The surface area of this prism is ##36″ in”^2##. Even though the explanation might seem slightly long, it is really easily broken down into pieces, so I highly recommend you read and learn from it. 🙂
In extremely simple terms, imagine you have a woodblock shaped like the triangular prism in picture you attached.
The surface area of that prism is basically all the areas that you can touch (total area of the outside/surface).
Let’s calculate each side separately, and then we can add them up at the end.
The triangular sides:
This triangular surface is 3 inches (height) by 4 inches (base). Using the area formula for triangles (##1/2bh## where ##b## is the base and ##h## is the height) , the area of the triangular surface is ##1/2bh=1/2(3″ in”)(4″ in”) = 1/2 (12″ in”^2)=6″ in”^2##.
There are two triangular surfaces (one in yellow and white), so multiply the area of the triangle by factor of ##2##: ##6″ in”^2 times 2 = color(blue)(12″ in”^2)##
The bottom of the prism:
The bottom of the prism is a rectangle since the angles are 90 degrees. The formula to calculate the area of the triangle is ##”length” times “width”##, and the rectangle at the bottom has base of ##4″ in”## and width of ##2″ in”##. Therefore, ##”length” times “width” = 4″ in” times 2″ in” = color(orange)(8″ in”^2)##
The “wall” of the prism:
This side of the prism (outlined in pink) is also a rectangle with a width of ##3″ in”## and length of ##2″ in”## (since the blue-circled sides are parallel and the prism structure is perpendicular).
Therefore, ##”length” times “width” = 2″ in” times 3″ in” = color(pink)(6″ in”^2)##
The slanted side:
This side is also a rectangle with length of ##5″ in”## and a width of ##2″ in”##. Therefore, ##”length” times “width” = 5″ in” times 2″ in” = color(green)(10″ in”^2)##
To find the surface area, add all the areas up:
##color(blue)(12″ in”^2) + color(orange)(8″ in”^2) + color(pink)(6″ in”^2) + color(green)(10″ in”^2) = 36″ in”^2##
If you have any questions, please do not hesitate to leave a comment!
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