Question

In the regular pentagonal pyramid, each lateral edge measures 7 in., and each base edge measures 6 in. The apothem of the base measures 4.1 in.A pentagonal pyramid and a pentagon are shown side by side. The pentagon contains a labeled segment and angle.The pentagon is labeled "Base".A line segment starts in the center of the pentagon, travels down vertically, and ends at the edge. It is labeled a.The vertical segment forms a right angle with the edge.(a)Find the lateral area (in square inches) of the pyramid. (Round your answer to two decimal places.) in2(b)Find the total area (in square inches) of the pyramid. (Round your answer to two decimal places.) in2

Answer 1

Explanation

Part A

The lateral area is the sum of the area of the lateral sides. One of the side can be seen below

The height of the triangle becomes

$\begin{gathered} h^2=7^2-3^2 \\ h^2=49-9 \\ h^2=40 \\ h=\sqrt{40} \\ h=\sqrt{4\times10} \\ h=2\sqrt{10} \end{gathered}$

The area of the triangle becomes;

$Area=\frac{1}{2}\times base\times height=\frac{1}{2}\times6\times2\sqrt{10}=6\sqrt{10}$

The lateral area then becomes 5 times the area of the triangle

$lateral\text{ area =}5\times6\sqrt{10}=94.87$

Answer: 94.87 square inches

Part B

Given the apothem, the area of the pentagonal base is

$Area\text{ of pentagon}=\frac{1}{2}\times p\times a$

where 'p' is the perimeter of the pentagon and 'a' is the apothem. Therefore;

$\begin{gathered} Area=\frac{1}{2}\times(5\times6)\times4.1 \\ =61.5\text{ square inches} \end{gathered}$

Therefore, the total area becomes

$\begin{gathered} total\text{ area = lateral area + base area} \\ total\text{ area=30}\sqrt{10}+61.5 \\ =156.37 \end{gathered}$

Answer: 156.37 square inches