Answer:
Area of triangle = 3 square units
Area of triangle = 6 square units
Area of regular pentagon = 15 square units
Area of sector = 5.181 square units
Total Area = 29.18 square units (nearest hundredth)
Step-by-step explanation:
Area of triangle and rectangle
As the pentagon is regular, this means that its sides are equal in length.
The base of the triangle and the length of the rectangle are both equal to the side length of the pentagon, which is 3 units. Therefore:
[tex]\textsf{Area of triangle}=\dfrac{1}{2} \cdot 3 \cdot 2=3\; \sf units^2[/tex]
[tex]\textsf{Area of rectangle}=3 \cdot 2=6\; \sf units^2[/tex]
[tex]\hrulefill[/tex]
Area of regular pentagon
To find the area of the regular pentagon, we can use the area of a regular polygon formula:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Area of a regular polygon}}\\\\A=\dfrac{n\cdot s\cdot a}{2}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$n$ is the number of sides.}\\ \phantom{ww}\bullet\;\textsf{$s$ is the length of one side.}\\ \phantom{ww}\bullet\;\textsf{$a$ is the apothem.}\end{array}}[/tex]
In this case:
Substitute these values into the formula:
[tex]\textsf{Area of regular pentagon}=\dfrac{5\cdot 3 \cdot 2}{2}=15\; \sf units^2[/tex]
[tex]\hrulefill[/tex]
Area of sector
To find the area of the sector, we can use the area of a sector formula:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Area of a Sector}}\\\\A= \left(\dfrac{\theta}{360^{\circ}}\right) \pi r^2\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$A$ is the area.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius.}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the central angle in degrees.}\end{array}}[/tex]
In this case the central angle is θ = 66° and the radius is equal to the side length of the pentagon, so r = 3.
Substitute the values into the formula, along with π = 3.14:
[tex]\textsf{Area of sector}=\left(\dfrac{66^{\circ}}{360^{\circ}}\right) \cdot 3.14\cdot3^2\\\\\\\textsf{Area of sector}=0.18333... \cdot 3.14\cdot 9\\\\\\\textsf{Area of sector}=5.181\; \sf units^2[/tex]
[tex]\hrulefill[/tex]
Total Area
To find the total area, sum the individual areas:
[tex]\textsf{Total Area}=3+6+15+5.181\\\\\\\textsf{Total Area}=29.181\\\\\\\textsf{Total Area}=29.18\;\sf units^2\;(nearest\;hundredth)[/tex]
Therefore, the total area of the composite shape is 29.18 square units, rounded to the nearest hundredth.
