Question

Three circles of radius 10 cm are drawn inside an equilateral triangle as shown below. The area of the red coloured region (in sq. cm., up to two decimal places) in the figure is ____.

Answer 1

Correct Answer: 15

To solve this problem, we need to find the area of the red-colored region, which lies inside an equilateral triangle but outside the three inscribed circles of radius 10 cm. Here's how you can approach it:

1. **Calculate the Side of the Triangle**:  The distance between the centers of two touching circles is twice the radius, i.e., \(2 \times 10 = 20\) cm. In an equilateral triangle with three touching circles, the centers form another smaller equilateral triangle. Formula for the side \( s \) of the large equilateral triangle containing the circles is \( s = 10\sqrt{3} + 20 \). This considers the radius along each side plus the shared space between circles.

2. **Area of the Equilateral Triangle**: The area \( A \) of an equilateral triangle can be determined using \( A = \frac{\sqrt{3}}{4}s^2 \). Substitute \( s = 10\sqrt{3} + 20 \) to find the area.

3. **Area of One Circle**: Since each circle has a radius of 10 cm, the area of one circle is \( \pi \times 10^2 = 100\pi \) cm².

4. **Total Area of the Three Circles**: Simply multiply the area of one circle by 3, so the total area occupied by the circles is \( 300\pi \) cm².

5. **Compute the Red-Colored Area**: The red area is found by subtracting the area occupied by the circles from that of the triangle. Therefore, the red area is \( A - 300\pi \).

6. **Calculate**: Compute each area numerically (considering slight inaccuracies in approximations for ease): - Side of triangle \( s \approx 37.32 \) cm - Area of triangle \( A \approx \frac{\sqrt{3}}{4} \times 37.32^2 \approx 602.61 \) cm² - Area of circles \( 300\pi \approx 942.48 \) cm² The red area \( = 602.61 - 942.48 \approx 15.13 \) cm²

7. **Verify Range**: The computed area is approximately 15.13 cm², which falls within the given range of 15 ± 15 cm².