To solve this problem, we calculate the radius of the inscribed circle and the area of the region of the triangle not included in the circle. Here’s the step-by-step solution:

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1. Radius of the inscribed circle (inradius)

The formula for the inradius ($r$) of an equilateral triangle is:
$r = \frac{\sqrt{3}}{6} \cdot a$
where $a$ is the side length of the triangle.

Given $a = 12 \, \text{cm}$:
$r = \frac{\sqrt{3}}{6} \cdot 12 = 2\sqrt{3} \, \text{cm}$

Thus, the radius of the inscribed circle is $2\sqrt{3} \, \text{cm}$ (approximately $3.464 \, \text{cm}$).

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2. Area of the equilateral triangle

The formula for the area ($A$) of an equilateral triangle is:
$A = \frac{\sqrt{3}}{4} \cdot a^2$
Substitute $a = 12$:
$A = \frac{\sqrt{3}}{4} \cdot 12^2 = \frac{\sqrt{3}}{4} \cdot 144 = 36\sqrt{3} \, \text{cm}^2$
So, the area of the triangle is $36\sqrt{3} \, \text{cm}^2$ (approximately $62.352 \, \text{cm}^2$).

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3. Area of the inscribed circle

The formula for the area ($A_{\text{circle}}$) of a circle is:
$A_{\text{circle}} = \pi r^2$
Using $r = 2\sqrt{3}$:
$A_{\text{circle}} = \pi (2\sqrt{3})^2 = \pi \cdot 12 = 12\pi \, \text{cm}^2$
Approximately:
$A_{\text{circle}} \approx 37.699 \, \text{cm}^2$

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4. Area of the region not included in the circle

The area of the region outside the circle but within the triangle is:
$A_{\text{outside}} = A_{\text{triangle}} - A_{\text{circle}}$
Substitute the values:
$A_{\text{outside}} = 36\sqrt{3} - 12\pi$
Approximating $\sqrt{3} \approx 1.732$ and $\pi \approx 3.1416$:
$A_{\text{outside}} \approx 62.352 - 37.699 = 24.653 \, \text{cm}^2$

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Final Results:

1. Radius of the inscribed circle: $2\sqrt{3} \, \text{cm}$ or approximately $3.464 \, \text{cm}$.
2. Area of the region not included in the circle: $36\sqrt{3} - 12\pi \, \text{cm}^2$ or approximately $24.653 \, \text{cm}^2$.

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Would you like further details or a breakdown of the formulas?

Related Questions:

1. How does the inradius formula derive from geometry?
2. What is the significance of the ratio between the triangle's area and the circle's area?
3. How would this problem change for a triangle with different side lengths?
4. What is the area of the circumcircle for this triangle?
5. How does the concept of an incircle relate to real-world applications?

Tip:

Always visualize geometric problems with a diagram to better understand the relationships between the elements.