# Finding the Surface Area of a Triangular Prism

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 Finding the Surface Area of a Triangular Prism Discussion: Mentor: Can anyone tell me what surface area means? Student 1: Surface area is the number of square units that are needed to cover the surface of a three dimensional figure. Student 2: Surface area is just what you see of a three-dimensional object, not what is inside. Mentor: Good, how do you find the surface area of a three dimensional object? Student 1: Well, there are a lot of three dimensional objects such as cones, rectangular prisms and triangular prisms. The formula for finding surface area cannot be the same for all of these figures. Mentor: That is true! Three-dimensional figures have different formulas for surface area depending on their shape. Let's examine the surface area specifically of a triangular prism. Mentor: Since we are finding the surface area of this shape, how many sides will we need to include? Student 1: This is a triangular prism so it has 5 sides (the two triangular sides and three rectangular sides). We have to make sure to count how many square units are on each of the five sides. That will give us the entire area that covers the shape. Mentor: Great! Lets first look at the top surface of this figure. How many square units cover this triangular surface? Student 2: You are asking me to find the area. I remember learning that the area of a triangle is 1/2 times the base times the height. Mentor: For this activity we are going to call the base of the trianglular prism the base width and we are going to call the height of the prism the base depth. Student 1: OK, then the base width is 4 units and the base depth is 6 units. Therefore, the area would be: 1/2 x 4 x 6 = 12 square units Mentor: That is right. Now, are there any other surfaces on this triangular prism that would be identical to the one we just worked with? Student 1: The surface directly below this (the other triangular side of the prism) should have the same amount of cubic units. Mentor: That is true, but why? Student 2: Well, the base width of the flat triangular shape will be the same (4 units) and the base depth of the shape will be the same (6 units) so that means that the area should be the same as well! Mentor: Good. Now we have two of the surfaces covered (each 12 square units). Let's move to the side of the shape facing towards the right. How many square units are on this surface? Student 1: Well, one side looks like it is exactly 3 units, but I am not sure how many units are on the other side of the shape. It looks slanted and I am not sure. Mentor: Good observation. The side that is 3 units measures the prism height. The other length that you are confused about is called the slant height. It can be hard to find the slant height on triangular prisms. For this exercise I am going to go ahead and give you the slant height: 6.32 units. Student 2: OK, if the slant height is 6.32 units and the prism height is 3 units then the area of that side would be 6.32 times 3, which equals 18.96 square units. Mentor: Good. And is there another surface identical to this one on the triangular prism? Student 2: Yes, the surface opposite of this one would be identical since it, too, would have a prism height of 3 units and a slant height of 6.32 units. Mentor: Excellent. Now we have four surfaces covered. Two of them are 12 square units each, and two of them are 18.96 square units each. Lets take a look at the last side that we can see of this three dimensional figure: Student 1: This surface is easy! There are 4 units on one side and 3 units on the other. Mentor: Right. The 4 units measure the base width and the 3 units measure the prism height. What will you do to find the area of this shape? Student 1: Since this is a rectangle all I have to do is multiply those two numbers. 4 (the base width) times 3 (the prism height) gives me the area of the flat rectangle: 12 square units! Student 2: Yes! And that means that we have found the area of five total sides. A triangular prism only has five sides so we have all of the areas that we need now. Mentor: Right! So we have: two surfaces that are 12 square units two surfaces that are 18.96 square units one surface that is 12 square units. Now, what is the total surface area of the triangular prism? Student 1: To find the total surface area I would need to add all of the separate areas that I found together. It would be: 12+12+ 18.96+18.96+ 12 = 73.92 square units! Mentor: Great job! You just found the surface area of a triangular prism! 