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To find the total surface area of the cone, we need to consider both the **curved surface area** and the **area of the base**.

1. **Curved surface area** formula:  
   \[
   \text{Curved Surface Area} = \pi r l
   \]
   where \( r = 4 \, \text{cm} \) (radius) and \( l = 5 \, \text{cm} \) (slant height).

2. **Area of the base** of the cone:  
   The base of the cone is a circle with area:
   \[
   \text{Base Area} = \pi r^2
   \]

### Step-by-Step Calculation:
1. **Curved surface area**:
   \[
   \pi r l = \pi \times 4 \times 5 = 20\pi \, \text{cm}^2
   \]

2. **Base area**:
   \[
   \pi r^2 = \pi \times 4^2 = 16\pi \, \text{cm}^2
   \]

3. **Total surface area**:
   The total surface area is the sum of the curved surface area and the base area:
   \[
   \text{Total Surface Area} = 20\pi + 16\pi = 36\pi \, \text{cm}^2
   \]

Thus, the total surface area of the cone is **36π cm²**.

Would you like more details on any part of this solution?

Here are five related questions to deepen your understanding:
1. How would the total surface area change if the radius were doubled but the slant height remained the same?
2. What is the difference between slant height and perpendicular height in a cone?
3. Can you derive the formula for the volume of a cone?
4. How does the surface area formula change for an open cone (without a base)?
5. How would you calculate the surface area of a cone if the slant height were unknown but the perpendicular height was given?

**Tip:** Always ensure that the radius and slant height are in the same units when calculating surface area.

Work out the total surface area of a cone with a radius of 4 cm and a slant height of 5 cm. Give your answer in terms of π.

Learn how to calculate the total surface area of a cone with a radius of 4 cm and a slant height of 5 cm. This problem demonstrates key geometric concepts, including the curved surface area and base area formulas. Suitable for Grades 7-9.

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