If an equilateral triangle of side 6 cm is rotated (by 360°) about an axis perpendicular to its plane and passing through its centroid, then what is the area of the figure that is formed?

(1) 36pi

(2) 12pi

(3) 3pi

(4) 9pi

(5) 15pi

When an equilateral triangle is rotated around an axis perpendicular to its plane and passing through its centroid, the figure formed is a **solid of revolution**, specifically a **double cone** or **bicone** (two cones joined at their bases).

### Steps to Solve:

1. **Height of the Cone (Distance from Centroid to Vertex):**
   - In an equilateral triangle, the distance from the centroid to the vertex is \( \frac{2}{3} \) of the altitude.
   - The altitude \( h \) of an equilateral triangle with side length \( a \) is given by:
   \[
   h = \frac{\sqrt{3}}{2} \times a = \frac{\sqrt{3}}{2} \times 6 = 3\sqrt{3} \, \text{cm}
   \]
   - The distance from the centroid to the vertex is then:
   \[
   \frac{2}{3} \times 3\sqrt{3} = 2\sqrt{3} \, \text{cm}
   \]

2. **Radius of the Base:**
   - The radius of the base of the cone is equal to the circumradius of the equilateral triangle.
   - The circumradius \( R \) of an equilateral triangle is given by:
   \[
   R = \frac{a}{\sqrt{3}} = \frac{6}{\sqrt{3}} = 2\sqrt{3} \, \text{cm}
   \]

3. **Volume of a Cone:**
   - The volume of a cone is given by the formula:
   \[
   V = \frac{1}{3} \pi r^2 h
   \]
   - Substituting the values:
   \[
   V = \frac{1}{3} \pi (2\sqrt{3})^2 (2\sqrt{3}) = \frac{1}{3} \pi (12)(2\sqrt{3}) = 36 \pi
   \]

Thus, the area of the figure formed is \( 36\pi \).

The correct answer is:
\[
\boxed{36\pi}
\]

---

Would you like more details or have any questions?

Here are 5 related questions:
1. How is the centroid of a triangle calculated?
2. What is a solid of revolution?
3. How is the circumradius of different triangles calculated?
4. What is the relation between altitude and side length in an equilateral triangle?
5. How does the volume of a cone change with varying height?

**Tip:** Always make sure to visualize geometric transformations to better understand the properties of the solid formed!

If an equilateral triangle of side 6 cm is rotated (by 360°) about an axis perpendicular to its plane and passing through its centroid, then what is the area of the figure that is formed? (1) 36pi (2) 12pi (3) 3pi (4) 9pi (5) 15pi

Explore the solid of revolution formed when an equilateral triangle with a side length of 6 cm is rotated about its centroid. Learn how to calculate the area of the resulting double cone using geometric principles such as the height and circumradius of an equilateral triangle. Suitable for students in Grades 10-12.

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