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The problem asks to calculate the area of an isosceles triangle with a base of 30 cm and the radius of the inscribed circle \( r = 10 \) cm.

### Solution Outline:
1. **Given Data:**
   - Base \( b = 30 \) cm.
   - Radius of the inscribed circle \( r = 10 \) cm.
   
2. **Formula for the area of an isosceles triangle using the radius of the inscribed circle:**
   The area \( A \) of a triangle can also be found by the formula:
   \[
   A = r \cdot s
   \]
   where \( r \) is the radius of the inscribed circle, and \( s \) is the semiperimeter of the triangle.

3. **Semiperimeter \( s \):**
   The semiperimeter is given by:
   \[
   s = \frac{a + a + b}{2} = \frac{2a + b}{2}
   \]
   where \( a \) is the length of each of the equal sides of the isosceles triangle, and \( b \) is the base.

4. **To calculate \( a \):**
   Using the Pythagorean theorem for the height of the triangle, the relation is:
   \[
   a^2 = \left(\frac{b}{2}\right)^2 + h^2
   \]
   where \( h \) is the height of the triangle. We also know that the area of the triangle can be expressed as:
   \[
   A = \frac{1}{2} b h
   \]
   
I will now calculate the side length \( a \), semiperimeter \( s \), and the area using the given values.

Let me perform these calculations.The calculations show that:

1. The semiperimeter \( s \) is \( a + 15 \), where \( a \) is the length of each of the equal sides of the isosceles triangle.
2. The area \( A \) of the triangle is \( 10a + 150 \), where \( a \) is still the unknown side length.

To proceed, we need to find \( a \), which can be determined using the Pythagorean theorem and the given data. Let's solve for \( a \).It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to guide you through the steps to manually find the side length \(a\) and complete the solution?

Here are some relative questions to expand your understanding:
1. How is the semiperimeter of a triangle defined and why is it useful?
2. What is the relationship between the radius of the inscribed circle and the area of a triangle?
3. How does the Pythagorean theorem help in finding unknown sides of a triangle?
4. What other methods can be used to calculate the area of an isosceles triangle?
5. Can the properties of a right triangle be applied to solve for unknowns in an isosceles triangle?

**Tip:** Knowing how to derive the side lengths of a triangle from the given geometric properties can help in solving many geometry problems more easily!

Izračunati površinu jednakokrakog trougla osnove 30 cm i poluprečnika upisane kružnice r = 10 cm.

Learn how to calculate the area of an isosceles triangle with a base of 30 cm and the radius of an inscribed circle being 10 cm. This solution uses key geometric concepts such as the semiperimeter, the Pythagorean theorem, and triangle area formulas.

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