Question

find the area of the shaded region. (each side of inner triangle is 1.1 cm) 3.8 cm 3.8 cm 3.8 cm the area of the shaded region is approximately . (simplify your answer. round the final answer to the nearest tenth as needed. round all intermediate values to the nearest thousandth as needed.)

Explanation:

Step1: Recall area formula for equilateral triangle

The area formula for an equilateral triangle is $A = \frac{\sqrt{3}}{4}s^{2}$, where $s$ is the side - length of the triangle.

Step2: Calculate area of outer triangle

The side - length of the outer triangle $s_{1}=3.8$ cm. So, $A_{1}=\frac{\sqrt{3}}{4}\times(3.8)^{2}=\frac{\sqrt{3}}{4}\times14.44\approx\frac{1.73205}{4}\times14.44 = 1.73205\times3.61\approx6.2527$.

Step3: Calculate area of inner triangle

The side - length of the inner triangle $s_{2}=1.1$ cm. So, $A_{2}=\frac{\sqrt{3}}{4}\times(1.1)^{2}=\frac{\sqrt{3}}{4}\times1.21\approx\frac{1.73205}{4}\times1.21 = 1.73205\times0.3025\approx0.5239$.

Step4: Calculate area of shaded region

The area of the shaded region $A = A_{1}-A_{2}$. So, $A\approx6.2527 - 0.5239=5.7288\approx5.7$ $cm^{2}$.

Answer:

$5.7$ $cm^{2}$