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To calculate the total surface area of the cone, we need to find both the curved (lateral) surface area and the base area. The total surface area \( A \) of a cone is given by the formula:

\[
A = \pi r l + \pi r^2
\]

Where:
- \( r \) is the radius of the base,
- \( l \) is the slant height, and
- \( \pi r^2 \) is the area of the base.

### Step 1: Find the slant height \( l \)
We can calculate the slant height \( l \) using the Pythagorean theorem. The slant height \( l \), radius \( r \), and height \( h \) form a right triangle, where:

\[
l = \sqrt{r^2 + h^2}
\]

Given:
- \( r = 24 \, \text{cm} \) (radius),
- \( h = 70 \, \text{cm} \) (height),

we can calculate \( l \) as:

\[
l = \sqrt{24^2 + 70^2} = \sqrt{576 + 4900} = \sqrt{5476} = 74 \, \text{cm}
\]

### Step 2: Calculate the total surface area
Now that we have the slant height, we can calculate the total surface area. Using the formula:

\[
A = \pi r l + \pi r^2
\]

Substituting the values:

\[
A = \pi (24)(74) + \pi (24)^2
\]
\[
A = 1776\pi + 576\pi = 2352\pi \, \text{cm}^2
\]

So, the total surface area of the cone is:

\[
A = 2352\pi \, \text{cm}^2
\]

Let me know if you need any further explanation!

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Here are 5 related questions for further exploration:
1. How would the total surface area change if the height were doubled?
2. What is the volume of this cone?
3. How can you find the lateral surface area using only the height and radius?
4. What would the total surface area be if the radius were halved but the height remained the same?
5. How does the surface area compare between a cone and a cylinder with the same radius and height?

**Tip:** When calculating surface areas of cones, always ensure you distinguish between the slant height and the vertical height to avoid confusion.

Determine the total surface area of a cone with a radius of 24 cm and a height of 70 cm. Give your answer in terms of π.

Learn how to calculate the total surface area of a cone with a radius of 24 cm and a height of 70 cm. This step-by-step guide explains the use of the Pythagorean theorem to find the slant height and applies surface area formulas. Suitable for students in Grades 9-10.

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