Calculating the surface area of a truncated cone, also known as a conical frustum, involves understanding a specific geometric formula. This article will guide you through the process using the formula:
\[ SA = \pi \cdot (R + r) \cdot l + \pi \cdot (R^2 + r^2) \]
Where:
- \( SA \) is the surface area
- \( R \) is the base radius
- \( r \) is the top radius
- \( l \) is the slant height
Explanation of the Formula
The formula consists of two parts:
1. The lateral surface area of the frustum: \( \pi \cdot (R + r) \cdot l \)
2. The surface areas of the two circular ends: \( \pi \cdot (R^2 + r^2) \)
Combining these gives the total surface area of the truncated cone.
Example 1: Calculating the Surface Area of a Flower Pot
Problem: A flower pot shaped like a truncated cone has a base radius \( R \) of 10 cm, a top radius \( r \) of 8 cm, and a slant height \( l \) of 15 cm. Calculate the surface area.
Calculation:
Given:
- \( R = 10 \, \text{cm} \)
- \( r = 8 \, \text{cm} \)
- \( l = 15 \, \text{cm} \)
Using the formula:
\[ SA = \pi \cdot (R + r) \cdot l + \pi \cdot (R^2 + r^2) \]
\[ SA = \pi \cdot (10 + 8) \cdot 15 + \pi \cdot (10^2 + 8^2) \]
\[ SA = \pi \cdot 18 \cdot 15 + \pi \cdot (100 + 64) \]
\[ SA = 270\pi + 164\pi \]
\[ SA = 434\pi \]
\[ SA \approx 1363.56 \, \text{cm}^2 \]
Answer: The surface area of the flower pot is approximately \( 1363.56 \, \text{cm}^2 \).
Example 2: Calculating the Surface Area of a Drinking Cup
Problem: A drinking cup shaped like a truncated cone has a base radius \( R \) of 5 cm, a top radius \( r \) of 3 cm, and a slant height \( l \) of 12 cm. Calculate the surface area.
Calculation:
Given:
- \( R = 5 \, \text{cm} \)
- \( r = 3 \, \text{cm} \)
- \( l = 12 \, \text{cm} \)
Using the formula:
\[ SA = \pi \cdot (R + r) \cdot l + \pi \cdot (R^2 + r^2) \]
\[ SA = \pi \cdot (5 + 3) \cdot 12 + \pi \cdot (5^2 + 3^2) \]
\[ SA = \pi \cdot 8 \cdot 12 + \pi \cdot (25 + 9) \]
\[ SA = 96\pi + 34\pi \]
\[ SA = 130\pi \]
\[ SA \approx 408.41 \, \text{cm}^2 \]
Answer: The surface area of the drinking cup is approximately \( 408.41 \, \text{cm}^2 \).
Example 3: Calculating the Surface Area of an Industrial Hopper
Problem: An industrial hopper shaped like a truncated cone has a base radius \( R \) of 20 cm, a top radius \( r \) of 10 cm, and a slant height \( l \) of 25 cm. Calculate the surface area.
Calculation:
Given:
- \( R = 20 \, \text{cm} \)
- \( r = 10 \, \text{cm} \)
- \( l = 25 \, \text{cm} \)
Using the formula:
\[ SA = \pi \cdot (R + r) \cdot l + \pi \cdot (R^2 + r^2) \]
\[ SA = \pi \cdot (20 + 10) \cdot 25 + \pi \cdot (20^2 + 10^2) \]
\[ SA = \pi \cdot 30 \cdot 25 + \pi \cdot (400 + 100) \]
\[ SA = 750\pi + 500\pi \]
\[ SA = 1250\pi \]
\[ SA \approx 3926.99 \, \text{cm}^2 \]
Answer: The surface area of the industrial hopper is approximately \( 3926.99 \, \text{cm}^2 \).
Conclusion
Calculating the surface area of a truncated cone involves combining the lateral surface area and the areas of the two circular ends using the formula \( SA = \pi \cdot (R + r) \cdot l + \pi \cdot (R^2 + r^2) \). The examples provided demonstrate how this formula can be applied to various real-life objects, providing a clear understanding of the calculations involved.
