 In this video practice problem 2, we're asked to find the exact lateral area and the exact surface area of a right triangular prism. I'm going to start out like I always do with the formula. Even though that this is a triangular prism, it is still a prism and so the formula surface area equals 2B plus P times H. I'm also going to start out by identifying my bases. The bases are triangles and that will make this a little different than the last problem. Before we begin this, let's just point out a couple things here that can clue you off on what's going to happen. We're asked to remember to find the lateral and the total surface area so we'll deal with that at the end. A couple things to point out. Any time you see the word exact in a problem, you know that chances are your final answer is either going to have Pi in it or a radical number in it, a square root number. The only time we're going to have Pi in a problem is when we're dealing with circles or curved areas. This one doesn't have any circles so we know we're probably going to have some square roots in this problem. That's kind of a tip. The other thing is we're given an angle measure of 30 degrees. That should tip you off that we're probably going to be using special right triangles. And if you're aware of these things as we get started that will kind of help you prepare for what we're going to be doing here. I'm going to just give myself a little bit more room because this is going to take a little bit more work than the last one. I'm starting out with a formula and like before I'm going to come down here and put all the values that I'm going to need B, P and H. And remember the B and the P are going to be referencing the base. The B is the area of my triangle base and the P is the perimeter of the distance around my triangle base. The H then we can fill in right away is just that distance between the bases of 7. So I have one of my values but in order to find the area of my triangle base and the perimeter of my triangle base I'm going to have to solve this triangle and find the missing sides. And that's why we're given that 30 degree angle measure. I'm just going to draw this triangle base a little bit bigger so I can give myself some room and see what's going on here. That right triangle piece is important because then I know that this is the hypotenuse of the triangle. This of course would be the short edge because it's opposite the 30 degree angle and then this is going to be the long edge of my triangle. And I certainly can put in that 60 degree measure too if that helps. So I'm going to find the missing sides of the triangle using my special right triangle rules. Opposite the short side is N. I know some of you guys probably use X or 1. I use N. Hypotenuse is 2N. Long side N writ 3. I know then my given side I set it equal to the rule N equals 4 and this one is pretty simple then. If N equals 4 then 2 times 4. The hypotenuse is going to be 8. And the long side if I plug in N equals 4 is going to be 4 writ 3. So I'm just using the special right triangle rules to solve the triangle. And now that I have all the missing sides I can go ahead and find the area of the triangle and the perimeter of the triangle. The area of the triangle of course the area is one half base times the height. And it's important to remember that the base and the height of the triangle no matter what way it's facing are always going to be the two pieces that make that perpendicular that are perpendicular to each other and make that right angle. It doesn't matter that this side is on the bottom. We want the two pieces that are perpendicular to each other. So one half times base times height becomes one half times 4 times 4 writ 3. It doesn't matter what order you put that in. And when I simplify that I can do 4 times 4 is 16 and then one half of 16 is 8. And then of course that writ 3 just stays alone as it is. So 8 writ 3 is going to be the area of my base. All I did was combine one half of 4 times 4. The perimeter of my base then I'm just going to add up all of the sides. I have 8 plus 4 plus 4 writ 3. And again combining like terms I can only combine the 8 and the 4 when I'm adding those up. My final answer for the perimeter I have to leave it as 12 plus 4 writ 3. I cannot combine those because I'm adding those two terms. And remember we're asked to find the exact values of the lateral area and surface area. So we want to leave those in square writ form. And so now I have all the values to plug into my surface area formula and I'll go ahead and simplify that. So my total surface area is 2 times B. 2 times 8 writ 3 is my B. Plus my perimeter. 12 plus 4 writ 3. And that perimeter times the height of 7 and I almost ran out of room. It's really important to put the parentheses here. Because this 12 plus 4 writ 3 times 7 I'm going to have to multiply the 7 times each of those. I can distribute that 7 even though it's on the right side of this term. And when I'm multiplying perimeter times height I have to distribute that 7 out to everything that is in the parentheses. So if I go ahead and simplify the first term 2 times 8 writ 3. I'm going to multiply the 2 times the 8 and leave the writ 3 as is. And then the second term I'm going to distribute that 7 and I'm going to say 7 times 12. 7 times 12 is 84. You can check that on your calculator. And then 7 times 4 writ 3. Just like before I am going to do 7 times 4 is 28 and leave the writ 3 on its own. So all I did here on that second term is take the 7 times 12 to get 84. And then again because I have to distribute 7 times 4 writ 3. I can say 7 times 4 is 28 and then the writ 3 is on its own. And now the final step. I have 3 terms here that I'm adding and I want to combine like terms. And because 16 writ 3 and 28 writ 3 both have writ 3 in it. I can go ahead and combine those. 16 plus 28 is 44 writ 3. That writ 3 stays in there. And then I'm just going to add the 84. This is my final answer for the total surface area. Sometimes the answers look kind of weird but I can't combine that anymore. Because I'm asked to find the exact surface area. I have to leave it in those two separate terms. I don't want to make this into a decimal and combine those out. I'm just going to have those two separate terms. Just like if I had pi in one of my terms. I'm going to leave them separate and not combine them. So that answered the question for the exact surface area. And now the final piece of this is to find the exact lateral area. And the lateral area, remember, I don't need to go back and recalculate all this. The lateral area is represented by that P times H. And that's why I like to do the total surface area first. Because then I can come back down here and say, my lateral area was this portion of the surface area. And so I don't need to recalculate anything. I can just go back here and say, my lateral area was 12 plus 4 root 3 times 7. And when I simplify that, I got 84 plus 28 root 3. And again, I can't combine those. And so my final answer for lateral area is 84 plus 28 root 3. And I want to add, because it's area, I'm going to say unit squared. I did not do that up above on the total surface area, unit squared, and those are my two answers.