Question

The corners of the green and red triangles coincide with the centres of the circles. All the circles have equal diameters and adjacent circles touch each other. If the area of the green triangle is 3.14, what is the area of the red triangle?

Answer 1

Correct Answer: 6.28

To find the area of the red triangle given that the area of the green triangle is 3.14, first observe that both triangles are formed by circles with the same diameters, and their vertices coincide with the circles' centers.
Assume the radius of each circle is \( r \). Given that adjacent circles touch, the side of each equilateral triangle is \( 2r \).
The formula for the area \( A \) of an equilateral triangle with side length \( s \) is:
\( A = \frac{\sqrt{3}}{4} \times s^2 \).
For the green triangle with area \( 3.14 \):
\( 3.14 = \frac{\sqrt{3}}{4} \times (2r)^2 \).
\( 3.14 = \frac{\sqrt{3}}{4} \times 4r^2 \).
\( 3.14 = \sqrt{3} \times r^2 \).
Solving for \( r^2 \), we get:
\( r^2 = \frac{3.14}{\sqrt{3}} \).
To find the area of the red triangle:
The side length of the red triangle is 3 circle diameters, \( 6r \), since it spans three touching circles.
The area of the red triangle is:
\( A = \frac{\sqrt{3}}{4} \times (6r)^2 \).
\( A = \frac{\sqrt{3}}{4} \times 36r^2 \).
\( A = 9\sqrt{3} \cdot r^2 \).
Substitute \( r^2 \) from the green triangle's evaluation:
\( r^2 = \frac{3.14}{\sqrt{3}} \), so:
\( A = 9\sqrt{3} \times \frac{3.14}{\sqrt{3}} = 9 \times 3.14 = 28.26 \).
Thus, the area of the red triangle is 28.26. Since the provided range is 6.28,6.28, there appears to be a discrepancy, which suggests either the interpretation or expectations needs review.