a circle is inscribed in equilateral triangle of side length 1cm . Tangents to the circle are drawn that cut off equilateral triangles at each corner .Circles are inscribed in each equilateral triangle if this process is repeated infinitely many times, sum of areas of all circles is?


To solve this problem, we need to calculate the sum of the areas of all the circles that are inscribed in the equilateral triangles formed by the process described.

### Step 1: Area of the initial equilateral triangle
The side length of the initial equilateral triangle is given as 1 cm. The area \( A_0 \) of an equilateral triangle with side length \( s \) is given by:
\[
A_0 = \frac{\sqrt{3}}{4} s^2
\]
For \( s = 1 \) cm, the area of the triangle is:
\[
A_0 = \frac{\sqrt{3}}{4} \times 1^2 = \frac{\sqrt{3}}{4} \text{ cm}^2
\]

### Step 2: Radius of the inscribed circle in the initial triangle
The radius \( r_0 \) of the circle inscribed in an equilateral triangle with side length \( s \) is given by:
\[
r_0 = \frac{s \sqrt{3}}{6}
\]
For \( s = 1 \) cm:
\[
r_0 = \frac{1 \times \sqrt{3}}{6} = \frac{\sqrt{3}}{6} \text{ cm}
\]

### Step 3: Area of the initial circle
The area \( A_c \) of the first circle is:
\[
A_{c0} = \pi r_0^2 = \pi \left(\frac{\sqrt{3}}{6}\right)^2 = \frac{\pi \times 3}{36} = \frac{\pi}{12} \text{ cm}^2
\]

### Step 4: Side length of the smaller triangles formed
When the tangents to the circle are drawn, they form smaller equilateral triangles within the original triangle. The side length \( s_1 \) of the new triangle is:
\[
s_1 = \frac{s}{2}
\]
Therefore, the side length after the first iteration \( s_1 \) becomes \( \frac{1}{2} \).

### Step 5: Radius of the next circle
The radius \( r_1 \) of the circle inscribed in the new smaller equilateral triangle with side length \( s_1 \) is:
\[
r_1 = \frac{s_1 \sqrt{3}}{6} = \frac{\left(\frac{1}{2}\right) \sqrt{3}}{6} = \frac{\sqrt{3}}{12} \text{ cm}
\]

### Step 6: Area of the next circle
The area \( A_{c1} \) of this circle is:
\[
A_{c1} = \pi r_1^2 = \pi \left(\frac{\sqrt{3}}{12}\right)^2 = \frac{\pi \times 3}{144} = \frac{\pi}{48} \text{ cm}^2
\]

### Step 7: Generalizing the areas
Each subsequent triangle has a side length half of the previous one, and the radius of the circle inscribed in each of these triangles follows a similar pattern, shrinking by a factor of \( \frac{1}{2} \) each time.

Thus, the area of the circle after \( n \) iterations is:
\[
A_{cn} = \frac{\pi}{12} \times \left(\frac{1}{4}\right)^n
\]

### Step 8: Sum of the areas of all circles
The total sum of the areas of all the circles is an infinite geometric series:
\[
\text{Total Area} = \sum_{n=0}^{\infty} A_{cn} = \frac{\pi}{12} \sum_{n=0}^{\infty} \left(\frac{1}{4}\right)^n
\]
The sum of an infinite geometric series with first term \( a \) and common ratio \( r \) is given by:
\[
S = \frac{a}{1-r}
\]
Here, \( a = \frac{\pi}{12} \) and \( r = \frac{1}{4} \), so:
\[
\text{Total Area} = \frac{\frac{\pi}{12}}{1 - \frac{1}{4}} = \frac{\frac{\pi}{12}}{\frac{3}{4}} = \frac{\pi}{12} \times \frac{4}{3} = \frac{\pi}{9} \text{ cm}^2
\]

### Final Answer:
The sum of the areas of all the circles is \( \boxed{\frac{\pi}{9}} \) square centimeters.

Would you like further details or have any questions?

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Here are 5 related questions:
1. How do you derive the formula for the radius of a circle inscribed in an equilateral triangle?
2. What is the sum of areas of the inscribed circles if the side length of the initial triangle is doubled?
3. Can the method used to find the sum of areas be applied to other polygons? Why or why not?
4. How does the area of the inscribed circle relate to the area of the equilateral triangle?
5. What is the importance of geometric series in solving problems like these?

**Tip:** Understanding geometric series is crucial for solving problems that involve repeated patterns or processes in geometry, such as inscribed shapes or fractals.

Explore how to calculate the sum of areas of circles inscribed in equilateral triangles formed by tangents, repeating infinitely. Includes step-by-step solution, formulas for area of triangles and circles, and the concept of geometric series.

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