Question

Find the surface area of the prism or regular pyramid.

Answer 1

Hint: The prism is composed of 2 triangles and 3 rectangles.

Step 1: Surface area of the pyramid is the sum of the areas of all its faces, or the sum of its lateral area and base area.

Step 2: $A_{total}=2A_T+A_1+A_2+A_3$

Where $A_T$ is the area of the triangles & $A_1,A_2,A_3$ are the area of the rectangles.

Now, $A_T=\frac{1}{2}b·h$

Where $b$ is the base of the triangle & $h$ is the height of the triangle.

Step 3: Now, $b=3\&h=4$

Substituting all these values in the formula,

Area of the triangle:

$$\begin{gathered} A_T=\frac{1}{3}\left(3\right)\left(4\right) \\ {A}_T=6 \end{gathered}$$

Remember There are two similar triangles in the prism.

So, $2A_T=12$

Step 4: Area of the base rectangle:

$A_1=l\times w$

Where $l$ is the length & $w$ is the width of the rectangle.

Here, $l=5\&w=8$, putting all the values in the formula

$$\begin{gathered} A_1=5\times 8 \\ {A}_1=40 \end{gathered}$$

Step 5: Area of the vertical leg which is rectangular in shape

$A_2=l\times w$

Where $l$ is the length & $w$ is the width of the rectangle.

Here, $l=4\&w=8$, putting all the values in the formula

$$\begin{gathered} A_2=4\times 8 \\ {A}_2=32 \end{gathered}$$

Step 6: Area of the diagonal leg which is rectangular in shape

$A_3=l\times w$

Where $l$ is the length & $w$ is the width of the rectangle.

Here, $l=3\&w=8$, putting all the values in the formula

$$\begin{gathered} A_3=3\times 8 \\ {A}_3=24 \end{gathered}$$

Step 7: $A_{total}=2A_T+A_1+A_2+A_3$

$$\begin{gathered} A_{total}=12+40+32+24 \\ {A}_{total}=108 \end{gathered}$$

Final Answer: The total surface area of the regular pyramid in $108m^2$.