Calculating the surface area of a cone involves understanding and applying the formula that combines both the lateral surface area and the base area of the cone. Here's a step-by-step guide to help you determine the surface area of a cone when the slant height \( l \) and the radius of the base \( r \) are given.
Formula to Calculate the Surface Area of a Cone
The surface area \( SA \) of a cone can be calculated using the following formula:
\[ SA = \pi \cdot r \cdot l + \pi \cdot r^2 \]
Where:
- \( SA \) is the surface area of the cone.
- \( r \) is the radius of the base of the cone.
- \( l \) is the slant height of the cone.
Explanation of the Formulas
The formula consists of two parts:
1. The lateral surface area: \( \pi \cdot r \cdot l \)
2. The base area: \( \pi \cdot r^2 \)
Step-by-Step Calculation
Let's go through an example to illustrate how to use these formulas.
Example:
Given:
- \( r = 5 \) units (the radius of the base)
- \( l = 10 \) units (the slant height)
We want to find the surface area of the cone.
Step 1: Identify the Given Values
Given:
- \( r = 5 \) units
- \( l = 10 \) units
Step 2: Use the Formula to Calculate the Surface Area
1. Calculate the lateral surface area:
\[\pi \cdot r \cdot l = \pi \cdot 5 \cdot 10\]
\[= 50\pi\]
2. Calculate the base area:
\[\pi \cdot r^2 = \pi \cdot 5^2\]
\[= 25\pi\]
Step 3: Add Both Areas to Find the Total Surface Area
\[SA = 50\pi + 25\pi\]
\[= 75\pi\]
Step 4: Calculate the Final Value
Using \( \pi \approx 3.14159 \):
\[SA \approx 75 \cdot 3.14159\]
\[SA \approx 235.62 \text{ units}^2\]
Final Value
The surface area of the cone with a radius of 5 units and a slant height of 10 units is approximately 235.62 square units.
By following these steps, you can easily calculate the surface area of a cone when you have the slant height and the radius of the base. This method combines the lateral surface area and the base area to give you the total surface area.
