High school geometry: surface area of a triangular prism

[Audrey7]

High school geometry: Resolved, thank you!

Hi everyone, my name is Audrey. I am a sophomore in high school and am currently in geometry.

I came across this relatively easy-seeming problem in my homework and was having some trouble with it.

Here is my best representation of the drawing:

Instructions: Find the lateral area and surface area of each right prism.

I easily found the lateral surface area by multiplying the perimeter by the height: (5+3+4)*2 = 24

I know that the total surface area is just found by adding the lateral surface area to twice the area of the base. That's my problem! I don't know how to find the area of the base. I'm sure there's something really easy that I'm missing (I apologize in advance ), but at this point, I'm definitely stuck.

In the textbook this is from, the answers to the odd problems are in the back of the book, so I do know that the answer should be lateral area = 24 cm^2 and total surface area = 36 cm^2. However, there's no work shown for the problem, so this wasn't much help to me in understanding it.

I did try to find it by using the law of cosines, but it didn't work out . Here is what I did for that:
5^2 = 3^2 + 4^2 - 2(3)(4) * cos(B) (B being the angle opposite the side with length 5)
25 = 25 - 1 cos(B)
0 = - cos(B) (I don't really know what to do with a negative cosine, so at this point I knew this wasn't right, but I just pretended it was positive for the sake of getting an answer )
cos^-1(B) = 90
m<B = 90 degrees
Next, I drew the angle intercept of <B that hit the opposite line at a perpendicular. Of course, that gave me two 45-45-90 triangles with different hypotenuses and equal legs, which is impossible, so I stopped there.

I'm pretty sure that entire process can be disregarded (there's got to be a more obvious way!), but I know I'm supposed to show all the work I did.

Any help is appreciated! This is my first time posting on here, so I'd be happy for any advice about making this more clear or anything like that.

Again, I'm sure the answer will be embarrassingly simple!

Thanks!

 

[MarkFL]

The way I see it, you have two congruent triangular faces, which are 3-4-5 right triangles, and so each of these faces has an area of 6 cm². You then have 3 rectangular faces, one of which is (2 cm)(5 cm) = 10 cm², one is (2 cm)(4 cm) = 8 cm², and the other is (2 cm)(3 cm) = 6 cm², and so the total surface area is:

S = (2(6) + 10 + 8 + 6) cm² = 36 cm²

 

[Audrey7]

Thanks Mark, I knew it would be simple! I totally forgot about Pythagorean triples