Calculating Cone Surface Area

To find the whole surface area of a cone, we need to calculate the sum of the area of its circular base and its lateral surface area. The formula for the total surface area (TSA) of a cone is given by:

TSA = Area of Base + Lateral Surface Area

The area of the circular base is $\pi r^2$, where $r$ is the base radius.

The lateral surface area is $\pi r l$, where $r$ is the base radius and $l$ is the slant height of the cone.

So, the total surface area formula is:

\( \text{TSA} = \pi r^2 + \pi r l = \pi r (r + l) \)

Determining the Slant Height of the Cone

We are given the base radius ($r$) and the height ($h$) of the cone, but not the slant height ($l$). The radius, height, and slant height of a cone form a right-angled triangle, with the slant height being the hypotenuse. We can use the Pythagorean theorem to find the slant height:

\( l^2 = r^2 + h^2 \)

\( l = \sqrt{r^2 + h^2} \)

Given:

• Base radius, $r = 6$ cm
• Height, $h = 8$ cm

Let's calculate the slant height:

\( l = \sqrt{(6 \text{ cm})^2 + (8 \text{ cm})^2} \)

\( l = \sqrt{36 \text{ cm}^2 + 64 \text{ cm}^2} \)

\( l = \sqrt{100 \text{ cm}^2} \)

\( l = 10 \text{ cm} \)

So, the slant height of the cone is 10 cm.

Calculating the Total Surface Area of the Cone

Now that we have the radius ($r = 6$ cm) and the slant height ($l = 10$ cm), we can calculate the total surface area using the formula:

\( \text{TSA} = \pi r (r + l) \)

Substitute the values:

\( \text{TSA} = \pi \times 6 \text{ cm} \times (6 \text{ cm} + 10 \text{ cm}) \)

\( \text{TSA} = \pi \times 6 \text{ cm} \times (16 \text{ cm}) \)

\( \text{TSA} = 96\pi \text{ cm}^2 \)

To get a numerical value, we use an approximate value for $\pi$. Using $\pi \approx 3.14159$:

\( \text{TSA} \approx 96 \times 3.14159 \text{ cm}^2 \)

\( \text{TSA} \approx 301.59264 \text{ cm}^2 \)

Using $\pi \approx \frac{22}{7} \approx 3.14286$:

\( \text{TSA} \approx 96 \times \frac{22}{7} \text{ cm}^2 \)

\( \text{TSA} \approx \frac{2112}{7} \text{ cm}^2 \)

\( \text{TSA} \approx 301.714... \text{ cm}^2 \)

Comparing our calculated value to the given options, the value calculated using $\pi \approx \frac{22}{7}$ is closest to one of the options.

Comparing Results with Options

Let's compare our calculated area ($301.714... \text{ cm}^2$) with the options provided:

• Option 1: 354.50 cm\(^2\)
• Option 2: 350.51 cm\(^2\)
• Option 3: 301.71 cm\(^2\)
• Option 4: 364.61 cm\(^2\)

The calculated total surface area of approximately 301.71 cm\(^2\) matches Option 3.

Revision Table: Key Cone Formulas

Additional Information on Cone Properties

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex. Here are some important terms related to cones:

• Base: The flat surface at the bottom of the cone. In a right circular cone, the base is a circle.
• Apex (or Vertex): The pointed top of the cone.
• Height ($h$): The perpendicular distance from the apex to the center of the base.
• Radius ($r$): The radius of the circular base.
• Slant Height ($l$): The distance from the apex to any point on the circumference of the base along the surface of the cone.
• Right Circular Cone: A cone where the apex is directly above the center of the base. This is the type of cone typically studied in basic geometry, and the one assumed in this problem.
• Oblique Cone: A cone where the apex is not directly above the center of the base. The formulas for surface area and volume are different or require calculus for calculation.

Understanding the relationship between the height, radius, and slant height via the Pythagorean theorem is crucial for solving many cone-related problems, especially those involving surface area and finding missing dimensions.