Question

The image features a yellow triangle that measures 22 cm in height and has a diameter of 25 cm. The triangle displays black text, and it is related to a cone.
What is the surface area of the cone? Use 3.14 for pi.

Answer 1

To calculate the surface area of a cone, we need to know both the radius of the base and the slant height. The formula for the surface area \( A \) of a cone is:

\[
A = \pi r (r + l)
\]

where:
- \( r \) is the radius of the base,
- \( l \) is the slant height of the cone.

From the information provided:

1. The diameter of the base of the cone is given as 25 cm. The radius \( r \) can be calculated as:

\[
r = \frac{\text{diameter}}{2} = \frac{25 \text{ cm}}{2} = 12.5 \text{ cm}
\]

2. The height of the cone \( h \) is given as 22 cm. We need to find the slant height \( l \) using the Pythagorean theorem, which is applicable since the height, radius, and slant height form a right triangle:

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

Substituting the values we have:

\[
l = \sqrt{(12.5 \text{ cm})^2 + (22 \text{ cm})^2}
\]

Calculating the squares:

\[
(12.5 \text{ cm})^2 = 156.25 \text{ cm}^2
\]

\[
(22 \text{ cm})^2 = 484 \text{ cm}^2
\]

Now add those together:

\[
l = \sqrt{156.25 \text{ cm}^2 + 484 \text{ cm}^2} = \sqrt{640.25 \text{ cm}^2} \approx 25.25 \text{ cm}
\]

Now we can calculate the surface area \( A \):

\[
A = \pi r (r + l)
\]

Substituting \( r = 12.5 \text{ cm} \) and \( l \approx 25.25 \text{ cm} \):

\[
A = 3.14 \times 12.5 \times (12.5 + 25.25)
\]

Calculating \( 12.5 + 25.25 \):

\[
12.5 + 25.25 = 37.75 \text{ cm}
\]

Now substitute that back into the area formula:

\[
A = 3.14 \times 12.5 \times 37.75
\]

Calculating:

\[
3.14 \times 12.5 \approx 39.25
\]

Now multiply by \( 37.75 \):

\[
A \approx 39.25 \times 37.75 \approx 1484.6875 \text{ cm}^2
\]

Rounded to two decimal places, the surface area of the cone is approximately:

\[
\boxed{1484.69 \text{ cm}^2}
\]