Calculating the surface area of a cone involves knowing the radius of the base and the slant height. When the height is given instead of the slant height, we can use trigonometry to find the slant height first. Here's a step-by-step guide to help you determine the surface area of a cone when the height and the base radius 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.
Finding the Slant Height
When the height \( h \) of the cone is given, we can find the slant height \( l \) using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
Step-by-Step Calculation
Let's go through an example to illustrate how to use these formulas.
Example:
Given:
- \( r = 3 \) units (the radius of the base)
- \( h = 4 \) units (the height of the cone)
We want to find the surface area of the cone.
Step 1: Identify the Given Values
Given:
- \( r = 3 \) units
- \( h = 4 \) units
Step 2: Use the Pythagorean Theorem to Find the Slant Height
\[ l = \sqrt{r^2 + h^2} \]
\[ l = \sqrt{3^2 + 4^2} \]
\[ l = \sqrt{9 + 16} \]
\[ l = \sqrt{25} \]
\[ l = 5 \] units
Step 3: Use the Surface Area Formula
1. Calculate the lateral surface area:
\[ \pi \cdot r \cdot l = \pi \cdot 3 \cdot 5 = 15 \pi \]
2. Calculate the base area:
\[\pi \cdot r^2 = \pi \cdot 3^2 = 9\pi\]
Step 4: Add Both Areas to Find the Total Surface Area
\[ SA = 15\pi + 9\pi = 24\pi \]
Step 5: Calculate the Final Value
Using \( \pi \approx 3.14159 \):
\[SA \approx 24 \cdot 3.14159\]
\[SA \approx 75.40 \text{ units}^2\]
Final Value
The surface area of the cone with a radius of 3 units and a height of 4 units is approximately 75.40 square units.
By following these steps, you can easily calculate the surface area of a cone when you have the height and the radius of the base. This method involves finding the slant height using the Pythagorean theorem and then applying the surface area formula.
