What is the surface area of this triangular prism?

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!