Question

If the radius is 5 and the height is 12. Find the volume and the total structure of a cone

Answer 1

To find the volume and surface area of a cone, you can use the following formulas:

1. **Volume (V)** of a cone:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.

2. **Surface Area (A)** of a cone:
\[
A = \pi r (r + l)
\]
where \( l \) is the slant height of the cone. The slant height can be found using the Pythagorean theorem:
\[
l = \sqrt{r^2 + h^2}
\]

Given:
- Radius \( r = 5 \)
- Height \( h = 12 \)

**Step 1: Calculate the volume.**
\[
V = \frac{1}{3} \pi (5)^2 (12) = \frac{1}{3} \pi (25) (12) = \frac{1}{3} \pi (300) = 100\pi
\]

Thus, the volume of the cone is approximately:
\[
V \approx 100 \times 3.14 \approx 314.16 \text{ cubic units}
\]

**Step 2: Calculate the slant height.**
\[
l = \sqrt{(5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]

**Step 3: Calculate the surface area.**
\[
A = \pi (5) (5 + 13) = \pi (5)(18) = 90\pi
\]

Thus, the total surface area of the cone is approximately:
\[
A \approx 90 \times 3.14 \approx 282.6 \text{ square units}
\]

**Summary:**
- Volume: \( 100\pi \) cubic units (approximately \( 314.16 \) cubic units)
- Total surface area: \( 90\pi \) square units (approximately \( 282.6 \) square units)