Answer:
65.79 sq units
Explanation:
We are given this:
Sides of triangle = 8 units
Interior angle between the sides = 72.6 degrees
The sum of the interior angles in a triangle is 180 degrees:
[tex]\begin{gathered} 72.6+x+x=180 \\ 72.6+2x=180 \\ \text{Subtract ''72.6'' from both sides, we have:} \\ 2x=180-72.6 \\ 2x=107.4 \\ x=\frac{107.4}{2}=53.7 \\ x=53.7^{\circ} \end{gathered}[/tex]We will obtain the value for the base of the triangle using the Sine rule, we have:
[tex]\begin{gathered} \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c} \\ \frac{\sin72.6^{\circ}}{a}=\frac{\sin53.7^{\circ}}{8}=\frac{\sin53.7^{\circ}}{8} \\ \frac{\sin72.6^{\circ}}{a}=\frac{\sin53.7^{\circ}}{8} \\ \text{Cross multiply, we have:} \\ a\cdot\sin 53.7^{\circ}=8\times\sin 53.7^{\circ} \\ a=\frac{8\times\sin72.6^{\circ}}{\sin53.7^{\circ}} \\ a=\frac{8\times0.9542}{0.8059} \\ a=9.4721\approx9.47 \\ a=9.47units \end{gathered}[/tex]The base of the triangle is 9.47 units
We will thus find the area of the figure as shown below:
[tex]\begin{gathered} Area=Area_{triangle}+Area_{semi-circle} \\ \text{ We will use Heron's formula to calculate the area of the triangle:} \\ Area_{triangle}=\sqrt[]{s(s-a)(s-b)(s-c)} \\ s=\frac{a+b+c}{2}=\frac{8+8+9.47}{2}=12.7360\approx12.74 \\ Area_{triangle}=\sqrt[]{12.74(12.74-9.47)(12.74-8)\mleft(12.74-8\mright)} \\ Area_{triangle}=\sqrt[]{12.74(3.27)(4.74)(4.74)} \\ Area_{triangle}=\sqrt[]{935.99572248} \\ Area_{triangle}=30.5940\approx30.59 \\ \\ Area_{semi-circle}=\frac{1}{2}\pi r^2 \\ Area_{semi-circle}=\frac{1}{2}\times3.14\times4.735^2 \\ Area_{semi-circle}=35.19975\approx35.20 \\ Area_{semi-circle}=35.20 \\ \\ Area=30.59+35.20=65.79 \\ \therefore Area=65.79units^2 \end{gathered}[/tex]Therefore, the area of the shape is 65.79 sq units
