Question

Find the lateral area of the cone with a diameter of 13 inches and a height of 32 inches. Round to the nearest whole number.

Answer 1

To find the lateral area (LA) of a cone, we use the formula:

\[
LA = \pi r l
\]

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

1. **Find the radius \( r \)**:
The diameter of the cone is given as 13 inches, so the radius \( r \) is:

\[
r = \frac{\text{diameter}}{2} = \frac{13}{2} = 6.5 \text{ inches}
\]

2. **Calculate the slant height \( l \)**:
The slant height can be found using the Pythagorean theorem. The formula to calculate the slant height \( l \) is:

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

where \( h \) is the height of the cone. Substituting the values, where \( h = 32 \) inches:

\[
l = \sqrt{(6.5)^2 + (32)^2}
\]
\[
= \sqrt{42.25 + 1024}
\]
\[
= \sqrt{1066.25}
\]
\[
l \approx 32.65 \text{ inches}
\]

3. **Calculate the lateral area**:
Now substitute \( r \) and \( l \) into the lateral area formula:

\[
LA = \pi (6.5)(32.65)
\]
\[
LA \approx \pi (212.225)
\]
\[
LA \approx 666.74 \text{ square inches}
\]

4. **Round to the nearest whole number**:

\[
LA \approx 667 \text{ square inches}
\]

Thus, the lateral area of the cone is approximately \( \boxed{667} \) square inches.