The shaded region below represents a “flag” in the shape of an isoceles triangle with a base of 40 and equal sides of length 25. Imagine rotating this flag about its "pole" and draw a diagram of the results three-dimensional figure. What is the volume of this figure?

To solve this problem, let's break it down into two parts: first, understanding the 3D shape formed by the rotation, and second, calculating the volume of that shape.

### Part 1: Understanding the 3D Shape

The flag is an isosceles triangle with a base of 40 units and equal sides of 25 units. When you rotate this triangle about its "pole" (which would be one of the equal sides of the triangle), the resulting 3D shape will be a **cone**.

In this scenario:
- The height of the triangle (which becomes the height of the cone) is the distance from the base of the triangle to the opposite vertex. Let's denote this height as \( h \).
- The radius of the base of the cone is half the base of the triangle since the base will sweep out a circular shape as it rotates. Therefore, the radius \( r \) will be \( 20 \) units.

### Part 2: Calculating the Height \( h \)

We can find the height \( h \) using the Pythagorean theorem in the right triangle formed by the height, half the base, and one of the equal sides:
\[
\left(\frac{\text{base}}{2}\right)^2 + h^2 = \text{side}^2
\]
Substituting the values:
\[
20^2 + h^2 = 25^2
\]
\[
400 + h^2 = 625
\]
\[
h^2 = 225
\]
\[
h = 15 \text{ units}
\]

### Part 3: Volume of the Cone

The volume \( V \) of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
Substituting the values \( r = 20 \) units and \( h = 15 \) units:
\[
V = \frac{1}{3} \pi (20)^2 (15)
\]
\[
V = \frac{1}{3} \pi (400)(15)
\]
\[
V = \frac{1}{3} \pi (6000)
\]
\[
V = 2000\pi \text{ cubic units}
\]

### Final Answer

The volume of the three-dimensional figure formed by rotating the flag around its pole is \( 2000\pi \) cubic units.

Would you like to see a diagram of the resulting 3D shape or need further details on any of these steps? Here are some related questions:

1. What is the formula for the volume of a cone, and why does it apply here?
2. How can the Pythagorean theorem be used to solve problems involving triangles and their heights?
3. What would happen if the flag were rotated around the base instead of the pole?
4. How does rotating a shape in 2D around an axis result in a 3D figure?
5. What are other common shapes that can be formed by rotating basic geometric figures?
6. How would the volume change if the dimensions of the flag were different?
7. What is the significance of using \(\pi\) in volume calculations involving circular shapes?
8. How can we verify the result using a different approach?

**Tip:** Always double-check whether you’re rotating around the correct axis in geometry problems to avoid calculating the wrong 3D figure's volume.

Discover the volume of the three-dimensional shape formed by rotating an isosceles triangle with a base of 40 units and equal sides of 25 units about its pole. Learn how to calculate the volume of a cone using the radius and height derived from the triangle's dimensions. Suitable for high school students studying geometry and 3D shapes.

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