To find the area of material required for the lampshade in the form of a frustum of a cone, we can use the formula for the lateral surface area of a frustum of a cone:
\[ A = \pi (r_1 + r_2) l \]
where:
- \( A \) is the lateral surface area,
- \( r_1 \) is the radius of the smaller base (top),
- \( r_2 \) is the radius of the larger base (bottom),
- \( l \) is the slant height of the frustum.
First, we need to find the radii of the frustum bases:
- The diameter of the top \( (d_1) = 20 , \text{cm} \) gives \( r_1 = \frac{d_1}{2} = 10 , \text{cm} \).
- The diameter of the bottom \( (d_2) = 30 , \text{cm} \) gives \( r_2 = \frac{d_2}{2} = 15 , \text{cm} \).
Now, we will find the slant height \( l \) of the frustum. The vertical height \( h \) is given as 12 cm. We can calculate the slant height using the Pythagorean theorem, where \( l \) is the hypotenuse:
\[ l = \sqrt{(r_2 - r_1)^2 + h^2} \]
Substituting the values:
\[ l = \sqrt{(15 - 10)^2 + 12^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 , \text{cm} \]
Now, we can substitute \( r_1 \), \( r_2 \), and \( l \) into the area formula:
\[ A = \pi (r_1 + r_2) l = \pi (10 + 15) \cdot 13 \] \[ A = \pi \cdot 25 \cdot 13 = 325\pi , \text{cm}^2 \]
To express the area in numerical form:
\[ A \approx 325 \cdot 3.14 \approx 1021.5 , \text{cm}^2 \]
Thus, the area of material required for the lampshade is approximately:
\[ \boxed{1021.5 , \text{cm}^2} \]