Slant Height of Frustum of Cone given Total Surface Area and Top Area Calculator

Formula Used:

\[ h_{slant} = \frac{TSA - A_{top} - A_{base}}{\pi (r_{top} + r_{base})} \]

Total Surface Area (TSA):

m²

Top Area (A_top):

m²

Base Area (A_base):

m²

Top Radius (r_top):

Base Radius (r_base):

Slant Height (h_slant):

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

The slant height of a frustum of a cone is the shortest distance between the edges of the two bases along the lateral surface. It is an important measurement in geometry and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ h_{slant} = \frac{TSA - A_{top} - A_{base}}{\pi (r_{top} + r_{base})} \]

Where:

\( h_{slant} \) — Slant height of the frustum
\( TSA \) — Total surface area of the frustum
\( A_{top} \) — Area of the top circular base
\( A_{base} \) — Area of the bottom circular base
\( r_{top} \) — Radius of the top circular base
\( r_{base} \) — Radius of the bottom circular base

Explanation: The formula calculates the slant height by subtracting the areas of the two bases from the total surface area and dividing by the circumference of the average radius.

3. Importance of Slant Height Calculation

Details: Calculating the slant height is crucial for determining the lateral surface area, volume, and other geometric properties of a frustum. It's essential in architectural design, manufacturing, and various engineering applications.

4. Using the Calculator

Tips: Enter all values in consistent units (meters recommended). Ensure all values are positive and the total surface area is greater than the sum of the top and base areas.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between slant height and height?
A: The height is the perpendicular distance between the two bases, while the slant height is the distance along the lateral surface between the edges of the two bases.

Q2: Can this formula be used for any frustum?
A: This specific formula applies only to frustums of right circular cones where the bases are parallel and concentric.

Q3: What if I only have the radii and height?
A: If you have the radii and perpendicular height, you can calculate slant height using the Pythagorean theorem: \( h_{slant} = \sqrt{h^2 + (r_{base} - r_{top})^2} \)

Q4: Why is π used in the formula?
A: π is used because we're dealing with circular bases, and the lateral surface area calculation involves circular geometry.

Q5: What units should I use?
A: Use consistent units throughout (all meters, all centimeters, etc.). The result will be in the same linear unit as the input measurements.