is the length of the straight line connecting any point on the base circumference to the corresponding point on the top circumference. It represents the shortest distance between the two circular edges along the lateral surface.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ hSlant = \frac{TSA - \pi \times (rBase^2 + rTop^2)}{\pi \times (rBase + rTop)} \]

Where:

\( hSlant \) — Slant Height of Truncated Cone (m)
\( TSA \) — Total Surface Area of Truncated Cone (m²)
\( rBase \) — Base Radius of Truncated Cone (m)
\( rTop \) — Top Radius of Truncated Cone (m)
\( \pi \) — Mathematical constant pi (≈3.14159)

Explanation: This formula derives the slant height from the total surface area by subtracting the areas of the top and bottom circles, then dividing by the average circumference.

3. Importance of Slant Height Calculation

Details: Calculating the slant height is essential for determining the lateral surface area, understanding the geometry of the truncated cone, and for practical applications in construction and manufacturing where conical frustums are used.

4. Using the Calculator

Tips: Enter all values in consistent units (meters recommended). Ensure all values are positive numbers. The top and base radii must be different for a valid truncated cone.

5. Frequently Asked Questions (FAQ)

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

Q2: Can the slant height be less than the height?
A: No, the slant height is always greater than or equal to the perpendicular height of the truncated cone, as it's the hypotenuse of a right triangle.

Q3: What if my top radius equals the base radius?
A: If top radius equals base radius, you have a cylinder, not a truncated cone. The formula would give division by zero in this case.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact, assuming perfect measurements and a perfect truncated cone shape.

Q5: Can I use different units?
A: Yes, but all measurements must use the same units (e.g., all in centimeters or all in inches).