The exact Total Surface Area (TSA) of a cone with a radius of 7 cm and a height of 24 cm is $224\pi$ cm².
Understanding Cone Surface Area Calculations
To calculate the Total Surface Area (TSA) of a cone, we need to consider both the area of its circular base and the area of its curved lateral surface. The formula for TSA is the sum of these two components: $TSA = \pi r^2 + \pi r l$, where $r$ is the radius and $l$ is the slant height of the cone.
Steps to Calculate the TSA
Here's a detailed breakdown of the calculation for a cone with a radius of 7 cm and a height of 24 cm:
-
Identify Given Dimensions:
- Radius ($r$): 7 cm
- Height ($h$): 24 cm
-
Calculate the Slant Height ($l$):
The slant height is the distance from the apex (tip) of the cone to any point on the circumference of its base. It forms the hypotenuse of a right-angled triangle, with the cone's radius and height as the other two sides. We use the Pythagorean theorem:
$l = \sqrt{r^2 + h^2}$
$l = \sqrt{7^2 + 24^2}$
$l = \sqrt{49 + 576}$
$l = \sqrt{625}$
$l = 25$ cm -
Calculate the Base Area ($A_{base}$):
The base of the cone is a circle. The area of a circle is given by $\pi r^2$.
$A{base} = \pi \times 7^2$
$A{base} = 49\pi$ cm² -
Calculate the Lateral Surface Area ($A_{lateral}$):
This is the area of the curved surface that connects the base to the apex. The formula for lateral surface area is $\pi r l$.
$A{lateral} = \pi \times 7 \times 25$
$A{lateral} = 175\pi$ cm² -
Calculate the Total Surface Area (TSA):
The Total Surface Area is the sum of the base area and the lateral surface area.
$TSA = A{base} + A{lateral}$
$TSA = 49\pi + 175\pi$
$TSA = 224\pi$ cm²
Summary of Cone Surface Area Calculation
The following table summarizes the key dimensions and calculated areas for the specified cone:
| Measurement | Formula | Value (cm or cm²) |
|---|---|---|
| Radius ($r$) | Given | 7 cm |
| Height ($h$) | Given | 24 cm |
| Slant Height ($l$) | $\sqrt{r^2 + h^2}$ | 25 cm |
| Base Area | $\pi r^2$ | $49\pi$ cm² |
| Lateral Surface Area | $\pi r l$ | $175\pi$ cm² |
| Total Surface Area (TSA) | $\pi r^2 + \pi r l$ or $\pi r (r+l)$ | $224\pi$ cm² |
For a practical approximation, using $\pi \approx 22/7$, the TSA would be $224 \times (22/7) = 32 \times 22 = 704$ cm².
Exploring Variations in Cone Surface Areas
The dimensions of a cone directly influence its surface area. While the cone in this question has a Total Surface Area of $224\pi$ cm², it's valuable to understand how different dimensions can lead to varied surface areas. For instance, a cone could have a lateral surface area of exactly 440 cm² if its radius was 7 cm and its slant height was 20 cm (calculated as $\pi \times 7 \times 20 \approx (22/7) \times 7 \times 20 = 22 \times 20 = 440$ cm²). This demonstrates the wide range of surface area values cones can possess, each uniquely determined by their specific radius and height.
For further information on the geometry of cones, including their volume and surface area, you can visit the Wikipedia page on Cones.
